Welcome to Abella 2.0.3-dev
Abella < Kind name, proc type.
Abella < Type taup proc -> proc.
Abella < Type inp name -> (name -> proc) -> proc.
Abella < Type oup name -> name -> proc -> proc.
Abella < Type nu (name -> proc) -> proc.
Abella < Type repl proc -> proc.
Abella < Type plus, par proc -> proc -> proc.
Abella < Type null proc.
Abella < Kind label type.
Abella < Type tau label.
Abella < Type up, dn name -> name -> label.
Abella < Define one : proc -> label -> proc -> prop,
oneb : proc -> (name -> label) -> (name -> proc) -> prop by
one (taup P) tau P;
one (oup X Y P) (up X Y) P;
one (plus P Q) L R := one P L R;
one (plus P Q) L R := one Q L R;
one (par P Q) L (par R Q) := one P L R;
one (par P Q) L (par P R) := one Q L R;
one (repl P) L (par (repl P) R) := one P L R;
one (nu P) L (nu Q) := nabla x, one (P x) L (Q x);
one (par P Q) tau (par PP QQ) := (exists X Y R, oneb P (dn X) R /\ one Q (up X Y) QQ /\ PP = R Y) \/
(exists X Y R, one P (up X Y) PP /\ oneb Q (dn X) R /\ QQ = R Y);
one (repl P) tau (par (repl P) (par PP QQ)) := exists X Y R, one P (up X Y) PP /\ oneb P (dn X) R /\ QQ = R Y;
one (par P Q) tau (nu (y\par (PP y) (QQ y))) := (exists X, oneb P (dn X) PP /\ oneb Q (up X) QQ) \/
(exists X, oneb P (up X) PP /\ oneb Q (dn X) QQ);
one (repl P) tau (par (repl P) (nu (y\par (PP y) (QQ y)))) := exists X, oneb P (up X) PP /\ oneb P (dn X) QQ;
oneb (nu P) (up X) R := nabla y, one (P y) (up X y) (R y);
oneb (inp X P) (dn X) P;
oneb (plus P Q) L R := oneb P L R;
oneb (plus P Q) L R := oneb Q L R;
oneb (par P Q) L (x\par (R x) Q) := oneb P L R;
oneb (par P Q) L (x\par P (R x)) := oneb Q L R;
oneb (repl P) L (x\par (repl P) (R x)) := oneb P L R;
oneb (nu P) L (y\nu (x\R x y)) := nabla x, oneb (P x) L (R x).
Abella < CoDefine bisim_up_to : (proc -> proc -> proc -> proc -> prop) -> proc -> proc -> prop by
bisim_up_to Tech P Q := (forall L P1, one P L P1 ->
(exists Q1, one Q L Q1 /\
(exists P2 Q2, Tech P1 P2 Q1 Q2 /\ bisim_up_to Tech P2 Q2))) /\
(forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb Q (dn X) Q1 /\
(exists P2 Q2, forall N, Tech (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to Tech (P2 N) (Q2 N)))) /\
(forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb Q (up X) Q1 /\
(exists P2 Q2, nabla x, Tech (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to Tech (P2 x) (Q2 x)))) /\
(forall L Q1, one Q L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, Tech P1 P2 Q1 Q2 /\ bisim_up_to Tech P2 Q2))) /\
(forall X Q1, oneb Q (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, Tech (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to Tech (P2 N) (Q2 N)))) /\
(forall X Q1, oneb Q (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, Tech (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to Tech (P2 x) (Q2 x)))).
Abella < Define refl_t : proc -> proc -> proc -> proc -> prop by
refl_t P P Q Q.
Abella < CoDefine bisim : proc -> proc -> prop by
bisim P Q := bisim_up_to refl_t P Q.
Abella < Define is_sound : (proc -> proc -> proc -> proc -> prop) -> prop by
is_sound Tech := forall P Q, bisim_up_to Tech P Q -> bisim_up_to refl_t P Q.
Warning: Definition can be used to defeat stratification
(higher-order argument "Tech" occurs to the left of ->)
Abella < Theorem bisim_reflexive :
forall P, bisim_up_to refl_t P P.
============================
forall P, bisim_up_to refl_t P P
bisim_reflexive < coinduction.
CH : forall P, bisim_up_to refl_t P P +
============================
forall P, bisim_up_to refl_t P P #
bisim_reflexive < intros.
Variables: P
CH : forall P, bisim_up_to refl_t P P +
============================
bisim_up_to refl_t P P #
bisim_reflexive < unfold.
