Welcome to Abella 2.0.5-dev.

Abella <Kind name, action, proc type.

Abella <Import "ccs_core".Importing from "ccs_core". Warning: Definition can be used to defeat stratification (higher-order argument "Tech" occurs to the left of ->) Warning: Definition can be used to defeat stratification (higher-order argument "Rel" occurs to the left of ->) Warning: Definition can be used to defeat stratification (higher-order argument "Rel" occurs to the left of ->)

Abella <Define inv : proc -> proc -> prop by inv P Q := bisim_up_to refl_t P Q; inv (par P1 Q1) (par P2 Q2) := inv P1 P2 /\ inv Q1 Q2.

Abella <Define bisim_inv : proc -> proc -> prop by bisim_inv P Q := (forall A P1, one P A P1 -> (exists Q1, one Q A Q1 /\ inv P1 Q1)) /\ (forall A Q1, one Q A Q1 -> (exists P1, one P A P1 /\ inv P1 Q1)).

Abella <Theorem inv_bisim_inv : forall P Q, inv P Q -> bisim_inv P Q.

============================ forall P Q, inv P Q -> bisim_inv P Q inv_bisim_inv <induction on 1.

IH : forall P Q, inv P Q * -> bisim_inv P Q ============================ forall P Q, inv P Q @ -> bisim_inv P Q inv_bisim_inv <intros.

Variables: P Q IH : forall P Q, inv P Q * -> bisim_inv P Q H1 : inv P Q @ ============================ bisim_inv P Q inv_bisim_inv <case H1.

Subgoal 1: Variables: P Q IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : bisim_up_to refl_t P Q ============================ bisim_inv P Q Subgoal 2 is: bisim_inv (par P1 Q1) (par P2 Q2) inv_bisim_inv <case H2.

Subgoal 1: Variables: P Q IH : forall P Q, inv P Q * -> bisim_inv P Q H3 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H4 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) ============================ bisim_inv P Q Subgoal 2 is: bisim_inv (par P1 Q1) (par P2 Q2) inv_bisim_inv <unfold.

Subgoal 1.1: Variables: P Q IH : forall P Q, inv P Q * -> bisim_inv P Q H3 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H4 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) ============================ forall A P1, one P A P1 -> (exists Q1, one Q A Q1 /\ inv P1 Q1) Subgoal 1.2 is: forall A Q1, one Q A Q1 -> (exists P1, one P A P1 /\ inv P1 Q1) Subgoal 2 is: bisim_inv (par P1 Q1) (par P2 Q2) inv_bisim_inv <intros.

Subgoal 1.1: Variables: P Q A P1 IH : forall P Q, inv P Q * -> bisim_inv P Q H3 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H4 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one P A P1 ============================ exists Q1, one Q A Q1 /\ inv P1 Q1 Subgoal 1.2 is: forall A Q1, one Q A Q1 -> (exists P1, one P A P1 /\ inv P1 Q1) Subgoal 2 is: bisim_inv (par P1 Q1) (par P2 Q2) inv_bisim_inv <apply H3 to H5.

Subgoal 1.1: Variables: P Q A P1 Q2 P3 Q3 IH : forall P Q, inv P Q * -> bisim_inv P Q H3 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H4 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one P A P1 H6 : one Q A Q2 H7 : refl_t P1 P3 Q2 Q3 H8 : bisim_up_to refl_t P3 Q3 ============================ exists Q1, one Q A Q1 /\ inv P1 Q1 Subgoal 1.2 is: forall A Q1, one Q A Q1 -> (exists P1, one P A P1 /\ inv P1 Q1) Subgoal 2 is: bisim_inv (par P1 Q1) (par P2 Q2) inv_bisim_inv <case H7.

Subgoal 1.1: Variables: P Q A P3 Q3 IH : forall P Q, inv P Q * -> bisim_inv P Q H3 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H4 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one P A P3 H6 : one Q A Q3 H8 : bisim_up_to refl_t P3 Q3 ============================ exists Q1, one Q A Q1 /\ inv P3 Q1 Subgoal 1.2 is: forall A Q1, one Q A Q1 -> (exists P1, one P A P1 /\ inv P1 Q1) Subgoal 2 is: bisim_inv (par P1 Q1) (par P2 Q2) inv_bisim_inv <witness Q3.

Subgoal 1.1: Variables: P Q A P3 Q3 IH : forall P Q, inv P Q * -> bisim_inv P Q H3 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H4 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one P A P3 H6 : one Q A Q3 H8 : bisim_up_to refl_t P3 Q3 ============================ one Q A Q3 /\ inv P3 Q3 Subgoal 1.2 is: forall A Q1, one Q A Q1 -> (exists P1, one P A P1 /\ inv P1 Q1) Subgoal 2 is: bisim_inv (par P1 Q1) (par P2 Q2) inv_bisim_inv <search.

Subgoal 1.2: Variables: P Q IH : forall P Q, inv P Q * -> bisim_inv P Q H3 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H4 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) ============================ forall A Q1, one Q A Q1 -> (exists P1, one P A P1 /\ inv P1 Q1) Subgoal 2 is: bisim_inv (par P1 Q1) (par P2 Q2) inv_bisim_inv <intros.

Subgoal 1.2: Variables: P Q A Q1 IH : forall P Q, inv P Q * -> bisim_inv P Q H3 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H4 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one Q A Q1 ============================ exists P1, one P A P1 /\ inv P1 Q1 Subgoal 2 is: bisim_inv (par P1 Q1) (par P2 Q2) inv_bisim_inv <apply H4 to H5.

Subgoal 1.2: Variables: P Q A Q1 P2 P3 Q3 IH : forall P Q, inv P Q * -> bisim_inv P Q H3 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H4 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one Q A Q1 H6 : one P A P2 H7 : refl_t P2 P3 Q1 Q3 H8 : bisim_up_to refl_t P3 Q3 ============================ exists P1, one P A P1 /\ inv P1 Q1 Subgoal 2 is: bisim_inv (par P1 Q1) (par P2 Q2) inv_bisim_inv <case H7.

Subgoal 1.2: Variables: P Q A P3 Q3 IH : forall P Q, inv P Q * -> bisim_inv P Q H3 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H4 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one Q A Q3 H6 : one P A P3 H8 : bisim_up_to refl_t P3 Q3 ============================ exists P1, one P A P1 /\ inv P1 Q3 Subgoal 2 is: bisim_inv (par P1 Q1) (par P2 Q2) inv_bisim_inv <witness P3.

Subgoal 1.2: Variables: P Q A P3 Q3 IH : forall P Q, inv P Q * -> bisim_inv P Q H3 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H4 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one Q A Q3 H6 : one P A P3 H8 : bisim_up_to refl_t P3 Q3 ============================ one P A P3 /\ inv P3 Q3 Subgoal 2 is: bisim_inv (par P1 Q1) (par P2 Q2) inv_bisim_inv <search.

Subgoal 2: Variables: Q2 P2 Q1 P1 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * ============================ bisim_inv (par P1 Q1) (par P2 Q2) inv_bisim_inv <apply IH to H2.

Subgoal 2: Variables: Q2 P2 Q1 P1 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H4 : bisim_inv P1 P2 ============================ bisim_inv (par P1 Q1) (par P2 Q2) inv_bisim_inv <apply IH to H3.

Subgoal 2: Variables: Q2 P2 Q1 P1 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H4 : bisim_inv P1 P2 H5 : bisim_inv Q1 Q2 ============================ bisim_inv (par P1 Q1) (par P2 Q2) inv_bisim_inv <case H4.

Subgoal 2: Variables: Q2 P2 Q1 P1 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H5 : bisim_inv Q1 Q2 H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) ============================ bisim_inv (par P1 Q1) (par P2 Q2) inv_bisim_inv <case H5.

Subgoal 2: Variables: Q2 P2 Q1 P1 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) ============================ bisim_inv (par P1 Q1) (par P2 Q2) inv_bisim_inv <unfold.

Subgoal 2.1: Variables: Q2 P2 Q1 P1 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) ============================ forall A P3, one (par P1 Q1) A P3 -> (exists Q3, one (par P2 Q2) A Q3 /\ inv P3 Q3) Subgoal 2.2 is: forall A Q3, one (par P2 Q2) A Q3 -> (exists P3, one (par P1 Q1) A P3 /\ inv P3 Q3) inv_bisim_inv <intros.

Subgoal 2.1: Variables: Q2 P2 Q1 P1 A P3 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) H10 : one (par P1 Q1) A P3 ============================ exists Q3, one (par P2 Q2) A Q3 /\ inv P3 Q3 Subgoal 2.2 is: forall A Q3, one (par P2 Q2) A Q3 -> (exists P3, one (par P1 Q1) A P3 /\ inv P3 Q3) inv_bisim_inv <case H10.

Subgoal 2.1.1: Variables: Q2 P2 Q1 P1 A P5 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) H11 : one P1 A P5 ============================ exists Q3, one (par P2 Q2) A Q3 /\ inv (par P5 Q1) Q3 Subgoal 2.1.2 is: exists Q3, one (par P2 Q2) A Q3 /\ inv (par P1 Q4) Q3 Subgoal 2.1.3 is: exists Q3, one (par P2 Q2) tau Q3 /\ inv (par P5 Q4) Q3 Subgoal 2.1.4 is: exists Q3, one (par P2 Q2) tau Q3 /\ inv (par P5 Q4) Q3 Subgoal 2.2 is: forall A Q3, one (par P2 Q2) A Q3 -> (exists P3, one (par P1 Q1) A P3 /\ inv P3 Q3) inv_bisim_inv <apply H6 to H11.

Subgoal 2.1.1: Variables: Q2 P2 Q1 P1 A P5 Q3 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) H11 : one P1 A P5 H12 : one P2 A Q3 H13 : inv P5 Q3 ============================ exists Q3, one (par P2 Q2) A Q3 /\ inv (par P5 Q1) Q3 Subgoal 2.1.2 is: exists Q3, one (par P2 Q2) A Q3 /\ inv (par P1 Q4) Q3 Subgoal 2.1.3 is: exists Q3, one (par P2 Q2) tau Q3 /\ inv (par P5 Q4) Q3 Subgoal 2.1.4 is: exists Q3, one (par P2 Q2) tau Q3 /\ inv (par P5 Q4) Q3 Subgoal 2.2 is: forall A Q3, one (par P2 Q2) A Q3 -> (exists P3, one (par P1 Q1) A P3 /\ inv P3 Q3) inv_bisim_inv <search.

Subgoal 2.1.2: Variables: Q2 P2 Q1 P1 A Q4 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) H11 : one Q1 A Q4 ============================ exists Q3, one (par P2 Q2) A Q3 /\ inv (par P1 Q4) Q3 Subgoal 2.1.3 is: exists Q3, one (par P2 Q2) tau Q3 /\ inv (par P5 Q4) Q3 Subgoal 2.1.4 is: exists Q3, one (par P2 Q2) tau Q3 /\ inv (par P5 Q4) Q3 Subgoal 2.2 is: forall A Q3, one (par P2 Q2) A Q3 -> (exists P3, one (par P1 Q1) A P3 /\ inv P3 Q3) inv_bisim_inv <apply H8 to H11.

Subgoal 2.1.2: Variables: Q2 P2 Q1 P1 A Q4 Q3 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) H11 : one Q1 A Q4 H12 : one Q2 A Q3 H13 : inv Q4 Q3 ============================ exists Q3, one (par P2 Q2) A Q3 /\ inv (par P1 Q4) Q3 Subgoal 2.1.3 is: exists Q3, one (par P2 Q2) tau Q3 /\ inv (par P5 Q4) Q3 Subgoal 2.1.4 is: exists Q3, one (par P2 Q2) tau Q3 /\ inv (par P5 Q4) Q3 Subgoal 2.2 is: forall A Q3, one (par P2 Q2) A Q3 -> (exists P3, one (par P1 Q1) A P3 /\ inv P3 Q3) inv_bisim_inv <search.

Subgoal 2.1.3: Variables: Q2 P2 Q1 P1 X Q4 P5 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) H11 : one P1 (up X) P5 H12 : one Q1 (dn X) Q4 ============================ exists Q3, one (par P2 Q2) tau Q3 /\ inv (par P5 Q4) Q3 Subgoal 2.1.4 is: exists Q3, one (par P2 Q2) tau Q3 /\ inv (par P5 Q4) Q3 Subgoal 2.2 is: forall A Q3, one (par P2 Q2) A Q3 -> (exists P3, one (par P1 Q1) A P3 /\ inv P3 Q3) inv_bisim_inv <apply H6 to H11.

Subgoal 2.1.3: Variables: Q2 P2 Q1 P1 X Q4 P5 Q3 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) H11 : one P1 (up X) P5 H12 : one Q1 (dn X) Q4 H13 : one P2 (up X) Q3 H14 : inv P5 Q3 ============================ exists Q3, one (par P2 Q2) tau Q3 /\ inv (par P5 Q4) Q3 Subgoal 2.1.4 is: exists Q3, one (par P2 Q2) tau Q3 /\ inv (par P5 Q4) Q3 Subgoal 2.2 is: forall A Q3, one (par P2 Q2) A Q3 -> (exists P3, one (par P1 Q1) A P3 /\ inv P3 Q3) inv_bisim_inv <apply H8 to H12.

Subgoal 2.1.3: Variables: Q2 P2 Q1 P1 X Q4 P5 Q3 Q5 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) H11 : one P1 (up X) P5 H12 : one Q1 (dn X) Q4 H13 : one P2 (up X) Q3 H14 : inv P5 Q3 H15 : one Q2 (dn X) Q5 H16 : inv Q4 Q5 ============================ exists Q3, one (par P2 Q2) tau Q3 /\ inv (par P5 Q4) Q3 Subgoal 2.1.4 is: exists Q3, one (par P2 Q2) tau Q3 /\ inv (par P5 Q4) Q3 Subgoal 2.2 is: forall A Q3, one (par P2 Q2) A Q3 -> (exists P3, one (par P1 Q1) A P3 /\ inv P3 Q3) inv_bisim_inv <search.

Subgoal 2.1.4: Variables: Q2 P2 Q1 P1 X Q4 P5 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) H11 : one P1 (dn X) P5 H12 : one Q1 (up X) Q4 ============================ exists Q3, one (par P2 Q2) tau Q3 /\ inv (par P5 Q4) Q3 Subgoal 2.2 is: forall A Q3, one (par P2 Q2) A Q3 -> (exists P3, one (par P1 Q1) A P3 /\ inv P3 Q3) inv_bisim_inv <apply H6 to H11.

Subgoal 2.1.4: Variables: Q2 P2 Q1 P1 X Q4 P5 Q3 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) H11 : one P1 (dn X) P5 H12 : one Q1 (up X) Q4 H13 : one P2 (dn X) Q3 H14 : inv P5 Q3 ============================ exists Q3, one (par P2 Q2) tau Q3 /\ inv (par P5 Q4) Q3 Subgoal 2.2 is: forall A Q3, one (par P2 Q2) A Q3 -> (exists P3, one (par P1 Q1) A P3 /\ inv P3 Q3) inv_bisim_inv <apply H8 to H12.

Subgoal 2.1.4: Variables: Q2 P2 Q1 P1 X Q4 P5 Q3 Q5 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) H11 : one P1 (dn X) P5 H12 : one Q1 (up X) Q4 H13 : one P2 (dn X) Q3 H14 : inv P5 Q3 H15 : one Q2 (up X) Q5 H16 : inv Q4 Q5 ============================ exists Q3, one (par P2 Q2) tau Q3 /\ inv (par P5 Q4) Q3 Subgoal 2.2 is: forall A Q3, one (par P2 Q2) A Q3 -> (exists P3, one (par P1 Q1) A P3 /\ inv P3 Q3) inv_bisim_inv <search.

Subgoal 2.2: Variables: Q2 P2 Q1 P1 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) ============================ forall A Q3, one (par P2 Q2) A Q3 -> (exists P3, one (par P1 Q1) A P3 /\ inv P3 Q3) inv_bisim_inv <intros.

Subgoal 2.2: Variables: Q2 P2 Q1 P1 A Q3 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) H10 : one (par P2 Q2) A Q3 ============================ exists P3, one (par P1 Q1) A P3 /\ inv P3 Q3 inv_bisim_inv <case H10.

Subgoal 2.2.1: Variables: Q2 P2 Q1 P1 A P4 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) H11 : one P2 A P4 ============================ exists P3, one (par P1 Q1) A P3 /\ inv P3 (par P4 Q2) Subgoal 2.2.2 is: exists P3, one (par P1 Q1) A P3 /\ inv P3 (par P2 Q5) Subgoal 2.2.3 is: exists P3, one (par P1 Q1) tau P3 /\ inv P3 (par P4 Q5) Subgoal 2.2.4 is: exists P3, one (par P1 Q1) tau P3 /\ inv P3 (par P4 Q5) inv_bisim_inv <apply H7 to H11.

Subgoal 2.2.1: Variables: Q2 P2 Q1 P1 A P4 P3 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) H11 : one P2 A P4 H12 : one P1 A P3 H13 : inv P3 P4 ============================ exists P3, one (par P1 Q1) A P3 /\ inv P3 (par P4 Q2) Subgoal 2.2.2 is: exists P3, one (par P1 Q1) A P3 /\ inv P3 (par P2 Q5) Subgoal 2.2.3 is: exists P3, one (par P1 Q1) tau P3 /\ inv P3 (par P4 Q5) Subgoal 2.2.4 is: exists P3, one (par P1 Q1) tau P3 /\ inv P3 (par P4 Q5) inv_bisim_inv <search.

Subgoal 2.2.2: Variables: Q2 P2 Q1 P1 A Q5 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) H11 : one Q2 A Q5 ============================ exists P3, one (par P1 Q1) A P3 /\ inv P3 (par P2 Q5) Subgoal 2.2.3 is: exists P3, one (par P1 Q1) tau P3 /\ inv P3 (par P4 Q5) Subgoal 2.2.4 is: exists P3, one (par P1 Q1) tau P3 /\ inv P3 (par P4 Q5) inv_bisim_inv <apply H9 to H11.

Subgoal 2.2.2: Variables: Q2 P2 Q1 P1 A Q5 P3 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) H11 : one Q2 A Q5 H12 : one Q1 A P3 H13 : inv P3 Q5 ============================ exists P3, one (par P1 Q1) A P3 /\ inv P3 (par P2 Q5) Subgoal 2.2.3 is: exists P3, one (par P1 Q1) tau P3 /\ inv P3 (par P4 Q5) Subgoal 2.2.4 is: exists P3, one (par P1 Q1) tau P3 /\ inv P3 (par P4 Q5) inv_bisim_inv <search.

Subgoal 2.2.3: Variables: Q2 P2 Q1 P1 X Q5 P4 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) H11 : one P2 (up X) P4 H12 : one Q2 (dn X) Q5 ============================ exists P3, one (par P1 Q1) tau P3 /\ inv P3 (par P4 Q5) Subgoal 2.2.4 is: exists P3, one (par P1 Q1) tau P3 /\ inv P3 (par P4 Q5) inv_bisim_inv <apply H7 to H11.

Subgoal 2.2.3: Variables: Q2 P2 Q1 P1 X Q5 P4 P3 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) H11 : one P2 (up X) P4 H12 : one Q2 (dn X) Q5 H13 : one P1 (up X) P3 H14 : inv P3 P4 ============================ exists P3, one (par P1 Q1) tau P3 /\ inv P3 (par P4 Q5) Subgoal 2.2.4 is: exists P3, one (par P1 Q1) tau P3 /\ inv P3 (par P4 Q5) inv_bisim_inv <apply H9 to H12.

Subgoal 2.2.3: Variables: Q2 P2 Q1 P1 X Q5 P4 P3 P5 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) H11 : one P2 (up X) P4 H12 : one Q2 (dn X) Q5 H13 : one P1 (up X) P3 H14 : inv P3 P4 H15 : one Q1 (dn X) P5 H16 : inv P5 Q5 ============================ exists P3, one (par P1 Q1) tau P3 /\ inv P3 (par P4 Q5) Subgoal 2.2.4 is: exists P3, one (par P1 Q1) tau P3 /\ inv P3 (par P4 Q5) inv_bisim_inv <search.