Subgoal 1:
Variables: P
CH : forall P, bisim_up_to refl_t P P +
============================
forall L P1, one P L P1 ->
(exists Q1, one P L Q1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 2 is:
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < intros.
Subgoal 1:
Variables: P L P1
CH : forall P, bisim_up_to refl_t P P +
H1 : one P L P1
============================
exists Q1, one P L Q1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)
Subgoal 2 is:
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < witness P1.
Subgoal 1:
Variables: P L P1
CH : forall P, bisim_up_to refl_t P P +
H1 : one P L P1
============================
one P L P1 /\
(exists P2 Q2, refl_t P1 P2 P1 Q2 /\ bisim_up_to refl_t P2 Q2 +)
Subgoal 2 is:
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < split.
Subgoal 1.1:
Variables: P L P1
CH : forall P, bisim_up_to refl_t P P +
H1 : one P L P1
============================
one P L P1
Subgoal 1.2 is:
exists P2 Q2, refl_t P1 P2 P1 Q2 /\ bisim_up_to refl_t P2 Q2 +
Subgoal 2 is:
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < search.
Subgoal 1.2:
Variables: P L P1
CH : forall P, bisim_up_to refl_t P P +
H1 : one P L P1
============================
exists P2 Q2, refl_t P1 P2 P1 Q2 /\ bisim_up_to refl_t P2 Q2 +
Subgoal 2 is:
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < witness P1.
Subgoal 1.2:
Variables: P L P1
CH : forall P, bisim_up_to refl_t P P +
H1 : one P L P1
============================
exists Q2, refl_t P1 P1 P1 Q2 /\ bisim_up_to refl_t P1 Q2 +
Subgoal 2 is:
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < witness P1.
Subgoal 1.2:
Variables: P L P1
CH : forall P, bisim_up_to refl_t P P +
H1 : one P L P1
============================
refl_t P1 P1 P1 P1 /\ bisim_up_to refl_t P1 P1 +
Subgoal 2 is:
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < split.
Subgoal 1.2.1:
Variables: P L P1
CH : forall P, bisim_up_to refl_t P P +
H1 : one P L P1
============================
refl_t P1 P1 P1 P1
Subgoal 1.2.2 is:
bisim_up_to refl_t P1 P1 +
Subgoal 2 is:
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < search.
Subgoal 1.2.2:
Variables: P L P1
CH : forall P, bisim_up_to refl_t P P +
H1 : one P L P1
============================
bisim_up_to refl_t P1 P1 +
Subgoal 2 is:
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < backchain CH.
Subgoal 2:
Variables: P
CH : forall P, bisim_up_to refl_t P P +
============================
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < intros.
Subgoal 2:
Variables: P X P1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (dn X) P1
============================
exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +)
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < witness P1.
Subgoal 2:
Variables: P X P1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (dn X) P1
============================
oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (P1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +)
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < split.
Subgoal 2.1:
Variables: P X P1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (dn X) P1
============================
oneb P (dn X) P1
Subgoal 2.2 is:
exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (P1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < search.
Subgoal 2.2:
Variables: P X P1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (dn X) P1
============================
exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (P1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < witness P1.
Subgoal 2.2:
Variables: P X P1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (dn X) P1
============================
exists Q2, forall N, refl_t (P1 N) (P1 N) (P1 N) (Q2 N) /\
bisim_up_to refl_t (P1 N) (Q2 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < witness P1.
Subgoal 2.2:
Variables: P X P1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (dn X) P1
============================
forall N, refl_t (P1 N) (P1 N) (P1 N) (P1 N) /\
bisim_up_to refl_t (P1 N) (P1 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < intros.
Subgoal 2.2:
Variables: P X P1 N
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (dn X) P1
============================
refl_t (P1 N) (P1 N) (P1 N) (P1 N) /\ bisim_up_to refl_t (P1 N) (P1 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < split.
Subgoal 2.2.1:
Variables: P X P1 N
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (dn X) P1
============================
refl_t (P1 N) (P1 N) (P1 N) (P1 N)
Subgoal 2.2.2 is:
bisim_up_to refl_t (P1 N) (P1 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < search.
Subgoal 2.2.2:
Variables: P X P1 N
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (dn X) P1
============================
bisim_up_to refl_t (P1 N) (P1 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < backchain CH.
Subgoal 3:
Variables: P
CH : forall P, bisim_up_to refl_t P P +
============================
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < intros.
Subgoal 3:
Variables: P X P1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (up X) P1
============================
exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +)
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < witness P1.
Subgoal 3:
Variables: P X P1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (up X) P1
============================
oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (P1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +)
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < split.
Subgoal 3.1:
Variables: P X P1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (up X) P1
============================
oneb P (up X) P1
Subgoal 3.2 is:
exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (P1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < search.
Subgoal 3.2:
Variables: P X P1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (up X) P1
============================
exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (P1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < witness P1.
Subgoal 3.2:
Variables: P X P1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (up X) P1
============================
exists Q2, nabla x, refl_t (P1 x) (P1 x) (P1 x) (Q2 x) /\
bisim_up_to refl_t (P1 x) (Q2 x) +
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < witness P1.
Subgoal 3.2:
Variables: P X P1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (up X) P1
============================
nabla x, refl_t (P1 x) (P1 x) (P1 x) (P1 x) /\
bisim_up_to refl_t (P1 x) (P1 x) +
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < intros.
Subgoal 3.2:
Variables: P X P1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (up X) P1
============================
refl_t (P1 n1) (P1 n1) (P1 n1) (P1 n1) /\
bisim_up_to refl_t (P1 n1) (P1 n1) +
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < split.
Subgoal 3.2.1:
Variables: P X P1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (up X) P1
============================
refl_t (P1 n1) (P1 n1) (P1 n1) (P1 n1)
Subgoal 3.2.2 is:
bisim_up_to refl_t (P1 n1) (P1 n1) +
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < search.
Subgoal 3.2.2:
Variables: P X P1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (up X) P1
============================
bisim_up_to refl_t (P1 n1) (P1 n1) +
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < backchain CH.
Subgoal 4:
Variables: P
CH : forall P, bisim_up_to refl_t P P +
============================
forall L Q1, one P L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < intros.
Subgoal 4:
Variables: P L Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : one P L Q1
============================
exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < witness Q1.
Subgoal 4:
Variables: P L Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : one P L Q1
============================
one P L Q1 /\
(exists P2 Q2, refl_t Q1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < split.
Subgoal 4.1:
Variables: P L Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : one P L Q1
============================
one P L Q1
Subgoal 4.2 is:
exists P2 Q2, refl_t Q1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < search.
Subgoal 4.2:
Variables: P L Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : one P L Q1
============================
exists P2 Q2, refl_t Q1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < witness Q1.
Subgoal 4.2:
Variables: P L Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : one P L Q1
============================
exists Q2, refl_t Q1 Q1 Q1 Q2 /\ bisim_up_to refl_t Q1 Q2 +
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < witness Q1.
Subgoal 4.2:
Variables: P L Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : one P L Q1
============================
refl_t Q1 Q1 Q1 Q1 /\ bisim_up_to refl_t Q1 Q1 +
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < intros.
Subgoal 4.2:
Variables: P L Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : one P L Q1
============================
refl_t Q1 Q1 Q1 Q1 /\ bisim_up_to refl_t Q1 Q1 +
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < split.
Subgoal 4.2.1:
Variables: P L Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : one P L Q1
============================
refl_t Q1 Q1 Q1 Q1
Subgoal 4.2.2 is:
bisim_up_to refl_t Q1 Q1 +
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < search.
Subgoal 4.2.2:
Variables: P L Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : one P L Q1
============================
bisim_up_to refl_t Q1 Q1 +
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < backchain CH.
Subgoal 5:
Variables: P
CH : forall P, bisim_up_to refl_t P P +
============================
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < intros.
Subgoal 5:
Variables: P X Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (dn X) Q1
============================
exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +)
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < witness Q1.
Subgoal 5:
Variables: P X Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (dn X) Q1
============================
oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (Q1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +)
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < split.
Subgoal 5.1:
Variables: P X Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (dn X) Q1
============================
oneb P (dn X) Q1
Subgoal 5.2 is:
exists P2 Q2, forall N, refl_t (Q1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < search.
Subgoal 5.2:
Variables: P X Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (dn X) Q1
============================
exists P2 Q2, forall N, refl_t (Q1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < witness Q1.
Subgoal 5.2:
Variables: P X Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (dn X) Q1
============================
exists Q2, forall N, refl_t (Q1 N) (Q1 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (Q1 N) (Q2 N) +
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < witness Q1.
Subgoal 5.2:
Variables: P X Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (dn X) Q1
============================
forall N, refl_t (Q1 N) (Q1 N) (Q1 N) (Q1 N) /\
bisim_up_to refl_t (Q1 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < intros.