Subgoal 2.2.4: Variables: Q2 P2 Q1 P1 X Q5 P4 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) H11 : one P2 (dn X) P4 H12 : one Q2 (up X) Q5 ============================ exists P3, one (par P1 Q1) tau P3 /\ inv P3 (par P4 Q5) inv_bisim_inv <apply H7 to H11.

Subgoal 2.2.4: Variables: Q2 P2 Q1 P1 X Q5 P4 P3 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) H11 : one P2 (dn X) P4 H12 : one Q2 (up X) Q5 H13 : one P1 (dn X) P3 H14 : inv P3 P4 ============================ exists P3, one (par P1 Q1) tau P3 /\ inv P3 (par P4 Q5) inv_bisim_inv <apply H9 to H12.

Subgoal 2.2.4: Variables: Q2 P2 Q1 P1 X Q5 P4 P3 P5 IH : forall P Q, inv P Q * -> bisim_inv P Q H2 : inv P1 P2 * H3 : inv Q1 Q2 * H6 : forall A P3, one P1 A P3 -> (exists Q1, one P2 A Q1 /\ inv P3 Q1) H7 : forall A Q1, one P2 A Q1 -> (exists P3, one P1 A P3 /\ inv P3 Q1) H8 : forall A P1, one Q1 A P1 -> (exists Q3, one Q2 A Q3 /\ inv P1 Q3) H9 : forall A Q3, one Q2 A Q3 -> (exists P1, one Q1 A P1 /\ inv P1 Q3) H11 : one P2 (dn X) P4 H12 : one Q2 (up X) Q5 H13 : one P1 (dn X) P3 H14 : inv P3 P4 H15 : one Q1 (up X) P5 H16 : inv P5 Q5 ============================ exists P3, one (par P1 Q1) tau P3 /\ inv P3 (par P4 Q5) inv_bisim_inv <search.Proof completed.

Abella <Theorem bisim_inv_bisim : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q.

============================ forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q bisim_inv_bisim <coinduction.

CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + ============================ forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q # bisim_inv_bisim <intros.

Variables: P Q CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H1 : bisim_inv P Q ============================ bisim_up_to refl_t P Q # bisim_inv_bisim <case H1.

Variables: P Q CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) ============================ bisim_up_to refl_t P Q # bisim_inv_bisim <unfold.

Subgoal 1: Variables: P Q CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) ============================ forall A P1, one P A P1 -> (exists Q1, one Q A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)) Subgoal 2 is: forall A Q1, one Q A Q1 -> (exists P1, one P A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)) bisim_inv_bisim <intros.

Subgoal 1: Variables: P Q A P1 CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) H4 : one P A P1 ============================ exists Q1, one Q A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +) Subgoal 2 is: forall A Q1, one Q A Q1 -> (exists P1, one P A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)) bisim_inv_bisim <apply H2 to H4.

Subgoal 1: Variables: P Q A P1 Q2 CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) H4 : one P A P1 H5 : one Q A Q2 H6 : inv P1 Q2 ============================ exists Q1, one Q A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +) Subgoal 2 is: forall A Q1, one Q A Q1 -> (exists P1, one P A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)) bisim_inv_bisim <witness Q2.

Subgoal 1: Variables: P Q A P1 Q2 CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) H4 : one P A P1 H5 : one Q A Q2 H6 : inv P1 Q2 ============================ one Q A Q2 /\ (exists P2 Q1, refl_t P1 P2 Q2 Q1 /\ bisim_up_to refl_t P2 Q1 +) Subgoal 2 is: forall A Q1, one Q A Q1 -> (exists P1, one P A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)) bisim_inv_bisim <split.

Subgoal 1.1: Variables: P Q A P1 Q2 CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) H4 : one P A P1 H5 : one Q A Q2 H6 : inv P1 Q2 ============================ one Q A Q2 Subgoal 1.2 is: exists P2 Q1, refl_t P1 P2 Q2 Q1 /\ bisim_up_to refl_t P2 Q1 + Subgoal 2 is: forall A Q1, one Q A Q1 -> (exists P1, one P A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)) bisim_inv_bisim <search.

Subgoal 1.2: Variables: P Q A P1 Q2 CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) H4 : one P A P1 H5 : one Q A Q2 H6 : inv P1 Q2 ============================ exists P2 Q1, refl_t P1 P2 Q2 Q1 /\ bisim_up_to refl_t P2 Q1 + Subgoal 2 is: forall A Q1, one Q A Q1 -> (exists P1, one P A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)) bisim_inv_bisim <witness P1.

Subgoal 1.2: Variables: P Q A P1 Q2 CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) H4 : one P A P1 H5 : one Q A Q2 H6 : inv P1 Q2 ============================ exists Q1, refl_t P1 P1 Q2 Q1 /\ bisim_up_to refl_t P1 Q1 + Subgoal 2 is: forall A Q1, one Q A Q1 -> (exists P1, one P A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)) bisim_inv_bisim <witness Q2.

Subgoal 1.2: Variables: P Q A P1 Q2 CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) H4 : one P A P1 H5 : one Q A Q2 H6 : inv P1 Q2 ============================ refl_t P1 P1 Q2 Q2 /\ bisim_up_to refl_t P1 Q2 + Subgoal 2 is: forall A Q1, one Q A Q1 -> (exists P1, one P A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)) bisim_inv_bisim <split.

Subgoal 1.2.1: Variables: P Q A P1 Q2 CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) H4 : one P A P1 H5 : one Q A Q2 H6 : inv P1 Q2 ============================ refl_t P1 P1 Q2 Q2 Subgoal 1.2.2 is: bisim_up_to refl_t P1 Q2 + Subgoal 2 is: forall A Q1, one Q A Q1 -> (exists P1, one P A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)) bisim_inv_bisim <search.

Subgoal 1.2.2: Variables: P Q A P1 Q2 CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) H4 : one P A P1 H5 : one Q A Q2 H6 : inv P1 Q2 ============================ bisim_up_to refl_t P1 Q2 + Subgoal 2 is: forall A Q1, one Q A Q1 -> (exists P1, one P A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)) bisim_inv_bisim <apply inv_bisim_inv to H6.

Subgoal 1.2.2: Variables: P Q A P1 Q2 CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) H4 : one P A P1 H5 : one Q A Q2 H6 : inv P1 Q2 H7 : bisim_inv P1 Q2 ============================ bisim_up_to refl_t P1 Q2 + Subgoal 2 is: forall A Q1, one Q A Q1 -> (exists P1, one P A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)) bisim_inv_bisim <backchain CH.

Subgoal 2: Variables: P Q CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) ============================ forall A Q1, one Q A Q1 -> (exists P1, one P A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)) bisim_inv_bisim <intros.

Subgoal 2: Variables: P Q A Q1 CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) H4 : one Q A Q1 ============================ exists P1, one P A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +) bisim_inv_bisim <apply H3 to H4.

Subgoal 2: Variables: P Q A Q1 P2 CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) H4 : one Q A Q1 H5 : one P A P2 H6 : inv P2 Q1 ============================ exists P1, one P A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +) bisim_inv_bisim <witness P2.

Subgoal 2: Variables: P Q A Q1 P2 CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) H4 : one Q A Q1 H5 : one P A P2 H6 : inv P2 Q1 ============================ one P A P2 /\ (exists P1 Q2, refl_t P2 P1 Q1 Q2 /\ bisim_up_to refl_t P1 Q2 +) bisim_inv_bisim <split.

Subgoal 2.1: Variables: P Q A Q1 P2 CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) H4 : one Q A Q1 H5 : one P A P2 H6 : inv P2 Q1 ============================ one P A P2 Subgoal 2.2 is: exists P1 Q2, refl_t P2 P1 Q1 Q2 /\ bisim_up_to refl_t P1 Q2 + bisim_inv_bisim <search.

Subgoal 2.2: Variables: P Q A Q1 P2 CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) H4 : one Q A Q1 H5 : one P A P2 H6 : inv P2 Q1 ============================ exists P1 Q2, refl_t P2 P1 Q1 Q2 /\ bisim_up_to refl_t P1 Q2 + bisim_inv_bisim <witness P2.

Subgoal 2.2: Variables: P Q A Q1 P2 CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) H4 : one Q A Q1 H5 : one P A P2 H6 : inv P2 Q1 ============================ exists Q2, refl_t P2 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 + bisim_inv_bisim <witness Q1.

Subgoal 2.2: Variables: P Q A Q1 P2 CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) H4 : one Q A Q1 H5 : one P A P2 H6 : inv P2 Q1 ============================ refl_t P2 P2 Q1 Q1 /\ bisim_up_to refl_t P2 Q1 + bisim_inv_bisim <split.

Subgoal 2.2.1: Variables: P Q A Q1 P2 CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) H4 : one Q A Q1 H5 : one P A P2 H6 : inv P2 Q1 ============================ refl_t P2 P2 Q1 Q1 Subgoal 2.2.2 is: bisim_up_to refl_t P2 Q1 + bisim_inv_bisim <search.

Subgoal 2.2.2: Variables: P Q A Q1 P2 CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) H4 : one Q A Q1 H5 : one P A P2 H6 : inv P2 Q1 ============================ bisim_up_to refl_t P2 Q1 + bisim_inv_bisim <apply inv_bisim_inv to H6.

Subgoal 2.2.2: Variables: P Q A Q1 P2 CH : forall P Q, bisim_inv P Q -> bisim_up_to refl_t P Q + H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ inv P2 Q2) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ inv P2 Q2) H4 : one Q A Q1 H5 : one P A P2 H6 : inv P2 Q1 H7 : bisim_inv P2 Q1 ============================ bisim_up_to refl_t P2 Q1 + bisim_inv_bisim <backchain CH.Proof completed.

Abella <Theorem bisim_par_subst_1 : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t (par P R) (par Q R).

============================ forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t (par P R) (par Q R) bisim_par_subst_1 <intros.

Variables: P Q R H1 : bisim_up_to refl_t P Q ============================ bisim_up_to refl_t (par P R) (par Q R) bisim_par_subst_1 <apply bisim_reflexive_ with P = R.

Variables: P Q R H1 : bisim_up_to refl_t P Q H2 : bisim_up_to refl_t R R ============================ bisim_up_to refl_t (par P R) (par Q R) bisim_par_subst_1 <backchain bisim_inv_bisim.

Variables: P Q R H1 : bisim_up_to refl_t P Q H2 : bisim_up_to refl_t R R ============================ bisim_inv (par P R) (par Q R) bisim_par_subst_1 <backchain inv_bisim_inv.Proof completed.

Abella <Theorem bisim_par_subst_2 : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t (par R P) (par R Q).

============================ forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t (par R P) (par R Q) bisim_par_subst_2 <intros.

Variables: P Q R H1 : bisim_up_to refl_t P Q ============================ bisim_up_to refl_t (par R P) (par R Q) bisim_par_subst_2 <apply bisim_reflexive_ with P = R.

Variables: P Q R H1 : bisim_up_to refl_t P Q H2 : bisim_up_to refl_t R R ============================ bisim_up_to refl_t (par R P) (par R Q) bisim_par_subst_2 <backchain bisim_inv_bisim.

Variables: P Q R H1 : bisim_up_to refl_t P Q H2 : bisim_up_to refl_t R R ============================ bisim_inv (par R P) (par R Q) bisim_par_subst_2 <backchain inv_bisim_inv.Proof completed.

Abella <Theorem bisim_plus_subst_1 : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t (plus P R) (plus Q R).

============================ forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t (plus P R) (plus Q R) bisim_plus_subst_1 <intros.

Variables: P Q R H1 : bisim_up_to refl_t P Q ============================ bisim_up_to refl_t (plus P R) (plus Q R) bisim_plus_subst_1 <case H1.

Variables: P Q R H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) ============================ bisim_up_to refl_t (plus P R) (plus Q R) bisim_plus_subst_1 <unfold.

Subgoal 1: Variables: P Q R H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) ============================ forall A P1, one (plus P R) A P1 -> (exists Q1, one (plus Q R) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) Subgoal 2 is: forall A Q1, one (plus Q R) A Q1 -> (exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_1 <intros.

Subgoal 1: Variables: P Q R A P1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H4 : one (plus P R) A P1 ============================ exists Q1, one (plus Q R) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus Q R) A Q1 -> (exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_1 <case H4.

Subgoal 1.1: Variables: P Q R A P1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one P A P1 ============================ exists Q1, one (plus Q R) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 1.2 is: exists Q1, one (plus Q R) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus Q R) A Q1 -> (exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_1 <apply H2 to H5.

Subgoal 1.1: Variables: P Q R A P1 Q2 P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one P A P1 H6 : one Q A Q2 H7 : refl_t P1 P3 Q2 Q3 H8 : bisim_up_to refl_t P3 Q3 ============================ exists Q1, one (plus Q R) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 1.2 is: exists Q1, one (plus Q R) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus Q R) A Q1 -> (exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_1 <case H7.

Subgoal 1.1: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one P A P3 H6 : one Q A Q3 H8 : bisim_up_to refl_t P3 Q3 ============================ exists Q1, one (plus Q R) A Q1 /\ (exists P2 Q2, refl_t P3 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 1.2 is: exists Q1, one (plus Q R) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus Q R) A Q1 -> (exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_1 <witness Q3.

Subgoal 1.1: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one P A P3 H6 : one Q A Q3 H8 : bisim_up_to refl_t P3 Q3 ============================ one (plus Q R) A Q3 /\ (exists P2 Q2, refl_t P3 P2 Q3 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 1.2 is: exists Q1, one (plus Q R) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus Q R) A Q1 -> (exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_1 <split.

Subgoal 1.1.1: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one P A P3 H6 : one Q A Q3 H8 : bisim_up_to refl_t P3 Q3 ============================ one (plus Q R) A Q3 Subgoal 1.1.2 is: exists P2 Q2, refl_t P3 P2 Q3 Q2 /\ bisim_up_to refl_t P2 Q2 Subgoal 1.2 is: exists Q1, one (plus Q R) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus Q R) A Q1 -> (exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_1 <search.

Subgoal 1.1.2: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one P A P3 H6 : one Q A Q3 H8 : bisim_up_to refl_t P3 Q3 ============================ exists P2 Q2, refl_t P3 P2 Q3 Q2 /\ bisim_up_to refl_t P2 Q2 Subgoal 1.2 is: exists Q1, one (plus Q R) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus Q R) A Q1 -> (exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_1 <witness P3.

Subgoal 1.1.2: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one P A P3 H6 : one Q A Q3 H8 : bisim_up_to refl_t P3 Q3 ============================ exists Q2, refl_t P3 P3 Q3 Q2 /\ bisim_up_to refl_t P3 Q2 Subgoal 1.2 is: exists Q1, one (plus Q R) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus Q R) A Q1 -> (exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_1 <witness Q3.

Subgoal 1.1.2: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one P A P3 H6 : one Q A Q3 H8 : bisim_up_to refl_t P3 Q3 ============================ refl_t P3 P3 Q3 Q3 /\ bisim_up_to refl_t P3 Q3 Subgoal 1.2 is: exists Q1, one (plus Q R) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus Q R) A Q1 -> (exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_1 <search.

Subgoal 1.2: Variables: P Q R A P1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A P1 ============================ exists Q1, one (plus Q R) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus Q R) A Q1 -> (exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_1 <witness P1.

Subgoal 1.2: Variables: P Q R A P1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A P1 ============================ one (plus Q R) A P1 /\ (exists P2 Q2, refl_t P1 P2 P1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus Q R) A Q1 -> (exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_1 <split.

Subgoal 1.2.1: Variables: P Q R A P1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A P1 ============================ one (plus Q R) A P1 Subgoal 1.2.2 is: exists P2 Q2, refl_t P1 P2 P1 Q2 /\ bisim_up_to refl_t P2 Q2 Subgoal 2 is: forall A Q1, one (plus Q R) A Q1 -> (exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_1 <search.

Subgoal 1.2.2: Variables: P Q R A P1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A P1 ============================ exists P2 Q2, refl_t P1 P2 P1 Q2 /\ bisim_up_to refl_t P2 Q2 Subgoal 2 is: forall A Q1, one (plus Q R) A Q1 -> (exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_1 <witness P1.

Subgoal 1.2.2: Variables: P Q R A P1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A P1 ============================ exists Q2, refl_t P1 P1 P1 Q2 /\ bisim_up_to refl_t P1 Q2 Subgoal 2 is: forall A Q1, one (plus Q R) A Q1 -> (exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_1 <witness P1.

Subgoal 1.2.2: Variables: P Q R A P1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A P1 ============================ refl_t P1 P1 P1 P1 /\ bisim_up_to refl_t P1 P1 Subgoal 2 is: forall A Q1, one (plus Q R) A Q1 -> (exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_1 <apply bisim_reflexive_ with P = P1.

Subgoal 1.2.2: Variables: P Q R A P1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A P1 H6 : bisim_up_to refl_t P1 P1 ============================ refl_t P1 P1 P1 P1 /\ bisim_up_to refl_t P1 P1 Subgoal 2 is: forall A Q1, one (plus Q R) A Q1 -> (exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_1 <search.

Subgoal 2: Variables: P Q R H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) ============================ forall A Q1, one (plus Q R) A Q1 -> (exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_1 <intros.

Subgoal 2: Variables: P Q R A Q1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H4 : one (plus Q R) A Q1 ============================ exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_1 <case H4.

Subgoal 2.1: Variables: P Q R A Q1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one Q A Q1 ============================ exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2.2 is: exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_1 <apply H3 to H5.

Subgoal 2.1: Variables: P Q R A Q1 P2 P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one Q A Q1 H6 : one P A P2 H7 : refl_t P2 P3 Q1 Q3 H8 : bisim_up_to refl_t P3 Q3 ============================ exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2.2 is: exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_1 <case H7.

Subgoal 2.1: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one Q A Q3 H6 : one P A P3 H8 : bisim_up_to refl_t P3 Q3 ============================ exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q3 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2.2 is: exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_1 <witness P3.

Subgoal 2.1: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one Q A Q3 H6 : one P A P3 H8 : bisim_up_to refl_t P3 Q3 ============================ one (plus P R) A P3 /\ (exists P2 Q2, refl_t P3 P2 Q3 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2.2 is: exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_1 <split.

Subgoal 2.1.1: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one Q A Q3 H6 : one P A P3 H8 : bisim_up_to refl_t P3 Q3 ============================ one (plus P R) A P3 Subgoal 2.1.2 is: exists P2 Q2, refl_t P3 P2 Q3 Q2 /\ bisim_up_to refl_t P2 Q2 Subgoal 2.2 is: exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_1 <search.

Subgoal 2.1.2: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one Q A Q3 H6 : one P A P3 H8 : bisim_up_to refl_t P3 Q3 ============================ exists P2 Q2, refl_t P3 P2 Q3 Q2 /\ bisim_up_to refl_t P2 Q2 Subgoal 2.2 is: exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_1 <witness P2.

Subgoal 2.1.2: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one Q A Q3 H6 : one P A P3 H8 : bisim_up_to refl_t P3 Q3 ============================ exists Q2, refl_t P3 P3 Q3 Q2 /\ bisim_up_to refl_t P3 Q2 Subgoal 2.2 is: exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_1 <witness Q3.

Subgoal 2.1.2: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one Q A Q3 H6 : one P A P3 H8 : bisim_up_to refl_t P3 Q3 ============================ refl_t P3 P3 Q3 Q3 /\ bisim_up_to refl_t P3 Q3 Subgoal 2.2 is: exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_1 <search.

Subgoal 2.2: Variables: P Q R A Q1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A Q1 ============================ exists P1, one (plus P R) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_1 <witness Q1.

Subgoal 2.2: Variables: P Q R A Q1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A Q1 ============================ one (plus P R) A Q1 /\ (exists P2 Q2, refl_t Q1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_1 <split.

Subgoal 2.2.1: Variables: P Q R A Q1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A Q1 ============================ one (plus P R) A Q1 Subgoal 2.2.2 is: exists P2 Q2, refl_t Q1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 bisim_plus_subst_1 <search.

Subgoal 2.2.2: Variables: P Q R A Q1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A Q1 ============================ exists P2 Q2, refl_t Q1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 bisim_plus_subst_1 <witness Q1.

Subgoal 2.2.2: Variables: P Q R A Q1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A Q1 ============================ exists Q2, refl_t Q1 Q1 Q1 Q2 /\ bisim_up_to refl_t Q1 Q2 bisim_plus_subst_1 <witness Q1.

Subgoal 2.2.2: Variables: P Q R A Q1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A Q1 ============================ refl_t Q1 Q1 Q1 Q1 /\ bisim_up_to refl_t Q1 Q1 bisim_plus_subst_1 <apply bisim_reflexive_ with P = Q1.

Subgoal 2.2.2: Variables: P Q R A Q1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A Q1 H6 : bisim_up_to refl_t Q1 Q1 ============================ refl_t Q1 Q1 Q1 Q1 /\ bisim_up_to refl_t Q1 Q1 bisim_plus_subst_1 <search.Proof completed.

Abella <Theorem bisim_plus_subst_2 : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t (plus R P) (plus R Q).

============================ forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t (plus R P) (plus R Q) bisim_plus_subst_2 <intros.

Variables: P Q R H1 : bisim_up_to refl_t P Q ============================ bisim_up_to refl_t (plus R P) (plus R Q) bisim_plus_subst_2 <case H1.

Variables: P Q R H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) ============================ bisim_up_to refl_t (plus R P) (plus R Q) bisim_plus_subst_2 <unfold.

Subgoal 1: Variables: P Q R H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) ============================ forall A P1, one (plus R P) A P1 -> (exists Q1, one (plus R Q) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) Subgoal 2 is: forall A Q1, one (plus R Q) A Q1 -> (exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_2 <intros.

Subgoal 1: Variables: P Q R A P1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H4 : one (plus R P) A P1 ============================ exists Q1, one (plus R Q) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus R Q) A Q1 -> (exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_2 <case H4.

Subgoal 1.1: Variables: P Q R A P1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A P1 ============================ exists Q1, one (plus R Q) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 1.2 is: exists Q1, one (plus R Q) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus R Q) A Q1 -> (exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_2 <witness P1.

Subgoal 1.1: Variables: P Q R A P1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A P1 ============================ one (plus R Q) A P1 /\ (exists P2 Q2, refl_t P1 P2 P1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 1.2 is: exists Q1, one (plus R Q) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus R Q) A Q1 -> (exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_2 <split.

Subgoal 1.1.1: Variables: P Q R A P1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A P1 ============================ one (plus R Q) A P1 Subgoal 1.1.2 is: exists P2 Q2, refl_t P1 P2 P1 Q2 /\ bisim_up_to refl_t P2 Q2 Subgoal 1.2 is: exists Q1, one (plus R Q) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus R Q) A Q1 -> (exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_2 <search.

Subgoal 1.1.2: Variables: P Q R A P1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A P1 ============================ exists P2 Q2, refl_t P1 P2 P1 Q2 /\ bisim_up_to refl_t P2 Q2 Subgoal 1.2 is: exists Q1, one (plus R Q) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus R Q) A Q1 -> (exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_2 <witness P1.

Subgoal 1.1.2: Variables: P Q R A P1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A P1 ============================ exists Q2, refl_t P1 P1 P1 Q2 /\ bisim_up_to refl_t P1 Q2 Subgoal 1.2 is: exists Q1, one (plus R Q) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus R Q) A Q1 -> (exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_2 <witness P1.

Subgoal 1.1.2: Variables: P Q R A P1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A P1 ============================ refl_t P1 P1 P1 P1 /\ bisim_up_to refl_t P1 P1 Subgoal 1.2 is: exists Q1, one (plus R Q) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus R Q) A Q1 -> (exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_2 <apply bisim_reflexive_ with P = P1.

Subgoal 1.1.2: Variables: P Q R A P1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A P1 H6 : bisim_up_to refl_t P1 P1 ============================ refl_t P1 P1 P1 P1 /\ bisim_up_to refl_t P1 P1 Subgoal 1.2 is: exists Q1, one (plus R Q) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus R Q) A Q1 -> (exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_2 <search.

Subgoal 1.2: Variables: P Q R A P1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one P A P1 ============================ exists Q1, one (plus R Q) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus R Q) A Q1 -> (exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_2 <apply H2 to H5.

Subgoal 1.2: Variables: P Q R A P1 Q2 P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one P A P1 H6 : one Q A Q2 H7 : refl_t P1 P3 Q2 Q3 H8 : bisim_up_to refl_t P3 Q3 ============================ exists Q1, one (plus R Q) A Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus R Q) A Q1 -> (exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_2 <case H7.

Subgoal 1.2: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one P A P3 H6 : one Q A Q3 H8 : bisim_up_to refl_t P3 Q3 ============================ exists Q1, one (plus R Q) A Q1 /\ (exists P2 Q2, refl_t P3 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus R Q) A Q1 -> (exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_2 <witness Q3.

Subgoal 1.2: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one P A P3 H6 : one Q A Q3 H8 : bisim_up_to refl_t P3 Q3 ============================ one (plus R Q) A Q3 /\ (exists P2 Q2, refl_t P3 P2 Q3 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A Q1, one (plus R Q) A Q1 -> (exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_2 <split.

Subgoal 1.2.1: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one P A P3 H6 : one Q A Q3 H8 : bisim_up_to refl_t P3 Q3 ============================ one (plus R Q) A Q3 Subgoal 1.2.2 is: exists P2 Q2, refl_t P3 P2 Q3 Q2 /\ bisim_up_to refl_t P2 Q2 Subgoal 2 is: forall A Q1, one (plus R Q) A Q1 -> (exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_2 <search.

Subgoal 1.2.2: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one P A P3 H6 : one Q A Q3 H8 : bisim_up_to refl_t P3 Q3 ============================ exists P2 Q2, refl_t P3 P2 Q3 Q2 /\ bisim_up_to refl_t P2 Q2 Subgoal 2 is: forall A Q1, one (plus R Q) A Q1 -> (exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_2 <witness P3.

Subgoal 1.2.2: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one P A P3 H6 : one Q A Q3 H8 : bisim_up_to refl_t P3 Q3 ============================ exists Q2, refl_t P3 P3 Q3 Q2 /\ bisim_up_to refl_t P3 Q2 Subgoal 2 is: forall A Q1, one (plus R Q) A Q1 -> (exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_2 <witness Q3.

Subgoal 1.2.2: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one P A P3 H6 : one Q A Q3 H8 : bisim_up_to refl_t P3 Q3 ============================ refl_t P3 P3 Q3 Q3 /\ bisim_up_to refl_t P3 Q3 Subgoal 2 is: forall A Q1, one (plus R Q) A Q1 -> (exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_2 <search.

Subgoal 2: Variables: P Q R H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) ============================ forall A Q1, one (plus R Q) A Q1 -> (exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_plus_subst_2 <intros.

Subgoal 2: Variables: P Q R A Q1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H4 : one (plus R Q) A Q1 ============================ exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_2 <case H4.

Subgoal 2.1: Variables: P Q R A Q1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A Q1 ============================ exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2.2 is: exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_2 <witness Q1.

Subgoal 2.1: Variables: P Q R A Q1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A Q1 ============================ one (plus R P) A Q1 /\ (exists P2 Q2, refl_t Q1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2.2 is: exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_2 <split.

Subgoal 2.1.1: Variables: P Q R A Q1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A Q1 ============================ one (plus R P) A Q1 Subgoal 2.1.2 is: exists P2 Q2, refl_t Q1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 Subgoal 2.2 is: exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_2 <search.

Subgoal 2.1.2: Variables: P Q R A Q1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A Q1 ============================ exists P2 Q2, refl_t Q1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 Subgoal 2.2 is: exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_2 <witness Q1.

Subgoal 2.1.2: Variables: P Q R A Q1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A Q1 ============================ exists Q2, refl_t Q1 Q1 Q1 Q2 /\ bisim_up_to refl_t Q1 Q2 Subgoal 2.2 is: exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_2 <witness Q1.

Subgoal 2.1.2: Variables: P Q R A Q1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A Q1 ============================ refl_t Q1 Q1 Q1 Q1 /\ bisim_up_to refl_t Q1 Q1 Subgoal 2.2 is: exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_2 <apply bisim_reflexive_ with P = Q1.

Subgoal 2.1.2: Variables: P Q R A Q1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one R A Q1 H6 : bisim_up_to refl_t Q1 Q1 ============================ refl_t Q1 Q1 Q1 Q1 /\ bisim_up_to refl_t Q1 Q1 Subgoal 2.2 is: exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_2 <search.

Subgoal 2.2: Variables: P Q R A Q1 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one Q A Q1 ============================ exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_2 <apply H3 to H5.

Subgoal 2.2: Variables: P Q R A Q1 P2 P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one Q A Q1 H6 : one P A P2 H7 : refl_t P2 P3 Q1 Q3 H8 : bisim_up_to refl_t P3 Q3 ============================ exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_2 <case H7.

Subgoal 2.2: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one Q A Q3 H6 : one P A P3 H8 : bisim_up_to refl_t P3 Q3 ============================ exists P1, one (plus R P) A P1 /\ (exists P2 Q2, refl_t P1 P2 Q3 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_2 <witness P3.

Subgoal 2.2: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one Q A Q3 H6 : one P A P3 H8 : bisim_up_to refl_t P3 Q3 ============================ one (plus R P) A P3 /\ (exists P2 Q2, refl_t P3 P2 Q3 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_plus_subst_2 <split.

Subgoal 2.2.1: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one Q A Q3 H6 : one P A P3 H8 : bisim_up_to refl_t P3 Q3 ============================ one (plus R P) A P3 Subgoal 2.2.2 is: exists P2 Q2, refl_t P3 P2 Q3 Q2 /\ bisim_up_to refl_t P2 Q2 bisim_plus_subst_2 <search.

Subgoal 2.2.2: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one Q A Q3 H6 : one P A P3 H8 : bisim_up_to refl_t P3 Q3 ============================ exists P2 Q2, refl_t P3 P2 Q3 Q2 /\ bisim_up_to refl_t P2 Q2 bisim_plus_subst_2 <witness P2.

Subgoal 2.2.2: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one Q A Q3 H6 : one P A P3 H8 : bisim_up_to refl_t P3 Q3 ============================ exists Q2, refl_t P3 P3 Q3 Q2 /\ bisim_up_to refl_t P3 Q2 bisim_plus_subst_2 <witness Q3.

Subgoal 2.2.2: Variables: P Q R A P3 Q3 H2 : forall A P2, one P A P2 -> (exists Q2, one Q A Q2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H3 : forall A Q2, one Q A Q2 -> (exists P2, one P A P2 /\ (exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3)) H5 : one Q A Q3 H6 : one P A P3 H8 : bisim_up_to refl_t P3 Q3 ============================ refl_t P3 P3 Q3 Q3 /\ bisim_up_to refl_t P3 Q3 bisim_plus_subst_2 <search.Proof completed.

Abella <Theorem bisim_act_subst : forall P Q A, bisim_up_to refl_t P Q -> bisim_up_to refl_t (act A P) (act A Q).

============================ forall P Q A, bisim_up_to refl_t P Q -> bisim_up_to refl_t (act A P) (act A Q) bisim_act_subst <intros.

Variables: P Q A H1 : bisim_up_to refl_t P Q ============================ bisim_up_to refl_t (act A P) (act A Q) bisim_act_subst <unfold.

Subgoal 1: Variables: P Q A H1 : bisim_up_to refl_t P Q ============================ forall A1 P1, one (act A P) A1 P1 -> (exists Q1, one (act A Q) A1 Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) Subgoal 2 is: forall A1 Q1, one (act A Q) A1 Q1 -> (exists P1, one (act A P) A1 P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_act_subst <intros.

Subgoal 1: Variables: P Q A A1 P1 H1 : bisim_up_to refl_t P Q H2 : one (act A P) A1 P1 ============================ exists Q1, one (act A Q) A1 Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A1 Q1, one (act A Q) A1 Q1 -> (exists P1, one (act A P) A1 P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_act_subst <case H2.

Subgoal 1: Variables: Q A1 P1 H1 : bisim_up_to refl_t P1 Q ============================ exists Q1, one (act A1 Q) A1 Q1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A1 Q1, one (act A Q) A1 Q1 -> (exists P1, one (act A P) A1 P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_act_subst <witness Q.

Subgoal 1: Variables: Q A1 P1 H1 : bisim_up_to refl_t P1 Q ============================ one (act A1 Q) A1 Q /\ (exists P2 Q2, refl_t P1 P2 Q Q2 /\ bisim_up_to refl_t P2 Q2) Subgoal 2 is: forall A1 Q1, one (act A Q) A1 Q1 -> (exists P1, one (act A P) A1 P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_act_subst <split.

Subgoal 1.1: Variables: Q A1 P1 H1 : bisim_up_to refl_t P1 Q ============================ one (act A1 Q) A1 Q Subgoal 1.2 is: exists P2 Q2, refl_t P1 P2 Q Q2 /\ bisim_up_to refl_t P2 Q2 Subgoal 2 is: forall A1 Q1, one (act A Q) A1 Q1 -> (exists P1, one (act A P) A1 P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_act_subst <search.

Subgoal 1.2: Variables: Q A1 P1 H1 : bisim_up_to refl_t P1 Q ============================ exists P2 Q2, refl_t P1 P2 Q Q2 /\ bisim_up_to refl_t P2 Q2 Subgoal 2 is: forall A1 Q1, one (act A Q) A1 Q1 -> (exists P1, one (act A P) A1 P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_act_subst <witness P1.

Subgoal 1.2: Variables: Q A1 P1 H1 : bisim_up_to refl_t P1 Q ============================ exists Q2, refl_t P1 P1 Q Q2 /\ bisim_up_to refl_t P1 Q2 Subgoal 2 is: forall A1 Q1, one (act A Q) A1 Q1 -> (exists P1, one (act A P) A1 P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_act_subst <witness Q.

Subgoal 1.2: Variables: Q A1 P1 H1 : bisim_up_to refl_t P1 Q ============================ refl_t P1 P1 Q Q /\ bisim_up_to refl_t P1 Q Subgoal 2 is: forall A1 Q1, one (act A Q) A1 Q1 -> (exists P1, one (act A P) A1 P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_act_subst <search.

Subgoal 2: Variables: P Q A H1 : bisim_up_to refl_t P Q ============================ forall A1 Q1, one (act A Q) A1 Q1 -> (exists P1, one (act A P) A1 P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2)) bisim_act_subst <intros.

Subgoal 2: Variables: P Q A A1 Q1 H1 : bisim_up_to refl_t P Q H2 : one (act A Q) A1 Q1 ============================ exists P1, one (act A P) A1 P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_act_subst <case H2.

Subgoal 2: Variables: P A1 Q1 H1 : bisim_up_to refl_t P Q1 ============================ exists P1, one (act A1 P) A1 P1 /\ (exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_act_subst <witness P.

Subgoal 2: Variables: P A1 Q1 H1 : bisim_up_to refl_t P Q1 ============================ one (act A1 P) A1 P /\ (exists P2 Q2, refl_t P P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2) bisim_act_subst <split.

Subgoal 2.1: Variables: P A1 Q1 H1 : bisim_up_to refl_t P Q1 ============================ one (act A1 P) A1 P Subgoal 2.2 is: exists P2 Q2, refl_t P P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 bisim_act_subst <search.

Subgoal 2.2: Variables: P A1 Q1 H1 : bisim_up_to refl_t P Q1 ============================ exists P2 Q2, refl_t P P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 bisim_act_subst <witness P.

Subgoal 2.2: Variables: P A1 Q1 H1 : bisim_up_to refl_t P Q1 ============================ exists Q2, refl_t P P Q1 Q2 /\ bisim_up_to refl_t P Q2 bisim_act_subst <witness Q1.

Subgoal 2.2: Variables: P A1 Q1 H1 : bisim_up_to refl_t P Q1 ============================ refl_t P P Q1 Q1 /\ bisim_up_to refl_t P Q1 bisim_act_subst <search.Proof completed.

Abella <Kind ctx type.

Abella <Type hole ctx.

Abella <Type plus_l, par_l ctx -> proc -> ctx.

Abella <Type plus_r, par_r proc -> ctx -> ctx.

Abella <Type act_d action -> ctx -> ctx.

Abella <Define at : ctx -> proc -> proc -> prop by at hole P P; at (plus_l C R) P (plus Q R) := at C P Q; at (plus_r R C) P (plus R Q) := at C P Q; at (par_l C R) P (par Q R) := at C P Q; at (par_r R C) P (par R Q) := at C P Q; at (act_d A C) P (act A Q) := at C P Q.

Abella <Define substitutive_rel : (proc -> proc -> prop) -> prop by substitutive_rel Rel := forall P1 P2 C Q1 Q2, at C P1 Q1 -> at C P2 Q2 -> Rel P1 P2 -> Rel Q1 Q2.Warning: Definition can be used to defeat stratification (higher-order argument "Rel" occurs to the left of ->)

Abella <Theorem bisim_substitutive : substitutive_rel (bisim_up_to refl_t).

============================ substitutive_rel (bisim_up_to refl_t) bisim_substitutive <unfold.

============================ forall P1 P2 C Q1 Q2, at C P1 Q1 -> at C P2 Q2 -> bisim_up_to refl_t P1 P2 -> bisim_up_to refl_t Q1 Q2 bisim_substitutive <induction on 1.

IH : forall P1 P2 C Q1 Q2, at C P1 Q1 * -> at C P2 Q2 -> bisim_up_to refl_t P1 P2 -> bisim_up_to refl_t Q1 Q2 ============================ forall P1 P2 C Q1 Q2, at C P1 Q1 @ -> at C P2 Q2 -> bisim_up_to refl_t P1 P2 -> bisim_up_to refl_t Q1 Q2 bisim_substitutive <intros.

Variables: P1 P2 C Q1 Q2 IH : forall P1 P2 C Q1 Q2, at C P1 Q1 * -> at C P2 Q2 -> bisim_up_to refl_t P1 P2 -> bisim_up_to refl_t Q1 Q2 H1 : at C P1 Q1 @ H2 : at C P2 Q2 H3 : bisim_up_to refl_t P1 P2 ============================ bisim_up_to refl_t Q1 Q2 bisim_substitutive <case H1.

Subgoal 1: Variables: P2 Q1 Q2 IH : forall P1 P2 C Q1 Q2, at C P1 Q1 * -> at C P2 Q2 -> bisim_up_to refl_t P1 P2 -> bisim_up_to refl_t Q1 Q2 H2 : at hole P2 Q2 H3 : bisim_up_to refl_t Q1 P2 ============================ bisim_up_to refl_t Q1 Q2 Subgoal 2 is: bisim_up_to refl_t (plus Q R) Q2 Subgoal 3 is: bisim_up_to refl_t (plus R Q) Q2 Subgoal 4 is: bisim_up_to refl_t (par Q R) Q2 Subgoal 5 is: bisim_up_to refl_t (par R Q) Q2 Subgoal 6 is: bisim_up_to refl_t (act A Q) Q2 bisim_substitutive <case H2.

Subgoal 1: Variables: Q1 Q2 IH : forall P1 P2 C Q1 Q2, at C P1 Q1 * -> at C P2 Q2 -> bisim_up_to refl_t P1 P2 -> bisim_up_to refl_t Q1 Q2 H3 : bisim_up_to refl_t Q1 Q2 ============================ bisim_up_to refl_t Q1 Q2 Subgoal 2 is: bisim_up_to refl_t (plus Q R) Q2 Subgoal 3 is: bisim_up_to refl_t (plus R Q) Q2 Subgoal 4 is: bisim_up_to refl_t (par Q R) Q2 Subgoal 5 is: bisim_up_to refl_t (par R Q) Q2 Subgoal 6 is: bisim_up_to refl_t (act A Q) Q2 bisim_substitutive <search.