Subgoal 5.2:
Variables: P X Q1 N
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (dn X) Q1
============================
refl_t (Q1 N) (Q1 N) (Q1 N) (Q1 N) /\ bisim_up_to refl_t (Q1 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < split.
Subgoal 5.2.1:
Variables: P X Q1 N
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (dn X) Q1
============================
refl_t (Q1 N) (Q1 N) (Q1 N) (Q1 N)
Subgoal 5.2.2 is:
bisim_up_to refl_t (Q1 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < search.
Subgoal 5.2.2:
Variables: P X Q1 N
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (dn X) Q1
============================
bisim_up_to refl_t (Q1 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < backchain CH.
Subgoal 6:
Variables: P
CH : forall P, bisim_up_to refl_t P P +
============================
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_reflexive < intros.
Subgoal 6:
Variables: P X Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (up X) Q1
============================
exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +)
bisim_reflexive < witness Q1.
Subgoal 6:
Variables: P X Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (up X) Q1
============================
oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (Q1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +)
bisim_reflexive < split.
Subgoal 6.1:
Variables: P X Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (up X) Q1
============================
oneb P (up X) Q1
Subgoal 6.2 is:
exists P2 Q2, nabla x, refl_t (Q1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +
bisim_reflexive < search.
Subgoal 6.2:
Variables: P X Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (up X) Q1
============================
exists P2 Q2, nabla x, refl_t (Q1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +
bisim_reflexive < witness Q1.
Subgoal 6.2:
Variables: P X Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (up X) Q1
============================
exists Q2, nabla x, refl_t (Q1 x) (Q1 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (Q1 x) (Q2 x) +
bisim_reflexive < witness Q1.
Subgoal 6.2:
Variables: P X Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (up X) Q1
============================
nabla x, refl_t (Q1 x) (Q1 x) (Q1 x) (Q1 x) /\
bisim_up_to refl_t (Q1 x) (Q1 x) +
bisim_reflexive < intros.
Subgoal 6.2:
Variables: P X Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (up X) Q1
============================
refl_t (Q1 n1) (Q1 n1) (Q1 n1) (Q1 n1) /\
bisim_up_to refl_t (Q1 n1) (Q1 n1) +
bisim_reflexive < split.
Subgoal 6.2.1:
Variables: P X Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (up X) Q1
============================
refl_t (Q1 n1) (Q1 n1) (Q1 n1) (Q1 n1)
Subgoal 6.2.2 is:
bisim_up_to refl_t (Q1 n1) (Q1 n1) +
bisim_reflexive < search.
Subgoal 6.2.2:
Variables: P X Q1
CH : forall P, bisim_up_to refl_t P P +
H1 : oneb P (up X) Q1
============================
bisim_up_to refl_t (Q1 n1) (Q1 n1) +
bisim_reflexive < backchain CH.
Proof completed.
Abella < Theorem bisim_eq_1L :
forall P1 P2 Q, bisim_up_to refl_t P1 Q -> P1 = P2 -> bisim_up_to refl_t P2 Q.
============================
forall P1 P2 Q, bisim_up_to refl_t P1 Q -> P1 = P2 ->
bisim_up_to refl_t P2 Q
bisim_eq_1L < intros.
Variables: P1 P2 Q
H1 : bisim_up_to refl_t P1 Q
H2 : P1 = P2
============================
bisim_up_to refl_t P2 Q
bisim_eq_1L < case H2.
Variables: P2 Q
H1 : bisim_up_to refl_t P2 Q
============================
bisim_up_to refl_t P2 Q
bisim_eq_1L < search.
Proof completed.
Abella < Theorem bisim_eq_1R :
forall P1 P2 Q, bisim_up_to refl_t P2 Q -> P1 = P2 -> bisim_up_to refl_t P1 Q.
============================
forall P1 P2 Q, bisim_up_to refl_t P2 Q -> P1 = P2 ->
bisim_up_to refl_t P1 Q
bisim_eq_1R < intros.
Variables: P1 P2 Q
H1 : bisim_up_to refl_t P2 Q
H2 : P1 = P2
============================
bisim_up_to refl_t P1 Q
bisim_eq_1R < case H2.
Variables: P2 Q
H1 : bisim_up_to refl_t P2 Q
============================
bisim_up_to refl_t P2 Q
bisim_eq_1R < search.
Proof completed.
Abella < Theorem bisim_eq_2L :
forall P Q1 Q2, bisim_up_to refl_t P Q1 -> Q1 = Q2 -> bisim_up_to refl_t P Q2.