Subgoal 2: Variables: P1 P2 Q2 R Q C1 IH : forall P1 P2 C Q1 Q2, at C P1 Q1 * -> at C P2 Q2 -> bisim_up_to refl_t P1 P2 -> bisim_up_to refl_t Q1 Q2 H2 : at (plus_l C1 R) P2 Q2 H3 : bisim_up_to refl_t P1 P2 H4 : at C1 P1 Q * ============================ bisim_up_to refl_t (plus Q R) Q2 Subgoal 3 is: bisim_up_to refl_t (plus R Q) Q2 Subgoal 4 is: bisim_up_to refl_t (par Q R) Q2 Subgoal 5 is: bisim_up_to refl_t (par R Q) Q2 Subgoal 6 is: bisim_up_to refl_t (act A Q) Q2 bisim_substitutive <case H2.

Subgoal 2: Variables: P1 P2 R Q C1 Q3 IH : forall P1 P2 C Q1 Q2, at C P1 Q1 * -> at C P2 Q2 -> bisim_up_to refl_t P1 P2 -> bisim_up_to refl_t Q1 Q2 H3 : bisim_up_to refl_t P1 P2 H4 : at C1 P1 Q * H5 : at C1 P2 Q3 ============================ bisim_up_to refl_t (plus Q R) (plus Q3 R) Subgoal 3 is: bisim_up_to refl_t (plus R Q) Q2 Subgoal 4 is: bisim_up_to refl_t (par Q R) Q2 Subgoal 5 is: bisim_up_to refl_t (par R Q) Q2 Subgoal 6 is: bisim_up_to refl_t (act A Q) Q2 bisim_substitutive <apply IH to H4 H5 H3.

Subgoal 2: Variables: P1 P2 R Q C1 Q3 IH : forall P1 P2 C Q1 Q2, at C P1 Q1 * -> at C P2 Q2 -> bisim_up_to refl_t P1 P2 -> bisim_up_to refl_t Q1 Q2 H3 : bisim_up_to refl_t P1 P2 H4 : at C1 P1 Q * H5 : at C1 P2 Q3 H6 : bisim_up_to refl_t Q Q3 ============================ bisim_up_to refl_t (plus Q R) (plus Q3 R) Subgoal 3 is: bisim_up_to refl_t (plus R Q) Q2 Subgoal 4 is: bisim_up_to refl_t (par Q R) Q2 Subgoal 5 is: bisim_up_to refl_t (par R Q) Q2 Subgoal 6 is: bisim_up_to refl_t (act A Q) Q2 bisim_substitutive <backchain bisim_plus_subst_1.

Subgoal 3: Variables: P1 P2 Q2 Q R C1 IH : forall P1 P2 C Q1 Q2, at C P1 Q1 * -> at C P2 Q2 -> bisim_up_to refl_t P1 P2 -> bisim_up_to refl_t Q1 Q2 H2 : at (plus_r R C1) P2 Q2 H3 : bisim_up_to refl_t P1 P2 H4 : at C1 P1 Q * ============================ bisim_up_to refl_t (plus R Q) Q2 Subgoal 4 is: bisim_up_to refl_t (par Q R) Q2 Subgoal 5 is: bisim_up_to refl_t (par R Q) Q2 Subgoal 6 is: bisim_up_to refl_t (act A Q) Q2 bisim_substitutive <case H2.

Subgoal 3: Variables: P1 P2 Q R C1 Q3 IH : forall P1 P2 C Q1 Q2, at C P1 Q1 * -> at C P2 Q2 -> bisim_up_to refl_t P1 P2 -> bisim_up_to refl_t Q1 Q2 H3 : bisim_up_to refl_t P1 P2 H4 : at C1 P1 Q * H5 : at C1 P2 Q3 ============================ bisim_up_to refl_t (plus R Q) (plus R Q3) Subgoal 4 is: bisim_up_to refl_t (par Q R) Q2 Subgoal 5 is: bisim_up_to refl_t (par R Q) Q2 Subgoal 6 is: bisim_up_to refl_t (act A Q) Q2 bisim_substitutive <apply IH to H4 H5 H3.

Subgoal 3: Variables: P1 P2 Q R C1 Q3 IH : forall P1 P2 C Q1 Q2, at C P1 Q1 * -> at C P2 Q2 -> bisim_up_to refl_t P1 P2 -> bisim_up_to refl_t Q1 Q2 H3 : bisim_up_to refl_t P1 P2 H4 : at C1 P1 Q * H5 : at C1 P2 Q3 H6 : bisim_up_to refl_t Q Q3 ============================ bisim_up_to refl_t (plus R Q) (plus R Q3) Subgoal 4 is: bisim_up_to refl_t (par Q R) Q2 Subgoal 5 is: bisim_up_to refl_t (par R Q) Q2 Subgoal 6 is: bisim_up_to refl_t (act A Q) Q2 bisim_substitutive <backchain bisim_plus_subst_2.

Subgoal 4: Variables: P1 P2 Q2 R Q C1 IH : forall P1 P2 C Q1 Q2, at C P1 Q1 * -> at C P2 Q2 -> bisim_up_to refl_t P1 P2 -> bisim_up_to refl_t Q1 Q2 H2 : at (par_l C1 R) P2 Q2 H3 : bisim_up_to refl_t P1 P2 H4 : at C1 P1 Q * ============================ bisim_up_to refl_t (par Q R) Q2 Subgoal 5 is: bisim_up_to refl_t (par R Q) Q2 Subgoal 6 is: bisim_up_to refl_t (act A Q) Q2 bisim_substitutive <case H2.

Subgoal 4: Variables: P1 P2 R Q C1 Q3 IH : forall P1 P2 C Q1 Q2, at C P1 Q1 * -> at C P2 Q2 -> bisim_up_to refl_t P1 P2 -> bisim_up_to refl_t Q1 Q2 H3 : bisim_up_to refl_t P1 P2 H4 : at C1 P1 Q * H5 : at C1 P2 Q3 ============================ bisim_up_to refl_t (par Q R) (par Q3 R) Subgoal 5 is: bisim_up_to refl_t (par R Q) Q2 Subgoal 6 is: bisim_up_to refl_t (act A Q) Q2 bisim_substitutive <apply IH to H4 H5 H3.

Subgoal 4: Variables: P1 P2 R Q C1 Q3 IH : forall P1 P2 C Q1 Q2, at C P1 Q1 * -> at C P2 Q2 -> bisim_up_to refl_t P1 P2 -> bisim_up_to refl_t Q1 Q2 H3 : bisim_up_to refl_t P1 P2 H4 : at C1 P1 Q * H5 : at C1 P2 Q3 H6 : bisim_up_to refl_t Q Q3 ============================ bisim_up_to refl_t (par Q R) (par Q3 R) Subgoal 5 is: bisim_up_to refl_t (par R Q) Q2 Subgoal 6 is: bisim_up_to refl_t (act A Q) Q2 bisim_substitutive <backchain bisim_par_subst_1.

Subgoal 5: Variables: P1 P2 Q2 Q R C1 IH : forall P1 P2 C Q1 Q2, at C P1 Q1 * -> at C P2 Q2 -> bisim_up_to refl_t P1 P2 -> bisim_up_to refl_t Q1 Q2 H2 : at (par_r R C1) P2 Q2 H3 : bisim_up_to refl_t P1 P2 H4 : at C1 P1 Q * ============================ bisim_up_to refl_t (par R Q) Q2 Subgoal 6 is: bisim_up_to refl_t (act A Q) Q2 bisim_substitutive <case H2.

Subgoal 5: Variables: P1 P2 Q R C1 Q3 IH : forall P1 P2 C Q1 Q2, at C P1 Q1 * -> at C P2 Q2 -> bisim_up_to refl_t P1 P2 -> bisim_up_to refl_t Q1 Q2 H3 : bisim_up_to refl_t P1 P2 H4 : at C1 P1 Q * H5 : at C1 P2 Q3 ============================ bisim_up_to refl_t (par R Q) (par R Q3) Subgoal 6 is: bisim_up_to refl_t (act A Q) Q2 bisim_substitutive <apply IH to H4 H5 H3.

Subgoal 5: Variables: P1 P2 Q R C1 Q3 IH : forall P1 P2 C Q1 Q2, at C P1 Q1 * -> at C P2 Q2 -> bisim_up_to refl_t P1 P2 -> bisim_up_to refl_t Q1 Q2 H3 : bisim_up_to refl_t P1 P2 H4 : at C1 P1 Q * H5 : at C1 P2 Q3 H6 : bisim_up_to refl_t Q Q3 ============================ bisim_up_to refl_t (par R Q) (par R Q3) Subgoal 6 is: bisim_up_to refl_t (act A Q) Q2 bisim_substitutive <backchain bisim_par_subst_2.

Subgoal 6: Variables: P1 P2 Q2 Q A C1 IH : forall P1 P2 C Q1 Q2, at C P1 Q1 * -> at C P2 Q2 -> bisim_up_to refl_t P1 P2 -> bisim_up_to refl_t Q1 Q2 H2 : at (act_d A C1) P2 Q2 H3 : bisim_up_to refl_t P1 P2 H4 : at C1 P1 Q * ============================ bisim_up_to refl_t (act A Q) Q2 bisim_substitutive <case H2.

Subgoal 6: Variables: P1 P2 Q A C1 Q3 IH : forall P1 P2 C Q1 Q2, at C P1 Q1 * -> at C P2 Q2 -> bisim_up_to refl_t P1 P2 -> bisim_up_to refl_t Q1 Q2 H3 : bisim_up_to refl_t P1 P2 H4 : at C1 P1 Q * H5 : at C1 P2 Q3 ============================ bisim_up_to refl_t (act A Q) (act A Q3) bisim_substitutive <apply IH to H4 H5 H3.

Subgoal 6: Variables: P1 P2 Q A C1 Q3 IH : forall P1 P2 C Q1 Q2, at C P1 Q1 * -> at C P2 Q2 -> bisim_up_to refl_t P1 P2 -> bisim_up_to refl_t Q1 Q2 H3 : bisim_up_to refl_t P1 P2 H4 : at C1 P1 Q * H5 : at C1 P2 Q3 H6 : bisim_up_to refl_t Q Q3 ============================ bisim_up_to refl_t (act A Q) (act A Q3) bisim_substitutive <backchain bisim_act_subst.Proof completed.

Abella <Theorem at_det3 : forall C P Q1 Q2, at C P Q1 -> at C P Q2 -> Q1 = Q2.

============================ forall C P Q1 Q2, at C P Q1 -> at C P Q2 -> Q1 = Q2 at_det3 <induction on 1.

IH : forall C P Q1 Q2, at C P Q1 * -> at C P Q2 -> Q1 = Q2 ============================ forall C P Q1 Q2, at C P Q1 @ -> at C P Q2 -> Q1 = Q2 at_det3 <intros.

Variables: C P Q1 Q2 IH : forall C P Q1 Q2, at C P Q1 * -> at C P Q2 -> Q1 = Q2 H1 : at C P Q1 @ H2 : at C P Q2 ============================ Q1 = Q2 at_det3 <case H1.

Subgoal 1: Variables: Q1 Q2 IH : forall C P Q1 Q2, at C P Q1 * -> at C P Q2 -> Q1 = Q2 H2 : at hole Q1 Q2 ============================ Q1 = Q2 Subgoal 2 is: plus Q R = Q2 Subgoal 3 is: plus R Q = Q2 Subgoal 4 is: par Q R = Q2 Subgoal 5 is: par R Q = Q2 Subgoal 6 is: act A Q = Q2 at_det3 <case H2.

Subgoal 1: Variables: Q2 IH : forall C P Q1 Q2, at C P Q1 * -> at C P Q2 -> Q1 = Q2 ============================ Q2 = Q2 Subgoal 2 is: plus Q R = Q2 Subgoal 3 is: plus R Q = Q2 Subgoal 4 is: par Q R = Q2 Subgoal 5 is: par R Q = Q2 Subgoal 6 is: act A Q = Q2 at_det3 <search.

Subgoal 2: Variables: P Q2 R Q C1 IH : forall C P Q1 Q2, at C P Q1 * -> at C P Q2 -> Q1 = Q2 H2 : at (plus_l C1 R) P Q2 H3 : at C1 P Q * ============================ plus Q R = Q2 Subgoal 3 is: plus R Q = Q2 Subgoal 4 is: par Q R = Q2 Subgoal 5 is: par R Q = Q2 Subgoal 6 is: act A Q = Q2 at_det3 <case H2.

Subgoal 2: Variables: P R Q C1 Q3 IH : forall C P Q1 Q2, at C P Q1 * -> at C P Q2 -> Q1 = Q2 H3 : at C1 P Q * H4 : at C1 P Q3 ============================ plus Q R = plus Q3 R Subgoal 3 is: plus R Q = Q2 Subgoal 4 is: par Q R = Q2 Subgoal 5 is: par R Q = Q2 Subgoal 6 is: act A Q = Q2 at_det3 <apply IH to H3 H4.

Subgoal 2: Variables: P R C1 Q3 IH : forall C P Q1 Q2, at C P Q1 * -> at C P Q2 -> Q1 = Q2 H3 : at C1 P Q3 * H4 : at C1 P Q3 ============================ plus Q3 R = plus Q3 R Subgoal 3 is: plus R Q = Q2 Subgoal 4 is: par Q R = Q2 Subgoal 5 is: par R Q = Q2 Subgoal 6 is: act A Q = Q2 at_det3 <search.

Subgoal 3: Variables: P Q2 Q R C1 IH : forall C P Q1 Q2, at C P Q1 * -> at C P Q2 -> Q1 = Q2 H2 : at (plus_r R C1) P Q2 H3 : at C1 P Q * ============================ plus R Q = Q2 Subgoal 4 is: par Q R = Q2 Subgoal 5 is: par R Q = Q2 Subgoal 6 is: act A Q = Q2 at_det3 <case H2.

Subgoal 3: Variables: P Q R C1 Q3 IH : forall C P Q1 Q2, at C P Q1 * -> at C P Q2 -> Q1 = Q2 H3 : at C1 P Q * H4 : at C1 P Q3 ============================ plus R Q = plus R Q3 Subgoal 4 is: par Q R = Q2 Subgoal 5 is: par R Q = Q2 Subgoal 6 is: act A Q = Q2 at_det3 <apply IH to H3 H4.

Subgoal 3: Variables: P R C1 Q3 IH : forall C P Q1 Q2, at C P Q1 * -> at C P Q2 -> Q1 = Q2 H3 : at C1 P Q3 * H4 : at C1 P Q3 ============================ plus R Q3 = plus R Q3 Subgoal 4 is: par Q R = Q2 Subgoal 5 is: par R Q = Q2 Subgoal 6 is: act A Q = Q2 at_det3 <search.

Subgoal 4: Variables: P Q2 R Q C1 IH : forall C P Q1 Q2, at C P Q1 * -> at C P Q2 -> Q1 = Q2 H2 : at (par_l C1 R) P Q2 H3 : at C1 P Q * ============================ par Q R = Q2 Subgoal 5 is: par R Q = Q2 Subgoal 6 is: act A Q = Q2 at_det3 <case H2.

Subgoal 4: Variables: P R Q C1 Q3 IH : forall C P Q1 Q2, at C P Q1 * -> at C P Q2 -> Q1 = Q2 H3 : at C1 P Q * H4 : at C1 P Q3 ============================ par Q R = par Q3 R Subgoal 5 is: par R Q = Q2 Subgoal 6 is: act A Q = Q2 at_det3 <apply IH to H3 H4.

Subgoal 4: Variables: P R C1 Q3 IH : forall C P Q1 Q2, at C P Q1 * -> at C P Q2 -> Q1 = Q2 H3 : at C1 P Q3 * H4 : at C1 P Q3 ============================ par Q3 R = par Q3 R Subgoal 5 is: par R Q = Q2 Subgoal 6 is: act A Q = Q2 at_det3 <search.

Subgoal 5: Variables: P Q2 Q R C1 IH : forall C P Q1 Q2, at C P Q1 * -> at C P Q2 -> Q1 = Q2 H2 : at (par_r R C1) P Q2 H3 : at C1 P Q * ============================ par R Q = Q2 Subgoal 6 is: act A Q = Q2 at_det3 <case H2.

Subgoal 5: Variables: P Q R C1 Q3 IH : forall C P Q1 Q2, at C P Q1 * -> at C P Q2 -> Q1 = Q2 H3 : at C1 P Q * H4 : at C1 P Q3 ============================ par R Q = par R Q3 Subgoal 6 is: act A Q = Q2 at_det3 <apply IH to H3 H4.

Subgoal 5: Variables: P R C1 Q3 IH : forall C P Q1 Q2, at C P Q1 * -> at C P Q2 -> Q1 = Q2 H3 : at C1 P Q3 * H4 : at C1 P Q3 ============================ par R Q3 = par R Q3 Subgoal 6 is: act A Q = Q2 at_det3 <search.

Subgoal 6: Variables: P Q2 Q A C1 IH : forall C P Q1 Q2, at C P Q1 * -> at C P Q2 -> Q1 = Q2 H2 : at (act_d A C1) P Q2 H3 : at C1 P Q * ============================ act A Q = Q2 at_det3 <case H2.

Subgoal 6: Variables: P Q A C1 Q3 IH : forall C P Q1 Q2, at C P Q1 * -> at C P Q2 -> Q1 = Q2 H3 : at C1 P Q * H4 : at C1 P Q3 ============================ act A Q = act A Q3 at_det3 <apply IH to H3 H4.

Subgoal 6: Variables: P A C1 Q3 IH : forall C P Q1 Q2, at C P Q1 * -> at C P Q2 -> Q1 = Q2 H3 : at C1 P Q3 * H4 : at C1 P Q3 ============================ act A Q3 = act A Q3 at_det3 <search.Proof completed.

Abella <Theorem at_det2 : forall C P1 P2 Q, at C P1 Q -> at C P2 Q -> P1 = P2.

============================ forall C P1 P2 Q, at C P1 Q -> at C P2 Q -> P1 = P2 at_det2 <induction on 1.

IH : forall C P1 P2 Q, at C P1 Q * -> at C P2 Q -> P1 = P2 ============================ forall C P1 P2 Q, at C P1 Q @ -> at C P2 Q -> P1 = P2 at_det2 <intros.

Variables: C P1 P2 Q IH : forall C P1 P2 Q, at C P1 Q * -> at C P2 Q -> P1 = P2 H1 : at C P1 Q @ H2 : at C P2 Q ============================ P1 = P2 at_det2 <case H1.

Subgoal 1: Variables: P2 Q IH : forall C P1 P2 Q, at C P1 Q * -> at C P2 Q -> P1 = P2 H2 : at hole P2 Q ============================ Q = P2 Subgoal 2 is: P1 = P2 Subgoal 3 is: P1 = P2 Subgoal 4 is: P1 = P2 Subgoal 5 is: P1 = P2 Subgoal 6 is: P1 = P2 at_det2 <case H2.

Subgoal 1: Variables: Q IH : forall C P1 P2 Q, at C P1 Q * -> at C P2 Q -> P1 = P2 ============================ Q = Q Subgoal 2 is: P1 = P2 Subgoal 3 is: P1 = P2 Subgoal 4 is: P1 = P2 Subgoal 5 is: P1 = P2 Subgoal 6 is: P1 = P2 at_det2 <search.

Subgoal 2: Variables: P1 P2 R Q1 C1 IH : forall C P1 P2 Q, at C P1 Q * -> at C P2 Q -> P1 = P2 H2 : at (plus_l C1 R) P2 (plus Q1 R) H3 : at C1 P1 Q1 * ============================ P1 = P2 Subgoal 3 is: P1 = P2 Subgoal 4 is: P1 = P2 Subgoal 5 is: P1 = P2 Subgoal 6 is: P1 = P2 at_det2 <case H2.

Subgoal 2: Variables: P1 P2 R Q1 C1 IH : forall C P1 P2 Q, at C P1 Q * -> at C P2 Q -> P1 = P2 H3 : at C1 P1 Q1 * H4 : at C1 P2 Q1 ============================ P1 = P2 Subgoal 3 is: P1 = P2 Subgoal 4 is: P1 = P2 Subgoal 5 is: P1 = P2 Subgoal 6 is: P1 = P2 at_det2 <apply IH to H3 H4.

Subgoal 2: Variables: P2 R Q1 C1 IH : forall C P1 P2 Q, at C P1 Q * -> at C P2 Q -> P1 = P2 H3 : at C1 P2 Q1 * H4 : at C1 P2 Q1 ============================ P2 = P2 Subgoal 3 is: P1 = P2 Subgoal 4 is: P1 = P2 Subgoal 5 is: P1 = P2 Subgoal 6 is: P1 = P2 at_det2 <search.