============================
forall P Q1 Q2, bisim_up_to refl_t P Q1 -> Q1 = Q2 ->
bisim_up_to refl_t P Q2
bisim_eq_2L < intros.
Variables: P Q1 Q2
H1 : bisim_up_to refl_t P Q1
H2 : Q1 = Q2
============================
bisim_up_to refl_t P Q2
bisim_eq_2L < case H2.
Variables: P Q2
H1 : bisim_up_to refl_t P Q2
============================
bisim_up_to refl_t P Q2
bisim_eq_2L < search.
Proof completed.
Abella < Theorem bisim_eq_2R :
forall P Q1 Q2, bisim_up_to refl_t P Q2 -> Q1 = Q2 -> bisim_up_to refl_t P Q1.
============================
forall P Q1 Q2, bisim_up_to refl_t P Q2 -> Q1 = Q2 ->
bisim_up_to refl_t P Q1
bisim_eq_2R < intros.
Variables: P Q1 Q2
H1 : bisim_up_to refl_t P Q2
H2 : Q1 = Q2
============================
bisim_up_to refl_t P Q1
bisim_eq_2R < case H2.
Variables: P Q2
H1 : bisim_up_to refl_t P Q2
============================
bisim_up_to refl_t P Q2
bisim_eq_2R < search.
Proof completed.
Abella < Theorem bisim_symmetric :
forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P.
============================
forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P
bisim_symmetric < coinduction.
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
============================
forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P #
bisim_symmetric < intros.
Variables: P Q
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
H1 : bisim_up_to refl_t P Q
============================
bisim_up_to refl_t Q P #
bisim_symmetric < Bis1 : case H1.
Variables: P Q
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
============================
bisim_up_to refl_t Q P #
bisim_symmetric < unfold.
Subgoal 1:
Variables: P Q
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
============================
forall L P1, one Q L P1 ->
(exists Q1, one P L Q1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 2 is:
forall X P1, oneb Q (dn X) P1 ->
(exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < intros.
Subgoal 1:
Variables: P Q L P1
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : one Q L P1
============================
exists Q1, one P L Q1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)
Subgoal 2 is:
forall X P1, oneb Q (dn X) P1 ->
(exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < apply Bis4 to H2.
Subgoal 1:
Variables: P Q L P1 P2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : one Q L P1
H3 : one P L P2
H4 : refl_t P2 P3 P1 Q3
H5 : bisim_up_to refl_t P3 Q3
============================
exists Q1, one P L Q1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)
Subgoal 2 is:
forall X P1, oneb Q (dn X) P1 ->
(exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < witness P2.
Subgoal 1:
Variables: P Q L P1 P2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : one Q L P1
H3 : one P L P2
H4 : refl_t P2 P3 P1 Q3
H5 : bisim_up_to refl_t P3 Q3
============================
one P L P2 /\
(exists P3 Q2, refl_t P1 P3 P2 Q2 /\ bisim_up_to refl_t P3 Q2 +)
Subgoal 2 is:
forall X P1, oneb Q (dn X) P1 ->
(exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < split.
Subgoal 1.1:
Variables: P Q L P1 P2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : one Q L P1
H3 : one P L P2
H4 : refl_t P2 P3 P1 Q3
H5 : bisim_up_to refl_t P3 Q3
============================
one P L P2
Subgoal 1.2 is:
exists P3 Q2, refl_t P1 P3 P2 Q2 /\ bisim_up_to refl_t P3 Q2 +
Subgoal 2 is:
forall X P1, oneb Q (dn X) P1 ->
(exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < search.
Subgoal 1.2:
Variables: P Q L P1 P2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : one Q L P1
H3 : one P L P2
H4 : refl_t P2 P3 P1 Q3
H5 : bisim_up_to refl_t P3 Q3
============================
exists P3 Q2, refl_t P1 P3 P2 Q2 /\ bisim_up_to refl_t P3 Q2 +
Subgoal 2 is:
forall X P1, oneb Q (dn X) P1 ->
(exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < witness P1.
Subgoal 1.2:
Variables: P Q L P1 P2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : one Q L P1
H3 : one P L P2
H4 : refl_t P2 P3 P1 Q3
H5 : bisim_up_to refl_t P3 Q3
============================
exists Q2, refl_t P1 P1 P2 Q2 /\ bisim_up_to refl_t P1 Q2 +
Subgoal 2 is:
forall X P1, oneb Q (dn X) P1 ->
(exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < witness P2.