Subgoal 3: Variables: P1 P2 Q1 R C1 IH : forall C P1 P2 Q, at C P1 Q * -> at C P2 Q -> P1 = P2 H2 : at (plus_r R C1) P2 (plus R Q1) H3 : at C1 P1 Q1 * ============================ P1 = P2 Subgoal 4 is: P1 = P2 Subgoal 5 is: P1 = P2 Subgoal 6 is: P1 = P2 at_det2 <case H2.

Subgoal 3: Variables: P1 P2 Q1 R C1 IH : forall C P1 P2 Q, at C P1 Q * -> at C P2 Q -> P1 = P2 H3 : at C1 P1 Q1 * H4 : at C1 P2 Q1 ============================ P1 = P2 Subgoal 4 is: P1 = P2 Subgoal 5 is: P1 = P2 Subgoal 6 is: P1 = P2 at_det2 <apply IH to H3 H4.

Subgoal 3: Variables: P2 Q1 R C1 IH : forall C P1 P2 Q, at C P1 Q * -> at C P2 Q -> P1 = P2 H3 : at C1 P2 Q1 * H4 : at C1 P2 Q1 ============================ P2 = P2 Subgoal 4 is: P1 = P2 Subgoal 5 is: P1 = P2 Subgoal 6 is: P1 = P2 at_det2 <search.

Subgoal 4: Variables: P1 P2 R Q1 C1 IH : forall C P1 P2 Q, at C P1 Q * -> at C P2 Q -> P1 = P2 H2 : at (par_l C1 R) P2 (par Q1 R) H3 : at C1 P1 Q1 * ============================ P1 = P2 Subgoal 5 is: P1 = P2 Subgoal 6 is: P1 = P2 at_det2 <case H2.

Subgoal 4: Variables: P1 P2 R Q1 C1 IH : forall C P1 P2 Q, at C P1 Q * -> at C P2 Q -> P1 = P2 H3 : at C1 P1 Q1 * H4 : at C1 P2 Q1 ============================ P1 = P2 Subgoal 5 is: P1 = P2 Subgoal 6 is: P1 = P2 at_det2 <apply IH to H3 H4.

Subgoal 4: Variables: P2 R Q1 C1 IH : forall C P1 P2 Q, at C P1 Q * -> at C P2 Q -> P1 = P2 H3 : at C1 P2 Q1 * H4 : at C1 P2 Q1 ============================ P2 = P2 Subgoal 5 is: P1 = P2 Subgoal 6 is: P1 = P2 at_det2 <search.

Subgoal 5: Variables: P1 P2 Q1 R C1 IH : forall C P1 P2 Q, at C P1 Q * -> at C P2 Q -> P1 = P2 H2 : at (par_r R C1) P2 (par R Q1) H3 : at C1 P1 Q1 * ============================ P1 = P2 Subgoal 6 is: P1 = P2 at_det2 <case H2.

Subgoal 5: Variables: P1 P2 Q1 R C1 IH : forall C P1 P2 Q, at C P1 Q * -> at C P2 Q -> P1 = P2 H3 : at C1 P1 Q1 * H4 : at C1 P2 Q1 ============================ P1 = P2 Subgoal 6 is: P1 = P2 at_det2 <apply IH to H3 H4.

Subgoal 5: Variables: P2 Q1 R C1 IH : forall C P1 P2 Q, at C P1 Q * -> at C P2 Q -> P1 = P2 H3 : at C1 P2 Q1 * H4 : at C1 P2 Q1 ============================ P2 = P2 Subgoal 6 is: P1 = P2 at_det2 <search.

Subgoal 6: Variables: P1 P2 Q1 A C1 IH : forall C P1 P2 Q, at C P1 Q * -> at C P2 Q -> P1 = P2 H2 : at (act_d A C1) P2 (act A Q1) H3 : at C1 P1 Q1 * ============================ P1 = P2 at_det2 <case H2.

Subgoal 6: Variables: P1 P2 Q1 A C1 IH : forall C P1 P2 Q, at C P1 Q * -> at C P2 Q -> P1 = P2 H3 : at C1 P1 Q1 * H4 : at C1 P2 Q1 ============================ P1 = P2 at_det2 <apply IH to H3 H4.

Subgoal 6: Variables: P2 Q1 A C1 IH : forall C P1 P2 Q, at C P1 Q * -> at C P2 Q -> P1 = P2 H3 : at C1 P2 Q1 * H4 : at C1 P2 Q1 ============================ P2 = P2 at_det2 <search.Proof completed.

Abella <Theorem concat_ctx : forall C1 P1 C2 P2 P3, at C1 P1 P2 -> at C2 P2 P3 -> (exists C3, at C3 P1 P3).

============================ forall C1 P1 C2 P2 P3, at C1 P1 P2 -> at C2 P2 P3 -> (exists C3, at C3 P1 P3) concat_ctx <induction on 2.

IH : forall C1 P1 C2 P2 P3, at C1 P1 P2 -> at C2 P2 P3 * -> (exists C3, at C3 P1 P3) ============================ forall C1 P1 C2 P2 P3, at C1 P1 P2 -> at C2 P2 P3 @ -> (exists C3, at C3 P1 P3) concat_ctx <intros.

Variables: C1 P1 C2 P2 P3 IH : forall C1 P1 C2 P2 P3, at C1 P1 P2 -> at C2 P2 P3 * -> (exists C3, at C3 P1 P3) H1 : at C1 P1 P2 H2 : at C2 P2 P3 @ ============================ exists C3, at C3 P1 P3 concat_ctx <case H2.

Subgoal 1: Variables: C1 P1 P3 IH : forall C1 P1 C2 P2 P3, at C1 P1 P2 -> at C2 P2 P3 * -> (exists C3, at C3 P1 P3) H1 : at C1 P1 P3 ============================ exists C3, at C3 P1 P3 Subgoal 2 is: exists C3, at C3 P1 (plus Q R) Subgoal 3 is: exists C3, at C3 P1 (plus R Q) Subgoal 4 is: exists C3, at C3 P1 (par Q R) Subgoal 5 is: exists C3, at C3 P1 (par R Q) Subgoal 6 is: exists C3, at C3 P1 (act A Q) concat_ctx <search.

Subgoal 2: Variables: C1 P1 P2 R Q C IH : forall C1 P1 C2 P2 P3, at C1 P1 P2 -> at C2 P2 P3 * -> (exists C3, at C3 P1 P3) H1 : at C1 P1 P2 H3 : at C P2 Q * ============================ exists C3, at C3 P1 (plus Q R) Subgoal 3 is: exists C3, at C3 P1 (plus R Q) Subgoal 4 is: exists C3, at C3 P1 (par Q R) Subgoal 5 is: exists C3, at C3 P1 (par R Q) Subgoal 6 is: exists C3, at C3 P1 (act A Q) concat_ctx <apply IH to H1 H3.

Subgoal 2: Variables: C1 P1 P2 R Q C C3 IH : forall C1 P1 C2 P2 P3, at C1 P1 P2 -> at C2 P2 P3 * -> (exists C3, at C3 P1 P3) H1 : at C1 P1 P2 H3 : at C P2 Q * H4 : at C3 P1 Q ============================ exists C3, at C3 P1 (plus Q R) Subgoal 3 is: exists C3, at C3 P1 (plus R Q) Subgoal 4 is: exists C3, at C3 P1 (par Q R) Subgoal 5 is: exists C3, at C3 P1 (par R Q) Subgoal 6 is: exists C3, at C3 P1 (act A Q) concat_ctx <search.

Subgoal 3: Variables: C1 P1 P2 Q R C IH : forall C1 P1 C2 P2 P3, at C1 P1 P2 -> at C2 P2 P3 * -> (exists C3, at C3 P1 P3) H1 : at C1 P1 P2 H3 : at C P2 Q * ============================ exists C3, at C3 P1 (plus R Q) Subgoal 4 is: exists C3, at C3 P1 (par Q R) Subgoal 5 is: exists C3, at C3 P1 (par R Q) Subgoal 6 is: exists C3, at C3 P1 (act A Q) concat_ctx <apply IH to H1 H3.

Subgoal 3: Variables: C1 P1 P2 Q R C C3 IH : forall C1 P1 C2 P2 P3, at C1 P1 P2 -> at C2 P2 P3 * -> (exists C3, at C3 P1 P3) H1 : at C1 P1 P2 H3 : at C P2 Q * H4 : at C3 P1 Q ============================ exists C3, at C3 P1 (plus R Q) Subgoal 4 is: exists C3, at C3 P1 (par Q R) Subgoal 5 is: exists C3, at C3 P1 (par R Q) Subgoal 6 is: exists C3, at C3 P1 (act A Q) concat_ctx <search.

Subgoal 4: Variables: C1 P1 P2 R Q C IH : forall C1 P1 C2 P2 P3, at C1 P1 P2 -> at C2 P2 P3 * -> (exists C3, at C3 P1 P3) H1 : at C1 P1 P2 H3 : at C P2 Q * ============================ exists C3, at C3 P1 (par Q R) Subgoal 5 is: exists C3, at C3 P1 (par R Q) Subgoal 6 is: exists C3, at C3 P1 (act A Q) concat_ctx <apply IH to H1 H3.

Subgoal 4: Variables: C1 P1 P2 R Q C C3 IH : forall C1 P1 C2 P2 P3, at C1 P1 P2 -> at C2 P2 P3 * -> (exists C3, at C3 P1 P3) H1 : at C1 P1 P2 H3 : at C P2 Q * H4 : at C3 P1 Q ============================ exists C3, at C3 P1 (par Q R) Subgoal 5 is: exists C3, at C3 P1 (par R Q) Subgoal 6 is: exists C3, at C3 P1 (act A Q) concat_ctx <search.

Subgoal 5: Variables: C1 P1 P2 Q R C IH : forall C1 P1 C2 P2 P3, at C1 P1 P2 -> at C2 P2 P3 * -> (exists C3, at C3 P1 P3) H1 : at C1 P1 P2 H3 : at C P2 Q * ============================ exists C3, at C3 P1 (par R Q) Subgoal 6 is: exists C3, at C3 P1 (act A Q) concat_ctx <apply IH to H1 H3.

Subgoal 5: Variables: C1 P1 P2 Q R C C3 IH : forall C1 P1 C2 P2 P3, at C1 P1 P2 -> at C2 P2 P3 * -> (exists C3, at C3 P1 P3) H1 : at C1 P1 P2 H3 : at C P2 Q * H4 : at C3 P1 Q ============================ exists C3, at C3 P1 (par R Q) Subgoal 6 is: exists C3, at C3 P1 (act A Q) concat_ctx <search.

Subgoal 6: Variables: C1 P1 P2 Q A C IH : forall C1 P1 C2 P2 P3, at C1 P1 P2 -> at C2 P2 P3 * -> (exists C3, at C3 P1 P3) H1 : at C1 P1 P2 H3 : at C P2 Q * ============================ exists C3, at C3 P1 (act A Q) concat_ctx <apply IH to H1 H3.

Subgoal 6: Variables: C1 P1 P2 Q A C C3 IH : forall C1 P1 C2 P2 P3, at C1 P1 P2 -> at C2 P2 P3 * -> (exists C3, at C3 P1 P3) H1 : at C1 P1 P2 H3 : at C P2 Q * H4 : at C3 P1 Q ============================ exists C3, at C3 P1 (act A Q) concat_ctx <search.Proof completed.

Abella <Theorem ctx_faithful : forall C P P0 A R, at C P P0 -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R).

============================ forall C P P0 A R, at C P P0 -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) ctx_faithful <induction on 1.

IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) ============================ forall C P P0 A R, at C P P0 @ -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) ctx_faithful <intros.

Variables: C P P0 A R IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at C P P0 @ H2 : one P0 A R ============================ (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) ctx_faithful <case H1 (keep).

Subgoal 1: Variables: P0 A R IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at hole P0 P0 @ H2 : one P0 A R ============================ (exists CC, at CC P0 R /\ (forall Q Q0, at hole Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P0 B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at hole Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at hole Q Q0 -> one Q0 A R) Subgoal 2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 1: Variables: P0 A R IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at hole P0 P0 @ H2 : one P0 A R ============================ (exists CC, at CC P0 R /\ (forall Q Q0, at hole Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P0 B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at hole Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) Subgoal 2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <right.

Subgoal 1: Variables: P0 A R IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at hole P0 P0 @ H2 : one P0 A R ============================ exists CC PBP B, one P0 B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at hole Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR))) Subgoal 2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness hole.

Subgoal 1: Variables: P0 A R IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at hole P0 P0 @ H2 : one P0 A R ============================ exists PBP B, one P0 B PBP /\ at hole PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at hole Q Q0 -> (exists RR, one Q0 A RR /\ at hole QBQ RR))) Subgoal 2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness R.

Subgoal 1: Variables: P0 A R IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at hole P0 P0 @ H2 : one P0 A R ============================ exists B, one P0 B R /\ at hole R R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at hole Q Q0 -> (exists RR, one Q0 A RR /\ at hole QBQ RR))) Subgoal 2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness A.

Subgoal 1: Variables: P0 A R IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at hole P0 P0 @ H2 : one P0 A R ============================ one P0 A R /\ at hole R R /\ (forall Q QBQ, one Q A QBQ -> (forall Q0, at hole Q Q0 -> (exists RR, one Q0 A RR /\ at hole QBQ RR))) Subgoal 2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <split.

Subgoal 1.1: Variables: P0 A R IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at hole P0 P0 @ H2 : one P0 A R ============================ one P0 A R Subgoal 1.2 is: at hole R R Subgoal 1.3 is: forall Q QBQ, one Q A QBQ -> (forall Q0, at hole Q Q0 -> (exists RR, one Q0 A RR /\ at hole QBQ RR)) Subgoal 2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 1.2: Variables: P0 A R IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at hole P0 P0 @ H2 : one P0 A R ============================ at hole R R Subgoal 1.3 is: forall Q QBQ, one Q A QBQ -> (forall Q0, at hole Q Q0 -> (exists RR, one Q0 A RR /\ at hole QBQ RR)) Subgoal 2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 1.3: Variables: P0 A R IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at hole P0 P0 @ H2 : one P0 A R ============================ forall Q QBQ, one Q A QBQ -> (forall Q0, at hole Q Q0 -> (exists RR, one Q0 A RR /\ at hole QBQ RR)) Subgoal 2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 1.3: Variables: P0 A R Q QBQ Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at hole P0 P0 @ H2 : one P0 A R H3 : one Q A QBQ H4 : at hole Q Q0 ============================ exists RR, one Q0 A RR /\ at hole QBQ RR Subgoal 2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H4.

Subgoal 1.3: Variables: P0 A R QBQ Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at hole P0 P0 @ H2 : one P0 A R H3 : one Q0 A QBQ ============================ exists RR, one Q0 A RR /\ at hole QBQ RR Subgoal 2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness QBQ.

Subgoal 1.3: Variables: P0 A R QBQ Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at hole P0 P0 @ H2 : one P0 A R H3 : one Q0 A QBQ ============================ one Q0 A QBQ /\ at hole QBQ QBQ Subgoal 2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 2: Variables: P A R R1 Q C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H2 : one (plus Q R1) A R H3 : at C1 P Q * ============================ (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H2.

Subgoal 2.1: Variables: P A R R1 Q C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R ============================ (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply IH to H3 H4.

Subgoal 2.1: Variables: P A R R1 Q C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H5 : (exists CC, at CC P R /\ (forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C1 Q Q0 -> one Q0 A R) ============================ (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H5.

Subgoal 2.1.1: Variables: P A R R1 Q C1 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : at CC P R H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.1.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.1.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 2.1.1: Variables: P A R R1 Q C1 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : at CC P R H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) Subgoal 2.1.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.1.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 2.1.1: Variables: P A R R1 Q C1 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : at CC P R H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR)) Subgoal 2.1.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.1.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness CC.

Subgoal 2.1.1: Variables: P A R R1 Q C1 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : at CC P R H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR)) Subgoal 2.1.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.1.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <split.

Subgoal 2.1.1.1: Variables: P A R R1 Q C1 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : at CC P R H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ at CC P R Subgoal 2.1.1.2 is: forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) Subgoal 2.1.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.1.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 2.1.1.2: Variables: P A R R1 Q C1 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : at CC P R H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) Subgoal 2.1.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.1.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 2.1.1.2: Variables: P A R R1 Q C1 CC Q1 Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : at CC P R H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) H8 : at (plus_l C1 R1) Q1 Q0 ============================ exists RR, one Q0 A RR /\ at CC Q1 RR Subgoal 2.1.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.1.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H8.

Subgoal 2.1.1.2: Variables: P A R R1 Q C1 CC Q1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : at CC P R H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) H9 : at C1 Q1 Q2 ============================ exists RR, one (plus Q2 R1) A RR /\ at CC Q1 RR Subgoal 2.1.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.1.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H7 to H9.

Subgoal 2.1.1.2: Variables: P A R R1 Q C1 CC Q1 Q2 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : at CC P R H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) H9 : at C1 Q1 Q2 H10 : one Q2 A RR H11 : at CC Q1 RR ============================ exists RR, one (plus Q2 R1) A RR /\ at CC Q1 RR Subgoal 2.1.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.1.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 2.1.2: Variables: P A R R1 Q C1 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.1.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 2.1.2: Variables: P A R R1 Q C1 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) Subgoal 2.1.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <right.

Subgoal 2.1.2: Variables: P A R R1 Q C1 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR))) Subgoal 2.1.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness CC.

Subgoal 2.1.2: Variables: P A R R1 Q C1 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ exists PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR))) Subgoal 2.1.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness PBP.

Subgoal 2.1.2: Variables: P A R R1 Q C1 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ exists B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR))) Subgoal 2.1.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness B.

Subgoal 2.1.2: Variables: P A R R1 Q C1 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR))) Subgoal 2.1.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <split.

Subgoal 2.1.2.1: Variables: P A R R1 Q C1 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ one P B PBP Subgoal 2.1.2.2 is: at CC PBP R Subgoal 2.1.2.3 is: forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) Subgoal 2.1.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 2.1.2.2: Variables: P A R R1 Q C1 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ at CC PBP R Subgoal 2.1.2.3 is: forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) Subgoal 2.1.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 2.1.2.3: Variables: P A R R1 Q C1 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) Subgoal 2.1.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 2.1.2.3: Variables: P A R R1 Q C1 CC PBP B Q1 QBQ Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) H9 : one Q1 B QBQ H10 : at (plus_l C1 R1) Q1 Q0 ============================ exists RR, one Q0 A RR /\ at CC QBQ RR Subgoal 2.1.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H8 to H9.

Subgoal 2.1.2.3: Variables: P A R R1 Q C1 CC PBP B Q1 QBQ Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) H9 : one Q1 B QBQ H10 : at (plus_l C1 R1) Q1 Q0 H11 : forall Q0, at C1 Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR) ============================ exists RR, one Q0 A RR /\ at CC QBQ RR Subgoal 2.1.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H10.

Subgoal 2.1.2.3: Variables: P A R R1 Q C1 CC PBP B Q1 QBQ Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) H9 : one Q1 B QBQ H11 : forall Q0, at C1 Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR) H12 : at C1 Q1 Q2 ============================ exists RR, one (plus Q2 R1) A RR /\ at CC QBQ RR Subgoal 2.1.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H11 to H12.

Subgoal 2.1.2.3: Variables: P A R R1 Q C1 CC PBP B Q1 QBQ Q2 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) H9 : one Q1 B QBQ H11 : forall Q0, at C1 Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR) H12 : at C1 Q1 Q2 H13 : one Q2 A RR H14 : at CC QBQ RR ============================ exists RR, one (plus Q2 R1) A RR /\ at CC QBQ RR Subgoal 2.1.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 2.1.3: Variables: P A R R1 Q C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : forall Q Q0, at C1 Q Q0 -> one Q0 A R ============================ (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <right.

Subgoal 2.1.3: Variables: P A R R1 Q C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : forall Q Q0, at C1 Q Q0 -> one Q0 A R ============================ forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 2.1.3: Variables: P A R R1 Q C1 Q1 Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : forall Q Q0, at C1 Q Q0 -> one Q0 A R H7 : at (plus_l C1 R1) Q1 Q0 ============================ one Q0 A R Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H7.

Subgoal 2.1.3: Variables: P A R R1 Q C1 Q1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : forall Q Q0, at C1 Q Q0 -> one Q0 A R H8 : at C1 Q1 Q2 ============================ one (plus Q2 R1) A R Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H6 to H8.

Subgoal 2.1.3: Variables: P A R R1 Q C1 Q1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one Q A R H6 : forall Q Q0, at C1 Q Q0 -> one Q0 A R H8 : at C1 Q1 Q2 H9 : one Q2 A R ============================ one (plus Q2 R1) A R Subgoal 2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 2.2: Variables: P A R R1 Q C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one R1 A R ============================ (exists CC, at CC P R /\ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <right.

Subgoal 2.2: Variables: P A R R1 Q C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one R1 A R ============================ forall Q Q0, at (plus_l C1 R1) Q Q0 -> one Q0 A R Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 2.2: Variables: P A R R1 Q C1 Q1 Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one R1 A R H5 : at (plus_l C1 R1) Q1 Q0 ============================ one Q0 A R Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H5.

Subgoal 2.2: Variables: P A R R1 Q C1 Q1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_l C1 R1) P (plus Q R1) @ H3 : at C1 P Q * H4 : one R1 A R H6 : at C1 Q1 Q2 ============================ one (plus Q2 R1) A R Subgoal 3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 3: Variables: P A R Q R1 C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H2 : one (plus R1 Q) A R H3 : at C1 P Q * ============================ (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H2.

Subgoal 3.1: Variables: P A R Q R1 C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one R1 A R ============================ (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 3.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <right.

Subgoal 3.1: Variables: P A R Q R1 C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one R1 A R ============================ forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R Subgoal 3.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 3.1: Variables: P A R Q R1 C1 Q1 Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one R1 A R H5 : at (plus_r R1 C1) Q1 Q0 ============================ one Q0 A R Subgoal 3.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H5.

Subgoal 3.1: Variables: P A R Q R1 C1 Q1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one R1 A R H6 : at C1 Q1 Q2 ============================ one (plus R1 Q2) A R Subgoal 3.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 3.2: Variables: P A R Q R1 C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R ============================ (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply IH to H3 H4.

Subgoal 3.2: Variables: P A R Q R1 C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H5 : (exists CC, at CC P R /\ (forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C1 Q Q0 -> one Q0 A R) ============================ (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H5.

Subgoal 3.2.1: Variables: P A R Q R1 C1 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : at CC P R H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 3.2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 3.2.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 3.2.1: Variables: P A R Q R1 C1 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : at CC P R H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) Subgoal 3.2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 3.2.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 3.2.1: Variables: P A R Q R1 C1 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : at CC P R H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR)) Subgoal 3.2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 3.2.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness CC.

Subgoal 3.2.1: Variables: P A R Q R1 C1 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : at CC P R H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR)) Subgoal 3.2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 3.2.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <split.

Subgoal 3.2.1.1: Variables: P A R Q R1 C1 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : at CC P R H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ at CC P R Subgoal 3.2.1.2 is: forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) Subgoal 3.2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 3.2.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 3.2.1.2: Variables: P A R Q R1 C1 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : at CC P R H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) Subgoal 3.2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 3.2.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 3.2.1.2: Variables: P A R Q R1 C1 CC Q1 Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : at CC P R H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) H8 : at (plus_r R1 C1) Q1 Q0 ============================ exists RR, one Q0 A RR /\ at CC Q1 RR Subgoal 3.2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 3.2.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H8.

Subgoal 3.2.1.2: Variables: P A R Q R1 C1 CC Q1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : at CC P R H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) H9 : at C1 Q1 Q2 ============================ exists RR, one (plus R1 Q2) A RR /\ at CC Q1 RR Subgoal 3.2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 3.2.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H7 to H9.

Subgoal 3.2.1.2: Variables: P A R Q R1 C1 CC Q1 Q2 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : at CC P R H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) H9 : at C1 Q1 Q2 H10 : one Q2 A RR H11 : at CC Q1 RR ============================ exists RR, one (plus R1 Q2) A RR /\ at CC Q1 RR Subgoal 3.2.2 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 3.2.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 3.2.2: Variables: P A R Q R1 C1 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 3.2.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 3.2.2: Variables: P A R Q R1 C1 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) Subgoal 3.2.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <right.

Subgoal 3.2.2: Variables: P A R Q R1 C1 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR))) Subgoal 3.2.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness CC.

Subgoal 3.2.2: Variables: P A R Q R1 C1 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ exists PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR))) Subgoal 3.2.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness PBP.

Subgoal 3.2.2: Variables: P A R Q R1 C1 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ exists B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR))) Subgoal 3.2.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness B.

Subgoal 3.2.2: Variables: P A R Q R1 C1 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR))) Subgoal 3.2.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <split.

Subgoal 3.2.2.1: Variables: P A R Q R1 C1 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ one P B PBP Subgoal 3.2.2.2 is: at CC PBP R Subgoal 3.2.2.3 is: forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) Subgoal 3.2.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 3.2.2.2: Variables: P A R Q R1 C1 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ at CC PBP R Subgoal 3.2.2.3 is: forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) Subgoal 3.2.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 3.2.2.3: Variables: P A R Q R1 C1 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) Subgoal 3.2.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 3.2.2.3: Variables: P A R Q R1 C1 CC PBP B Q1 QBQ Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) H9 : one Q1 B QBQ H10 : at (plus_r R1 C1) Q1 Q0 ============================ exists RR, one Q0 A RR /\ at CC QBQ RR Subgoal 3.2.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H8 to H9.

Subgoal 3.2.2.3: Variables: P A R Q R1 C1 CC PBP B Q1 QBQ Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) H9 : one Q1 B QBQ H10 : at (plus_r R1 C1) Q1 Q0 H11 : forall Q0, at C1 Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR) ============================ exists RR, one Q0 A RR /\ at CC QBQ RR Subgoal 3.2.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H10.

Subgoal 3.2.2.3: Variables: P A R Q R1 C1 CC PBP B Q1 QBQ Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) H9 : one Q1 B QBQ H11 : forall Q0, at C1 Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR) H12 : at C1 Q1 Q2 ============================ exists RR, one (plus R1 Q2) A RR /\ at CC QBQ RR Subgoal 3.2.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H11 to H12.

Subgoal 3.2.2.3: Variables: P A R Q R1 C1 CC PBP B Q1 QBQ Q2 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : one P B PBP H7 : at CC PBP R H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) H9 : one Q1 B QBQ H11 : forall Q0, at C1 Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR) H12 : at C1 Q1 Q2 H13 : one Q2 A RR H14 : at CC QBQ RR ============================ exists RR, one (plus R1 Q2) A RR /\ at CC QBQ RR Subgoal 3.2.3 is: (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 3.2.3: Variables: P A R Q R1 C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : forall Q Q0, at C1 Q Q0 -> one Q0 A R ============================ (exists CC, at CC P R /\ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (plus_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <right.

Subgoal 3.2.3: Variables: P A R Q R1 C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : forall Q Q0, at C1 Q Q0 -> one Q0 A R ============================ forall Q Q0, at (plus_r R1 C1) Q Q0 -> one Q0 A R Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 3.2.3: Variables: P A R Q R1 C1 Q1 Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : forall Q Q0, at C1 Q Q0 -> one Q0 A R H7 : at (plus_r R1 C1) Q1 Q0 ============================ one Q0 A R Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H7.

Subgoal 3.2.3: Variables: P A R Q R1 C1 Q1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : forall Q Q0, at C1 Q Q0 -> one Q0 A R H8 : at C1 Q1 Q2 ============================ one (plus R1 Q2) A R Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H6 to H8.

Subgoal 3.2.3: Variables: P A R Q R1 C1 Q1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (plus_r R1 C1) P (plus R1 Q) @ H3 : at C1 P Q * H4 : one Q A R H6 : forall Q Q0, at C1 Q Q0 -> one Q0 A R H8 : at C1 Q1 Q2 H9 : one Q2 A R ============================ one (plus R1 Q2) A R Subgoal 4 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 4: Variables: P A R R1 Q C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H2 : one (par Q R1) A R H3 : at C1 P Q * ============================ (exists CC, at CC P R /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A R) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H2.

Subgoal 4.1: Variables: P A R1 Q C1 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 ============================ (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply IH to H3 H4.

Subgoal 4.1: Variables: P A R1 Q C1 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H5 : (exists CC, at CC P P2 /\ (forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP P2 /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C1 Q Q0 -> one Q0 A P2) ============================ (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H5.

Subgoal 4.1.1: Variables: P A R1 Q C1 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : at CC P P2 H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.1.2 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 4.1.1: Variables: P A R1 Q C1 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : at CC P P2 H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) Subgoal 4.1.2 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 4.1.1: Variables: P A R1 Q C1 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : at CC P P2 H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR)) Subgoal 4.1.2 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par_l CC R1.

Subgoal 4.1.1: Variables: P A R1 Q C1 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : at CC P P2 H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ at (par_l CC R1) P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at (par_l CC R1) Q RR)) Subgoal 4.1.2 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <split.

Subgoal 4.1.1.1: Variables: P A R1 Q C1 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : at CC P P2 H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ at (par_l CC R1) P (par P2 R1) Subgoal 4.1.1.2 is: forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at (par_l CC R1) Q RR) Subgoal 4.1.2 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 4.1.1.2: Variables: P A R1 Q C1 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : at CC P P2 H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at (par_l CC R1) Q RR) Subgoal 4.1.2 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 4.1.1.2: Variables: P A R1 Q C1 P2 CC Q1 Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : at CC P P2 H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) H8 : at (par_l C1 R1) Q1 Q0 ============================ exists RR, one Q0 A RR /\ at (par_l CC R1) Q1 RR Subgoal 4.1.2 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H8.

Subgoal 4.1.1.2: Variables: P A R1 Q C1 P2 CC Q1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : at CC P P2 H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) H9 : at C1 Q1 Q2 ============================ exists RR, one (par Q2 R1) A RR /\ at (par_l CC R1) Q1 RR Subgoal 4.1.2 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H7 to H9.

Subgoal 4.1.1.2: Variables: P A R1 Q C1 P2 CC Q1 Q2 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : at CC P P2 H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) H9 : at C1 Q1 Q2 H10 : one Q2 A RR H11 : at CC Q1 RR ============================ exists RR, one (par Q2 R1) A RR /\ at (par_l CC R1) Q1 RR Subgoal 4.1.2 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par RR R1.

Subgoal 4.1.1.2: Variables: P A R1 Q C1 P2 CC Q1 Q2 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : at CC P P2 H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) H9 : at C1 Q1 Q2 H10 : one Q2 A RR H11 : at CC Q1 RR ============================ one (par Q2 R1) A (par RR R1) /\ at (par_l CC R1) Q1 (par RR R1) Subgoal 4.1.2 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 4.1.2: Variables: P A R1 Q C1 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : one P B PBP H7 : at CC PBP P2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 4.1.2: Variables: P A R1 Q C1 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : one P B PBP H7 : at CC PBP P2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <right.

Subgoal 4.1.2: Variables: P A R1 Q C1 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : one P B PBP H7 : at CC PBP P2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR))) Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par_l CC R1.

Subgoal 4.1.2: Variables: P A R1 Q C1 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : one P B PBP H7 : at CC PBP P2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ exists PBP B, one P B PBP /\ at (par_l CC R1) PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at (par_l CC R1) QBQ RR))) Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness PBP.

Subgoal 4.1.2: Variables: P A R1 Q C1 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : one P B PBP H7 : at CC PBP P2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ exists B, one P B PBP /\ at (par_l CC R1) PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at (par_l CC R1) QBQ RR))) Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness B.

Subgoal 4.1.2: Variables: P A R1 Q C1 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : one P B PBP H7 : at CC PBP P2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ one P B PBP /\ at (par_l CC R1) PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at (par_l CC R1) QBQ RR))) Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <split.

Subgoal 4.1.2.1: Variables: P A R1 Q C1 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : one P B PBP H7 : at CC PBP P2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ one P B PBP Subgoal 4.1.2.2 is: at (par_l CC R1) PBP (par P2 R1) Subgoal 4.1.2.3 is: forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at (par_l CC R1) QBQ RR)) Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 4.1.2.2: Variables: P A R1 Q C1 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : one P B PBP H7 : at CC PBP P2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ at (par_l CC R1) PBP (par P2 R1) Subgoal 4.1.2.3 is: forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at (par_l CC R1) QBQ RR)) Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 4.1.2.3: Variables: P A R1 Q C1 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : one P B PBP H7 : at CC PBP P2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at (par_l CC R1) QBQ RR)) Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 4.1.2.3: Variables: P A R1 Q C1 P2 CC PBP B Q1 QBQ Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : one P B PBP H7 : at CC PBP P2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) H9 : one Q1 B QBQ H10 : at (par_l C1 R1) Q1 Q0 ============================ exists RR, one Q0 A RR /\ at (par_l CC R1) QBQ RR Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H10.

Subgoal 4.1.2.3: Variables: P A R1 Q C1 P2 CC PBP B Q1 QBQ Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : one P B PBP H7 : at CC PBP P2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) H9 : one Q1 B QBQ H11 : at C1 Q1 Q2 ============================ exists RR, one (par Q2 R1) A RR /\ at (par_l CC R1) QBQ RR Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H8 to H9.

Subgoal 4.1.2.3: Variables: P A R1 Q C1 P2 CC PBP B Q1 QBQ Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : one P B PBP H7 : at CC PBP P2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) H9 : one Q1 B QBQ H11 : at C1 Q1 Q2 H12 : forall Q0, at C1 Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR) ============================ exists RR, one (par Q2 R1) A RR /\ at (par_l CC R1) QBQ RR Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H12 to H11.

Subgoal 4.1.2.3: Variables: P A R1 Q C1 P2 CC PBP B Q1 QBQ Q2 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : one P B PBP H7 : at CC PBP P2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) H9 : one Q1 B QBQ H11 : at C1 Q1 Q2 H12 : forall Q0, at C1 Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR) H13 : one Q2 A RR H14 : at CC QBQ RR ============================ exists RR, one (par Q2 R1) A RR /\ at (par_l CC R1) QBQ RR Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par RR R1.

Subgoal 4.1.2.3: Variables: P A R1 Q C1 P2 CC PBP B Q1 QBQ Q2 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : one P B PBP H7 : at CC PBP P2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) H9 : one Q1 B QBQ H11 : at C1 Q1 Q2 H12 : forall Q0, at C1 Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR) H13 : one Q2 A RR H14 : at CC QBQ RR ============================ one (par Q2 R1) A (par RR R1) /\ at (par_l CC R1) QBQ (par RR R1) Subgoal 4.1.3 is: (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 4.1.3: Variables: P A R1 Q C1 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : forall Q Q0, at C1 Q Q0 -> one Q0 A P2 ============================ (exists CC, at CC P (par P2 R1) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 R1) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1)) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <right.

Subgoal 4.1.3: Variables: P A R1 Q C1 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : forall Q Q0, at C1 Q Q0 -> one Q0 A P2 ============================ forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 A (par P2 R1) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 4.1.3: Variables: P A R1 Q C1 P2 Q1 Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : forall Q Q0, at C1 Q Q0 -> one Q0 A P2 H7 : at (par_l C1 R1) Q1 Q0 ============================ one Q0 A (par P2 R1) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H7.

Subgoal 4.1.3: Variables: P A R1 Q C1 P2 Q1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : forall Q Q0, at C1 Q Q0 -> one Q0 A P2 H8 : at C1 Q1 Q2 ============================ one (par Q2 R1) A (par P2 R1) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H6 to H8.

Subgoal 4.1.3: Variables: P A R1 Q C1 P2 Q1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q A P2 H6 : forall Q Q0, at C1 Q Q0 -> one Q0 A P2 H8 : at C1 Q1 Q2 H9 : one Q2 A P2 ============================ one (par Q2 R1) A (par P2 R1) Subgoal 4.2 is: (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 4.2: Variables: P A R1 Q C1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one R1 A Q2 ============================ (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> one Q0 A (par Q Q2)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 4.2: Variables: P A R1 Q C1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one R1 A Q2 ============================ (exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par Q Q2) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 4.2: Variables: P A R1 Q C1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one R1 A Q2 ============================ exists CC, at CC P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par_l C1 Q2.

Subgoal 4.2: Variables: P A R1 Q C1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one R1 A Q2 ============================ at (par_l C1 Q2) P (par Q Q2) /\ (forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at (par_l C1 Q2) Q1 RR)) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <split.

Subgoal 4.2.1: Variables: P A R1 Q C1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one R1 A Q2 ============================ at (par_l C1 Q2) P (par Q Q2) Subgoal 4.2.2 is: forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at (par_l C1 Q2) Q1 RR) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 4.2.2: Variables: P A R1 Q C1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one R1 A Q2 ============================ forall Q1 Q0, at (par_l C1 R1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at (par_l C1 Q2) Q1 RR) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 4.2.2: Variables: P A R1 Q C1 Q2 Q1 Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one R1 A Q2 H5 : at (par_l C1 R1) Q1 Q0 ============================ exists RR, one Q0 A RR /\ at (par_l C1 Q2) Q1 RR Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H5.

Subgoal 4.2.2: Variables: P A R1 Q C1 Q2 Q1 Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one R1 A Q2 H6 : at C1 Q1 Q3 ============================ exists RR, one (par Q3 R1) A RR /\ at (par_l C1 Q2) Q1 RR Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par Q3 Q2.

Subgoal 4.2.2: Variables: P A R1 Q C1 Q2 Q1 Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one R1 A Q2 H6 : at C1 Q1 Q3 ============================ one (par Q3 R1) A (par Q3 Q2) /\ at (par_l C1 Q2) Q1 (par Q3 Q2) Subgoal 4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 4.3: Variables: P R1 Q C1 X Q2 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply IH to H3 H4.

Subgoal 4.3: Variables: P R1 Q C1 X Q2 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H6 : (exists CC, at CC P P2 /\ (forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP P2 /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C1 Q Q0 -> one Q0 (up X) P2) ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H6.

Subgoal 4.3.1: Variables: P R1 Q C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : at CC P P2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC Q RR) ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.3.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 4.3.1: Variables: P R1 Q C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : at CC P P2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC Q RR) ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) Subgoal 4.3.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 4.3.1: Variables: P R1 Q C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : at CC P P2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC Q RR) ============================ exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR)) Subgoal 4.3.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par_l CC Q2.

Subgoal 4.3.1: Variables: P R1 Q C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : at CC P P2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC Q RR) ============================ at (par_l CC Q2) P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_l CC Q2) Q RR)) Subgoal 4.3.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <split.

Subgoal 4.3.1.1: Variables: P R1 Q C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : at CC P P2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC Q RR) ============================ at (par_l CC Q2) P (par P2 Q2) Subgoal 4.3.1.2 is: forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_l CC Q2) Q RR) Subgoal 4.3.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 4.3.1.2: Variables: P R1 Q C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : at CC P P2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC Q RR) ============================ forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_l CC Q2) Q RR) Subgoal 4.3.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 4.3.1.2: Variables: P R1 Q C1 X Q2 P2 CC Q1 Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : at CC P P2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC Q RR) H9 : at (par_l C1 R1) Q1 Q0 ============================ exists RR, one Q0 tau RR /\ at (par_l CC Q2) Q1 RR Subgoal 4.3.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H9.

Subgoal 4.3.1.2: Variables: P R1 Q C1 X Q2 P2 CC Q1 Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : at CC P P2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC Q RR) H10 : at C1 Q1 Q3 ============================ exists RR, one (par Q3 R1) tau RR /\ at (par_l CC Q2) Q1 RR Subgoal 4.3.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H8 to H10.

Subgoal 4.3.1.2: Variables: P R1 Q C1 X Q2 P2 CC Q1 Q3 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : at CC P P2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC Q RR) H10 : at C1 Q1 Q3 H11 : one Q3 (up X) RR H12 : at CC Q1 RR ============================ exists RR, one (par Q3 R1) tau RR /\ at (par_l CC Q2) Q1 RR Subgoal 4.3.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par RR Q2.

Subgoal 4.3.1.2: Variables: P R1 Q C1 X Q2 P2 CC Q1 Q3 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : at CC P P2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC Q RR) H10 : at C1 Q1 Q3 H11 : one Q3 (up X) RR H12 : at CC Q1 RR ============================ one (par Q3 R1) tau (par RR Q2) /\ at (par_l CC Q2) Q1 (par RR Q2) Subgoal 4.3.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 4.3.2: Variables: P R1 Q C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 4.3.2: Variables: P R1 Q C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <right.

Subgoal 4.3.2: Variables: P R1 Q C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) ============================ exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR))) Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par_l CC Q2.

Subgoal 4.3.2: Variables: P R1 Q C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) ============================ exists PBP B, one P B PBP /\ at (par_l CC Q2) PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_l CC Q2) QBQ RR))) Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness PBP.

Subgoal 4.3.2: Variables: P R1 Q C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) ============================ exists B, one P B PBP /\ at (par_l CC Q2) PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_l CC Q2) QBQ RR))) Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness B.

Subgoal 4.3.2: Variables: P R1 Q C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) ============================ one P B PBP /\ at (par_l CC Q2) PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_l CC Q2) QBQ RR))) Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <split.

Subgoal 4.3.2.1: Variables: P R1 Q C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) ============================ one P B PBP Subgoal 4.3.2.2 is: at (par_l CC Q2) PBP (par P2 Q2) Subgoal 4.3.2.3 is: forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_l CC Q2) QBQ RR)) Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 4.3.2.2: Variables: P R1 Q C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) ============================ at (par_l CC Q2) PBP (par P2 Q2) Subgoal 4.3.2.3 is: forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_l CC Q2) QBQ RR)) Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 4.3.2.3: Variables: P R1 Q C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) ============================ forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_l CC Q2) QBQ RR)) Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 4.3.2.3: Variables: P R1 Q C1 X Q2 P2 CC PBP B Q1 QBQ Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) H10 : one Q1 B QBQ H11 : at (par_l C1 R1) Q1 Q0 ============================ exists RR, one Q0 tau RR /\ at (par_l CC Q2) QBQ RR Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H11.

Subgoal 4.3.2.3: Variables: P R1 Q C1 X Q2 P2 CC PBP B Q1 QBQ Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) H10 : one Q1 B QBQ H12 : at C1 Q1 Q3 ============================ exists RR, one (par Q3 R1) tau RR /\ at (par_l CC Q2) QBQ RR Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H9 to H10.

Subgoal 4.3.2.3: Variables: P R1 Q C1 X Q2 P2 CC PBP B Q1 QBQ Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) H10 : one Q1 B QBQ H12 : at C1 Q1 Q3 H13 : forall Q0, at C1 Q1 Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR) ============================ exists RR, one (par Q3 R1) tau RR /\ at (par_l CC Q2) QBQ RR Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H13 to H12.

Subgoal 4.3.2.3: Variables: P R1 Q C1 X Q2 P2 CC PBP B Q1 QBQ Q3 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) H10 : one Q1 B QBQ H12 : at C1 Q1 Q3 H13 : forall Q0, at C1 Q1 Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR) H14 : one Q3 (up X) RR H15 : at CC QBQ RR ============================ exists RR, one (par Q3 R1) tau RR /\ at (par_l CC Q2) QBQ RR Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par RR Q2.

Subgoal 4.3.2.3: Variables: P R1 Q C1 X Q2 P2 CC PBP B Q1 QBQ Q3 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) H10 : one Q1 B QBQ H12 : at C1 Q1 Q3 H13 : forall Q0, at C1 Q1 Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR) H14 : one Q3 (up X) RR H15 : at CC QBQ RR ============================ one (par Q3 R1) tau (par RR Q2) /\ at (par_l CC Q2) QBQ (par RR Q2) Subgoal 4.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 4.3.3: Variables: P R1 Q C1 X Q2 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : forall Q Q0, at C1 Q Q0 -> one Q0 (up X) P2 ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <right.

Subgoal 4.3.3: Variables: P R1 Q C1 X Q2 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : forall Q Q0, at C1 Q Q0 -> one Q0 (up X) P2 ============================ forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 4.3.3: Variables: P R1 Q C1 X Q2 P2 Q1 Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : forall Q Q0, at C1 Q Q0 -> one Q0 (up X) P2 H8 : at (par_l C1 R1) Q1 Q0 ============================ one Q0 tau (par P2 Q2) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H8.

Subgoal 4.3.3: Variables: P R1 Q C1 X Q2 P2 Q1 Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : forall Q Q0, at C1 Q Q0 -> one Q0 (up X) P2 H9 : at C1 Q1 Q3 ============================ one (par Q3 R1) tau (par P2 Q2) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H7 to H9.

Subgoal 4.3.3: Variables: P R1 Q C1 X Q2 P2 Q1 Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (up X) P2 H5 : one R1 (dn X) Q2 H7 : forall Q Q0, at C1 Q Q0 -> one Q0 (up X) P2 H9 : at C1 Q1 Q3 H10 : one Q3 (up X) P2 ============================ one (par Q3 R1) tau (par P2 Q2) Subgoal 4.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 4.4: Variables: P R1 Q C1 X Q2 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply IH to H3 H4.

Subgoal 4.4: Variables: P R1 Q C1 X Q2 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H6 : (exists CC, at CC P P2 /\ (forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP P2 /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C1 Q Q0 -> one Q0 (dn X) P2) ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H6.

Subgoal 4.4.1: Variables: P R1 Q C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : at CC P P2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC Q RR) ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 4.4.1: Variables: P R1 Q C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : at CC P P2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC Q RR) ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) Subgoal 4.4.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 4.4.1: Variables: P R1 Q C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : at CC P P2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC Q RR) ============================ exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR)) Subgoal 4.4.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par_l CC Q2.

Subgoal 4.4.1: Variables: P R1 Q C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : at CC P P2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC Q RR) ============================ at (par_l CC Q2) P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_l CC Q2) Q RR)) Subgoal 4.4.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <split.

Subgoal 4.4.1.1: Variables: P R1 Q C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : at CC P P2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC Q RR) ============================ at (par_l CC Q2) P (par P2 Q2) Subgoal 4.4.1.2 is: forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_l CC Q2) Q RR) Subgoal 4.4.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 4.4.1.2: Variables: P R1 Q C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : at CC P P2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC Q RR) ============================ forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_l CC Q2) Q RR) Subgoal 4.4.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 4.4.1.2: Variables: P R1 Q C1 X Q2 P2 CC Q1 Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : at CC P P2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC Q RR) H9 : at (par_l C1 R1) Q1 Q0 ============================ exists RR, one Q0 tau RR /\ at (par_l CC Q2) Q1 RR Subgoal 4.4.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H9.

Subgoal 4.4.1.2: Variables: P R1 Q C1 X Q2 P2 CC Q1 Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : at CC P P2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC Q RR) H10 : at C1 Q1 Q3 ============================ exists RR, one (par Q3 R1) tau RR /\ at (par_l CC Q2) Q1 RR Subgoal 4.4.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H8 to H10.

Subgoal 4.4.1.2: Variables: P R1 Q C1 X Q2 P2 CC Q1 Q3 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : at CC P P2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC Q RR) H10 : at C1 Q1 Q3 H11 : one Q3 (dn X) RR H12 : at CC Q1 RR ============================ exists RR, one (par Q3 R1) tau RR /\ at (par_l CC Q2) Q1 RR Subgoal 4.4.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par RR Q2.

Subgoal 4.4.1.2: Variables: P R1 Q C1 X Q2 P2 CC Q1 Q3 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : at CC P P2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC Q RR) H10 : at C1 Q1 Q3 H11 : one Q3 (dn X) RR H12 : at CC Q1 RR ============================ one (par Q3 R1) tau (par RR Q2) /\ at (par_l CC Q2) Q1 (par RR Q2) Subgoal 4.4.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 4.4.2: Variables: P R1 Q C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 4.4.2: Variables: P R1 Q C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <right.

Subgoal 4.4.2: Variables: P R1 Q C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) ============================ exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR))) Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par_l CC Q2.

Subgoal 4.4.2: Variables: P R1 Q C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) ============================ exists PBP B, one P B PBP /\ at (par_l CC Q2) PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_l CC Q2) QBQ RR))) Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness PBP.

Subgoal 4.4.2: Variables: P R1 Q C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) ============================ exists B, one P B PBP /\ at (par_l CC Q2) PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_l CC Q2) QBQ RR))) Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness B.

Subgoal 4.4.2: Variables: P R1 Q C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) ============================ one P B PBP /\ at (par_l CC Q2) PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_l CC Q2) QBQ RR))) Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <split.

Subgoal 4.4.2.1: Variables: P R1 Q C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) ============================ one P B PBP Subgoal 4.4.2.2 is: at (par_l CC Q2) PBP (par P2 Q2) Subgoal 4.4.2.3 is: forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_l CC Q2) QBQ RR)) Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 4.4.2.2: Variables: P R1 Q C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) ============================ at (par_l CC Q2) PBP (par P2 Q2) Subgoal 4.4.2.3 is: forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_l CC Q2) QBQ RR)) Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 4.4.2.3: Variables: P R1 Q C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) ============================ forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_l CC Q2) QBQ RR)) Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 4.4.2.3: Variables: P R1 Q C1 X Q2 P2 CC PBP B Q1 QBQ Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) H10 : one Q1 B QBQ H11 : at (par_l C1 R1) Q1 Q0 ============================ exists RR, one Q0 tau RR /\ at (par_l CC Q2) QBQ RR Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H11.

Subgoal 4.4.2.3: Variables: P R1 Q C1 X Q2 P2 CC PBP B Q1 QBQ Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) H10 : one Q1 B QBQ H12 : at C1 Q1 Q3 ============================ exists RR, one (par Q3 R1) tau RR /\ at (par_l CC Q2) QBQ RR Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H9 to H10.

Subgoal 4.4.2.3: Variables: P R1 Q C1 X Q2 P2 CC PBP B Q1 QBQ Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) H10 : one Q1 B QBQ H12 : at C1 Q1 Q3 H13 : forall Q0, at C1 Q1 Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR) ============================ exists RR, one (par Q3 R1) tau RR /\ at (par_l CC Q2) QBQ RR Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H13 to H12.

Subgoal 4.4.2.3: Variables: P R1 Q C1 X Q2 P2 CC PBP B Q1 QBQ Q3 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) H10 : one Q1 B QBQ H12 : at C1 Q1 Q3 H13 : forall Q0, at C1 Q1 Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR) H14 : one Q3 (dn X) RR H15 : at CC QBQ RR ============================ exists RR, one (par Q3 R1) tau RR /\ at (par_l CC Q2) QBQ RR Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par RR Q2.

Subgoal 4.4.2.3: Variables: P R1 Q C1 X Q2 P2 CC PBP B Q1 QBQ Q3 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : one P B PBP H8 : at CC PBP P2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) H10 : one Q1 B QBQ H12 : at C1 Q1 Q3 H13 : forall Q0, at C1 Q1 Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR) H14 : one Q3 (dn X) RR H15 : at CC QBQ RR ============================ one (par Q3 R1) tau (par RR Q2) /\ at (par_l CC Q2) QBQ (par RR Q2) Subgoal 4.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 4.4.3: Variables: P R1 Q C1 X Q2 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : forall Q Q0, at C1 Q Q0 -> one Q0 (dn X) P2 ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_l C1 R1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <right.

Subgoal 4.4.3: Variables: P R1 Q C1 X Q2 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : forall Q Q0, at C1 Q Q0 -> one Q0 (dn X) P2 ============================ forall Q Q0, at (par_l C1 R1) Q Q0 -> one Q0 tau (par P2 Q2) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 4.4.3: Variables: P R1 Q C1 X Q2 P2 Q1 Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : forall Q Q0, at C1 Q Q0 -> one Q0 (dn X) P2 H8 : at (par_l C1 R1) Q1 Q0 ============================ one Q0 tau (par P2 Q2) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H8.

Subgoal 4.4.3: Variables: P R1 Q C1 X Q2 P2 Q1 Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : forall Q Q0, at C1 Q Q0 -> one Q0 (dn X) P2 H9 : at C1 Q1 Q3 ============================ one (par Q3 R1) tau (par P2 Q2) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H7 to H9.

Subgoal 4.4.3: Variables: P R1 Q C1 X Q2 P2 Q1 Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_l C1 R1) P (par Q R1) @ H3 : at C1 P Q * H4 : one Q (dn X) P2 H5 : one R1 (up X) Q2 H7 : forall Q Q0, at C1 Q Q0 -> one Q0 (dn X) P2 H9 : at C1 Q1 Q3 H10 : one Q3 (dn X) P2 ============================ one (par Q3 R1) tau (par P2 Q2) Subgoal 5 is: (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 5: Variables: P A R Q R1 C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H2 : one (par R1 Q) A R H3 : at C1 P Q * ============================ (exists CC, at CC P R /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A R) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H2.

Subgoal 5.1: Variables: P A Q R1 C1 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 A P2 ============================ (exists CC, at CC P (par P2 Q) /\ (forall Q1 Q0, at (par_r R1 C1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_r R1 C1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q1 Q0, at (par_r R1 C1) Q1 Q0 -> one Q0 A (par P2 Q)) Subgoal 5.2 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 5.1: Variables: P A Q R1 C1 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 A P2 ============================ (exists CC, at CC P (par P2 Q) /\ (forall Q1 Q0, at (par_r R1 C1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q) /\ (forall Q1 QBQ, one Q1 B QBQ -> (forall Q0, at (par_r R1 C1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) Subgoal 5.2 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 5.1: Variables: P A Q R1 C1 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 A P2 ============================ exists CC, at CC P (par P2 Q) /\ (forall Q1 Q0, at (par_r R1 C1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC Q1 RR)) Subgoal 5.2 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par_r P2 C1.

Subgoal 5.1: Variables: P A Q R1 C1 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 A P2 ============================ at (par_r P2 C1) P (par P2 Q) /\ (forall Q1 Q0, at (par_r R1 C1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at (par_r P2 C1) Q1 RR)) Subgoal 5.2 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <split.

Subgoal 5.1.1: Variables: P A Q R1 C1 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 A P2 ============================ at (par_r P2 C1) P (par P2 Q) Subgoal 5.1.2 is: forall Q1 Q0, at (par_r R1 C1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at (par_r P2 C1) Q1 RR) Subgoal 5.2 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 5.1.2: Variables: P A Q R1 C1 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 A P2 ============================ forall Q1 Q0, at (par_r R1 C1) Q1 Q0 -> (exists RR, one Q0 A RR /\ at (par_r P2 C1) Q1 RR) Subgoal 5.2 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 5.1.2: Variables: P A Q R1 C1 P2 Q1 Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 A P2 H5 : at (par_r R1 C1) Q1 Q0 ============================ exists RR, one Q0 A RR /\ at (par_r P2 C1) Q1 RR Subgoal 5.2 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H5.

Subgoal 5.1.2: Variables: P A Q R1 C1 P2 Q1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 A P2 H6 : at C1 Q1 Q2 ============================ exists RR, one (par R1 Q2) A RR /\ at (par_r P2 C1) Q1 RR Subgoal 5.2 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par P2 Q2.

Subgoal 5.1.2: Variables: P A Q R1 C1 P2 Q1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 A P2 H6 : at C1 Q1 Q2 ============================ one (par R1 Q2) A (par P2 Q2) /\ at (par_r P2 C1) Q1 (par P2 Q2) Subgoal 5.2 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 5.2: Variables: P A Q R1 C1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 ============================ (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply IH to H3 H4.

Subgoal 5.2: Variables: P A Q R1 C1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H5 : (exists CC, at CC P Q2 /\ (forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP Q2 /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C1 Q Q0 -> one Q0 A Q2) ============================ (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H5.

Subgoal 5.2.1: Variables: P A Q R1 C1 Q2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : at CC P Q2 H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.2.2 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 5.2.1: Variables: P A Q R1 C1 Q2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : at CC P Q2 H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) Subgoal 5.2.2 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 5.2.1: Variables: P A Q R1 C1 Q2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : at CC P Q2 H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR)) Subgoal 5.2.2 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par_r R1 CC.

Subgoal 5.2.1: Variables: P A Q R1 C1 Q2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : at CC P Q2 H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ at (par_r R1 CC) P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at (par_r R1 CC) Q RR)) Subgoal 5.2.2 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <split.

Subgoal 5.2.1.1: Variables: P A Q R1 C1 Q2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : at CC P Q2 H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ at (par_r R1 CC) P (par R1 Q2) Subgoal 5.2.1.2 is: forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at (par_r R1 CC) Q RR) Subgoal 5.2.2 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 5.2.1.2: Variables: P A Q R1 C1 Q2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : at CC P Q2 H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) ============================ forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at (par_r R1 CC) Q RR) Subgoal 5.2.2 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 5.2.1.2: Variables: P A Q R1 C1 Q2 CC Q1 Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : at CC P Q2 H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) H8 : at (par_r R1 C1) Q1 Q0 ============================ exists RR, one Q0 A RR /\ at (par_r R1 CC) Q1 RR Subgoal 5.2.2 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H8.

Subgoal 5.2.1.2: Variables: P A Q R1 C1 Q2 CC Q1 Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : at CC P Q2 H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) H9 : at C1 Q1 Q3 ============================ exists RR, one (par R1 Q3) A RR /\ at (par_r R1 CC) Q1 RR Subgoal 5.2.2 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H7 to H9.

Subgoal 5.2.1.2: Variables: P A Q R1 C1 Q2 CC Q1 Q3 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : at CC P Q2 H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) H9 : at C1 Q1 Q3 H10 : one Q3 A RR H11 : at CC Q1 RR ============================ exists RR, one (par R1 Q3) A RR /\ at (par_r R1 CC) Q1 RR Subgoal 5.2.2 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par R1 RR.

Subgoal 5.2.1.2: Variables: P A Q R1 C1 Q2 CC Q1 Q3 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : at CC P Q2 H7 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR) H9 : at C1 Q1 Q3 H10 : one Q3 A RR H11 : at CC Q1 RR ============================ one (par R1 Q3) A (par R1 RR) /\ at (par_r R1 CC) Q1 (par R1 RR) Subgoal 5.2.2 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 5.2.2: Variables: P A Q R1 C1 Q2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : one P B PBP H7 : at CC PBP Q2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 5.2.2: Variables: P A Q R1 C1 Q2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : one P B PBP H7 : at CC PBP Q2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <right.

Subgoal 5.2.2: Variables: P A Q R1 C1 Q2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : one P B PBP H7 : at CC PBP Q2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR))) Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par_r R1 CC.

Subgoal 5.2.2: Variables: P A Q R1 C1 Q2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : one P B PBP H7 : at CC PBP Q2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ exists PBP B, one P B PBP /\ at (par_r R1 CC) PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at (par_r R1 CC) QBQ RR))) Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness PBP.

Subgoal 5.2.2: Variables: P A Q R1 C1 Q2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : one P B PBP H7 : at CC PBP Q2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ exists B, one P B PBP /\ at (par_r R1 CC) PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at (par_r R1 CC) QBQ RR))) Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness B.

Subgoal 5.2.2: Variables: P A Q R1 C1 Q2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : one P B PBP H7 : at CC PBP Q2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ one P B PBP /\ at (par_r R1 CC) PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at (par_r R1 CC) QBQ RR))) Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <split.

Subgoal 5.2.2.1: Variables: P A Q R1 C1 Q2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : one P B PBP H7 : at CC PBP Q2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ one P B PBP Subgoal 5.2.2.2 is: at (par_r R1 CC) PBP (par R1 Q2) Subgoal 5.2.2.3 is: forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at (par_r R1 CC) QBQ RR)) Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 5.2.2.2: Variables: P A Q R1 C1 Q2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : one P B PBP H7 : at CC PBP Q2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ at (par_r R1 CC) PBP (par R1 Q2) Subgoal 5.2.2.3 is: forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at (par_r R1 CC) QBQ RR)) Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 5.2.2.3: Variables: P A Q R1 C1 Q2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : one P B PBP H7 : at CC PBP Q2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) ============================ forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at (par_r R1 CC) QBQ RR)) Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 5.2.2.3: Variables: P A Q R1 C1 Q2 CC PBP B Q1 QBQ Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : one P B PBP H7 : at CC PBP Q2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) H9 : one Q1 B QBQ H10 : at (par_r R1 C1) Q1 Q0 ============================ exists RR, one Q0 A RR /\ at (par_r R1 CC) QBQ RR Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H10.

Subgoal 5.2.2.3: Variables: P A Q R1 C1 Q2 CC PBP B Q1 QBQ Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : one P B PBP H7 : at CC PBP Q2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) H9 : one Q1 B QBQ H11 : at C1 Q1 Q3 ============================ exists RR, one (par R1 Q3) A RR /\ at (par_r R1 CC) QBQ RR Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H8 to H9.

Subgoal 5.2.2.3: Variables: P A Q R1 C1 Q2 CC PBP B Q1 QBQ Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : one P B PBP H7 : at CC PBP Q2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) H9 : one Q1 B QBQ H11 : at C1 Q1 Q3 H12 : forall Q0, at C1 Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR) ============================ exists RR, one (par R1 Q3) A RR /\ at (par_r R1 CC) QBQ RR Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H12 to H11.

Subgoal 5.2.2.3: Variables: P A Q R1 C1 Q2 CC PBP B Q1 QBQ Q3 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : one P B PBP H7 : at CC PBP Q2 H8 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)) H9 : one Q1 B QBQ H11 : at C1 Q1 Q3 H12 : forall Q0, at C1 Q1 Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR) H13 : one Q3 A RR H14 : at CC QBQ RR ============================ exists RR, one (par R1 Q3) A RR /\ at (par_r R1 CC) QBQ RR Subgoal 5.2.3 is: (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 5.2.3: Variables: P A Q R1 C1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : forall Q Q0, at C1 Q Q0 -> one Q0 A Q2 ============================ (exists CC, at CC P (par R1 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par R1 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2)) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <right.

Subgoal 5.2.3: Variables: P A Q R1 C1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : forall Q Q0, at C1 Q Q0 -> one Q0 A Q2 ============================ forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 A (par R1 Q2) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 5.2.3: Variables: P A Q R1 C1 Q2 Q1 Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : forall Q Q0, at C1 Q Q0 -> one Q0 A Q2 H7 : at (par_r R1 C1) Q1 Q0 ============================ one Q0 A (par R1 Q2) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H7.

Subgoal 5.2.3: Variables: P A Q R1 C1 Q2 Q1 Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : forall Q Q0, at C1 Q Q0 -> one Q0 A Q2 H8 : at C1 Q1 Q3 ============================ one (par R1 Q3) A (par R1 Q2) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H6 to H8.

Subgoal 5.2.3: Variables: P A Q R1 C1 Q2 Q1 Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one Q A Q2 H6 : forall Q Q0, at C1 Q Q0 -> one Q0 A Q2 H8 : at C1 Q1 Q3 H9 : one Q3 A Q2 ============================ one (par R1 Q3) A (par R1 Q2) Subgoal 5.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 5.3: Variables: P Q R1 C1 X Q2 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply IH to H3 H5.

Subgoal 5.3: Variables: P Q R1 C1 X Q2 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H6 : (exists CC, at CC P Q2 /\ (forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP Q2 /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C1 Q Q0 -> one Q0 (dn X) Q2) ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H6.

Subgoal 5.3.1: Variables: P Q R1 C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : at CC P Q2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC Q RR) ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.3.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 5.3.1: Variables: P Q R1 C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : at CC P Q2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC Q RR) ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) Subgoal 5.3.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 5.3.1: Variables: P Q R1 C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : at CC P Q2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC Q RR) ============================ exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR)) Subgoal 5.3.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par_r P2 CC.

Subgoal 5.3.1: Variables: P Q R1 C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : at CC P Q2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC Q RR) ============================ at (par_r P2 CC) P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_r P2 CC) Q RR)) Subgoal 5.3.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <split.

Subgoal 5.3.1.1: Variables: P Q R1 C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : at CC P Q2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC Q RR) ============================ at (par_r P2 CC) P (par P2 Q2) Subgoal 5.3.1.2 is: forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_r P2 CC) Q RR) Subgoal 5.3.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 5.3.1.2: Variables: P Q R1 C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : at CC P Q2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC Q RR) ============================ forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_r P2 CC) Q RR) Subgoal 5.3.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 5.3.1.2: Variables: P Q R1 C1 X Q2 P2 CC Q1 Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : at CC P Q2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC Q RR) H9 : at (par_r R1 C1) Q1 Q0 ============================ exists RR, one Q0 tau RR /\ at (par_r P2 CC) Q1 RR Subgoal 5.3.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H9.

Subgoal 5.3.1.2: Variables: P Q R1 C1 X Q2 P2 CC Q1 Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : at CC P Q2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC Q RR) H10 : at C1 Q1 Q3 ============================ exists RR, one (par R1 Q3) tau RR /\ at (par_r P2 CC) Q1 RR Subgoal 5.3.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H8 to H10.

Subgoal 5.3.1.2: Variables: P Q R1 C1 X Q2 P2 CC Q1 Q3 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : at CC P Q2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC Q RR) H10 : at C1 Q1 Q3 H11 : one Q3 (dn X) RR H12 : at CC Q1 RR ============================ exists RR, one (par R1 Q3) tau RR /\ at (par_r P2 CC) Q1 RR Subgoal 5.3.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 5.3.2: Variables: P Q R1 C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 5.3.2: Variables: P Q R1 C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) Subgoal 5.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <right.

Subgoal 5.3.2: Variables: P Q R1 C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) ============================ exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR))) Subgoal 5.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par_r P2 CC.

Subgoal 5.3.2: Variables: P Q R1 C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) ============================ exists PBP B, one P B PBP /\ at (par_r P2 CC) PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_r P2 CC) QBQ RR))) Subgoal 5.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness PBP.

Subgoal 5.3.2: Variables: P Q R1 C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) ============================ exists B, one P B PBP /\ at (par_r P2 CC) PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_r P2 CC) QBQ RR))) Subgoal 5.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness B.

Subgoal 5.3.2: Variables: P Q R1 C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) ============================ one P B PBP /\ at (par_r P2 CC) PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_r P2 CC) QBQ RR))) Subgoal 5.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <split.

Subgoal 5.3.2.1: Variables: P Q R1 C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) ============================ one P B PBP Subgoal 5.3.2.2 is: at (par_r P2 CC) PBP (par P2 Q2) Subgoal 5.3.2.3 is: forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_r P2 CC) QBQ RR)) Subgoal 5.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 5.3.2.2: Variables: P Q R1 C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) ============================ at (par_r P2 CC) PBP (par P2 Q2) Subgoal 5.3.2.3 is: forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_r P2 CC) QBQ RR)) Subgoal 5.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 5.3.2.3: Variables: P Q R1 C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) ============================ forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_r P2 CC) QBQ RR)) Subgoal 5.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 5.3.2.3: Variables: P Q R1 C1 X Q2 P2 CC PBP B Q1 QBQ Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) H10 : one Q1 B QBQ H11 : at (par_r R1 C1) Q1 Q0 ============================ exists RR, one Q0 tau RR /\ at (par_r P2 CC) QBQ RR Subgoal 5.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H11.

Subgoal 5.3.2.3: Variables: P Q R1 C1 X Q2 P2 CC PBP B Q1 QBQ Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) H10 : one Q1 B QBQ H12 : at C1 Q1 Q3 ============================ exists RR, one (par R1 Q3) tau RR /\ at (par_r P2 CC) QBQ RR Subgoal 5.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H9 to H10.

Subgoal 5.3.2.3: Variables: P Q R1 C1 X Q2 P2 CC PBP B Q1 QBQ Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) H10 : one Q1 B QBQ H12 : at C1 Q1 Q3 H13 : forall Q0, at C1 Q1 Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR) ============================ exists RR, one (par R1 Q3) tau RR /\ at (par_r P2 CC) QBQ RR Subgoal 5.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H13 to H12.

Subgoal 5.3.2.3: Variables: P Q R1 C1 X Q2 P2 CC PBP B Q1 QBQ Q3 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR)) H10 : one Q1 B QBQ H12 : at C1 Q1 Q3 H13 : forall Q0, at C1 Q1 Q0 -> (exists RR, one Q0 (dn X) RR /\ at CC QBQ RR) H14 : one Q3 (dn X) RR H15 : at CC QBQ RR ============================ exists RR, one (par R1 Q3) tau RR /\ at (par_r P2 CC) QBQ RR Subgoal 5.3.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 5.3.3: Variables: P Q R1 C1 X Q2 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : forall Q Q0, at C1 Q Q0 -> one Q0 (dn X) Q2 ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <right.

Subgoal 5.3.3: Variables: P Q R1 C1 X Q2 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : forall Q Q0, at C1 Q Q0 -> one Q0 (dn X) Q2 ============================ forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 5.3.3: Variables: P Q R1 C1 X Q2 P2 Q1 Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : forall Q Q0, at C1 Q Q0 -> one Q0 (dn X) Q2 H8 : at (par_r R1 C1) Q1 Q0 ============================ one Q0 tau (par P2 Q2) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H8.

Subgoal 5.3.3: Variables: P Q R1 C1 X Q2 P2 Q1 Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : forall Q Q0, at C1 Q Q0 -> one Q0 (dn X) Q2 H9 : at C1 Q1 Q3 ============================ one (par R1 Q3) tau (par P2 Q2) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H7 to H9.

Subgoal 5.3.3: Variables: P Q R1 C1 X Q2 P2 Q1 Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (up X) P2 H5 : one Q (dn X) Q2 H7 : forall Q Q0, at C1 Q Q0 -> one Q0 (dn X) Q2 H9 : at C1 Q1 Q3 H10 : one Q3 (dn X) Q2 ============================ one (par R1 Q3) tau (par P2 Q2) Subgoal 5.4 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 5.4: Variables: P Q R1 C1 X Q2 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply IH to H3 H5.

Subgoal 5.4: Variables: P Q R1 C1 X Q2 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H6 : (exists CC, at CC P Q2 /\ (forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP Q2 /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C1 Q Q0 -> one Q0 (up X) Q2) ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H6.

Subgoal 5.4.1: Variables: P Q R1 C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : at CC P Q2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC Q RR) ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 5.4.1: Variables: P Q R1 C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : at CC P Q2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC Q RR) ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) Subgoal 5.4.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 5.4.1: Variables: P Q R1 C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : at CC P Q2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC Q RR) ============================ exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR)) Subgoal 5.4.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par_r P2 CC.

Subgoal 5.4.1: Variables: P Q R1 C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : at CC P Q2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC Q RR) ============================ at (par_r P2 CC) P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_r P2 CC) Q RR)) Subgoal 5.4.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <split.

Subgoal 5.4.1.1: Variables: P Q R1 C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : at CC P Q2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC Q RR) ============================ at (par_r P2 CC) P (par P2 Q2) Subgoal 5.4.1.2 is: forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_r P2 CC) Q RR) Subgoal 5.4.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 5.4.1.2: Variables: P Q R1 C1 X Q2 P2 CC IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : at CC P Q2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC Q RR) ============================ forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_r P2 CC) Q RR) Subgoal 5.4.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 5.4.1.2: Variables: P Q R1 C1 X Q2 P2 CC Q1 Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : at CC P Q2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC Q RR) H9 : at (par_r R1 C1) Q1 Q0 ============================ exists RR, one Q0 tau RR /\ at (par_r P2 CC) Q1 RR Subgoal 5.4.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H9.

Subgoal 5.4.1.2: Variables: P Q R1 C1 X Q2 P2 CC Q1 Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : at CC P Q2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC Q RR) H10 : at C1 Q1 Q3 ============================ exists RR, one (par R1 Q3) tau RR /\ at (par_r P2 CC) Q1 RR Subgoal 5.4.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H8 to H10.

Subgoal 5.4.1.2: Variables: P Q R1 C1 X Q2 P2 CC Q1 Q3 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : at CC P Q2 H8 : forall Q Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC Q RR) H10 : at C1 Q1 Q3 H11 : one Q3 (up X) RR H12 : at CC Q1 RR ============================ exists RR, one (par R1 Q3) tau RR /\ at (par_r P2 CC) Q1 RR Subgoal 5.4.2 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 5.4.2: Variables: P Q R1 C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 5.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 5.4.2: Variables: P Q R1 C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) Subgoal 5.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <right.

Subgoal 5.4.2: Variables: P Q R1 C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) ============================ exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR))) Subgoal 5.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness par_r P2 CC.

Subgoal 5.4.2: Variables: P Q R1 C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) ============================ exists PBP B, one P B PBP /\ at (par_r P2 CC) PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_r P2 CC) QBQ RR))) Subgoal 5.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness PBP.

Subgoal 5.4.2: Variables: P Q R1 C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) ============================ exists B, one P B PBP /\ at (par_r P2 CC) PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_r P2 CC) QBQ RR))) Subgoal 5.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <witness B.

Subgoal 5.4.2: Variables: P Q R1 C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) ============================ one P B PBP /\ at (par_r P2 CC) PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_r P2 CC) QBQ RR))) Subgoal 5.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <split.

Subgoal 5.4.2.1: Variables: P Q R1 C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) ============================ one P B PBP Subgoal 5.4.2.2 is: at (par_r P2 CC) PBP (par P2 Q2) Subgoal 5.4.2.3 is: forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_r P2 CC) QBQ RR)) Subgoal 5.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 5.4.2.2: Variables: P Q R1 C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) ============================ at (par_r P2 CC) PBP (par P2 Q2) Subgoal 5.4.2.3 is: forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_r P2 CC) QBQ RR)) Subgoal 5.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 5.4.2.3: Variables: P Q R1 C1 X Q2 P2 CC PBP B IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) ============================ forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at (par_r P2 CC) QBQ RR)) Subgoal 5.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 5.4.2.3: Variables: P Q R1 C1 X Q2 P2 CC PBP B Q1 QBQ Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) H10 : one Q1 B QBQ H11 : at (par_r R1 C1) Q1 Q0 ============================ exists RR, one Q0 tau RR /\ at (par_r P2 CC) QBQ RR Subgoal 5.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H11.

Subgoal 5.4.2.3: Variables: P Q R1 C1 X Q2 P2 CC PBP B Q1 QBQ Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) H10 : one Q1 B QBQ H12 : at C1 Q1 Q3 ============================ exists RR, one (par R1 Q3) tau RR /\ at (par_r P2 CC) QBQ RR Subgoal 5.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H9 to H10.

Subgoal 5.4.2.3: Variables: P Q R1 C1 X Q2 P2 CC PBP B Q1 QBQ Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) H10 : one Q1 B QBQ H12 : at C1 Q1 Q3 H13 : forall Q0, at C1 Q1 Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR) ============================ exists RR, one (par R1 Q3) tau RR /\ at (par_r P2 CC) QBQ RR Subgoal 5.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H13 to H12.

Subgoal 5.4.2.3: Variables: P Q R1 C1 X Q2 P2 CC PBP B Q1 QBQ Q3 RR IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : one P B PBP H8 : at CC PBP Q2 H9 : forall Q QBQ, one Q B QBQ -> (forall Q0, at C1 Q Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR)) H10 : one Q1 B QBQ H12 : at C1 Q1 Q3 H13 : forall Q0, at C1 Q1 Q0 -> (exists RR, one Q0 (up X) RR /\ at CC QBQ RR) H14 : one Q3 (up X) RR H15 : at CC QBQ RR ============================ exists RR, one (par R1 Q3) tau RR /\ at (par_r P2 CC) QBQ RR Subgoal 5.4.3 is: (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 5.4.3: Variables: P Q R1 C1 X Q2 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : forall Q Q0, at C1 Q Q0 -> one Q0 (up X) Q2 ============================ (exists CC, at CC P (par P2 Q2) /\ (forall Q Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP (par P2 Q2) /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (par_r R1 C1) Q Q0 -> (exists RR, one Q0 tau RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2)) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <right.

Subgoal 5.4.3: Variables: P Q R1 C1 X Q2 P2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : forall Q Q0, at C1 Q Q0 -> one Q0 (up X) Q2 ============================ forall Q Q0, at (par_r R1 C1) Q Q0 -> one Q0 tau (par P2 Q2) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <intros.

Subgoal 5.4.3: Variables: P Q R1 C1 X Q2 P2 Q1 Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : forall Q Q0, at C1 Q Q0 -> one Q0 (up X) Q2 H8 : at (par_r R1 C1) Q1 Q0 ============================ one Q0 tau (par P2 Q2) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H8.

Subgoal 5.4.3: Variables: P Q R1 C1 X Q2 P2 Q1 Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : forall Q Q0, at C1 Q Q0 -> one Q0 (up X) Q2 H9 : at C1 Q1 Q3 ============================ one (par R1 Q3) tau (par P2 Q2) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <apply H7 to H9.

Subgoal 5.4.3: Variables: P Q R1 C1 X Q2 P2 Q1 Q3 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (par_r R1 C1) P (par R1 Q) @ H3 : at C1 P Q * H4 : one R1 (dn X) P2 H5 : one Q (up X) Q2 H7 : forall Q Q0, at C1 Q Q0 -> one Q0 (up X) Q2 H9 : at C1 Q1 Q3 H10 : one Q3 (up X) Q2 ============================ one (par R1 Q3) tau (par P2 Q2) Subgoal 6 is: (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <search.

Subgoal 6: Variables: P A R Q A1 C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (act_d A1 C1) P (act A1 Q) @ H2 : one (act A1 Q) A R H3 : at C1 P Q * ============================ (exists CC, at CC P R /\ (forall Q Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A1 C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A1 C1) Q Q0 -> one Q0 A R) ctx_faithful <case H2.

Subgoal 6: Variables: P A R C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (act_d A C1) P (act A R) @ H3 : at C1 P R * ============================ (exists CC, at CC P R /\ (forall Q Q0, at (act_d A C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at (act_d A C1) Q Q0 -> one Q0 A R) ctx_faithful <left.

Subgoal 6: Variables: P A R C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (act_d A C1) P (act A R) @ H3 : at C1 P R * ============================ (exists CC, at CC P R /\ (forall Q Q0, at (act_d A C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at (act_d A C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) ctx_faithful <left.

Subgoal 6: Variables: P A R C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (act_d A C1) P (act A R) @ H3 : at C1 P R * ============================ exists CC, at CC P R /\ (forall Q Q0, at (act_d A C1) Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR)) ctx_faithful <witness C1.

Subgoal 6: Variables: P A R C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (act_d A C1) P (act A R) @ H3 : at C1 P R * ============================ at C1 P R /\ (forall Q Q0, at (act_d A C1) Q Q0 -> (exists RR, one Q0 A RR /\ at C1 Q RR)) ctx_faithful <split.

Subgoal 6.1: Variables: P A R C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (act_d A C1) P (act A R) @ H3 : at C1 P R * ============================ at C1 P R Subgoal 6.2 is: forall Q Q0, at (act_d A C1) Q Q0 -> (exists RR, one Q0 A RR /\ at C1 Q RR) ctx_faithful <search.

Subgoal 6.2: Variables: P A R C1 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (act_d A C1) P (act A R) @ H3 : at C1 P R * ============================ forall Q Q0, at (act_d A C1) Q Q0 -> (exists RR, one Q0 A RR /\ at C1 Q RR) ctx_faithful <intros.

Subgoal 6.2: Variables: P A R C1 Q1 Q0 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (act_d A C1) P (act A R) @ H3 : at C1 P R * H4 : at (act_d A C1) Q1 Q0 ============================ exists RR, one Q0 A RR /\ at C1 Q1 RR ctx_faithful <case H4.

Subgoal 6.2: Variables: P A R C1 Q1 Q2 IH : forall C P P0 A R, at C P P0 * -> one P0 A R -> (exists CC, at CC P R /\ (forall Q Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC Q RR))) \/ (exists CC PBP B, one P B PBP /\ at CC PBP R /\ (forall Q QBQ, one Q B QBQ -> (forall Q0, at C Q Q0 -> (exists RR, one Q0 A RR /\ at CC QBQ RR)))) \/ (forall Q Q0, at C Q Q0 -> one Q0 A R) H1 : at (act_d A C1) P (act A R) @ H3 : at C1 P R * H5 : at C1 Q1 Q2 ============================ exists RR, one (act A Q2) A RR /\ at C1 Q1 RR ctx_faithful <search.Proof completed.

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