Subgoal 1.2:
Variables: P Q L P1 P2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : one Q L P1
H3 : one P L P2
H4 : refl_t P2 P3 P1 Q3
H5 : bisim_up_to refl_t P3 Q3
============================
refl_t P1 P1 P2 P2 /\ bisim_up_to refl_t P1 P2 +
Subgoal 2 is:
forall X P1, oneb Q (dn X) P1 ->
(exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < case H4.
Subgoal 1.2:
Variables: P Q L P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : one Q L Q3
H3 : one P L P3
H5 : bisim_up_to refl_t P3 Q3
============================
refl_t Q3 Q3 P3 P3 /\ bisim_up_to refl_t Q3 P3 +
Subgoal 2 is:
forall X P1, oneb Q (dn X) P1 ->
(exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < split.
Subgoal 1.2.1:
Variables: P Q L P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : one Q L Q3
H3 : one P L P3
H5 : bisim_up_to refl_t P3 Q3
============================
refl_t Q3 Q3 P3 P3
Subgoal 1.2.2 is:
bisim_up_to refl_t Q3 P3 +
Subgoal 2 is:
forall X P1, oneb Q (dn X) P1 ->
(exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < search.
Subgoal 1.2.2:
Variables: P Q L P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : one Q L Q3
H3 : one P L P3
H5 : bisim_up_to refl_t P3 Q3
============================
bisim_up_to refl_t Q3 P3 +
Subgoal 2 is:
forall X P1, oneb Q (dn X) P1 ->
(exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < backchain CH.
Subgoal 2:
Variables: P Q
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
============================
forall X P1, oneb Q (dn X) P1 ->
(exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < intros.
Subgoal 2:
Variables: P Q X P1
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (dn X) P1
============================
exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +)
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < apply Bis5 to H2.
Subgoal 2:
Variables: P Q X P1 P2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (dn X) P1
H3 : oneb P (dn X) P2
H4 : forall N, refl_t (P2 N) (P3 N) (P1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
============================
exists Q1, oneb P (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +)
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < witness P2.
Subgoal 2:
Variables: P Q X P1 P2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (dn X) P1
H3 : oneb P (dn X) P2
H4 : forall N, refl_t (P2 N) (P3 N) (P1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
============================
oneb P (dn X) P2 /\
(exists P3 Q2, forall N, refl_t (P1 N) (P3 N) (P2 N) (Q2 N) /\
bisim_up_to refl_t (P3 N) (Q2 N) +)
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < split.
Subgoal 2.1:
Variables: P Q X P1 P2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (dn X) P1
H3 : oneb P (dn X) P2
H4 : forall N, refl_t (P2 N) (P3 N) (P1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
============================
oneb P (dn X) P2
Subgoal 2.2 is:
exists P3 Q2, forall N, refl_t (P1 N) (P3 N) (P2 N) (Q2 N) /\
bisim_up_to refl_t (P3 N) (Q2 N) +
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < search.
Subgoal 2.2:
Variables: P Q X P1 P2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (dn X) P1
H3 : oneb P (dn X) P2
H4 : forall N, refl_t (P2 N) (P3 N) (P1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
============================
exists P3 Q2, forall N, refl_t (P1 N) (P3 N) (P2 N) (Q2 N) /\
bisim_up_to refl_t (P3 N) (Q2 N) +
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < witness P1.
Subgoal 2.2:
Variables: P Q X P1 P2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (dn X) P1
H3 : oneb P (dn X) P2
H4 : forall N, refl_t (P2 N) (P3 N) (P1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
============================
exists Q2, forall N, refl_t (P1 N) (P1 N) (P2 N) (Q2 N) /\
bisim_up_to refl_t (P1 N) (Q2 N) +
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < witness P2.
Subgoal 2.2:
Variables: P Q X P1 P2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (dn X) P1
H3 : oneb P (dn X) P2
H4 : forall N, refl_t (P2 N) (P3 N) (P1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
============================
forall N, refl_t (P1 N) (P1 N) (P2 N) (P2 N) /\
bisim_up_to refl_t (P1 N) (P2 N) +
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < intros.
Subgoal 2.2:
Variables: P Q X P1 P2 P3 Q3 N
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (dn X) P1
H3 : oneb P (dn X) P2
H4 : forall N, refl_t (P2 N) (P3 N) (P1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
============================
refl_t (P1 N) (P1 N) (P2 N) (P2 N) /\ bisim_up_to refl_t (P1 N) (P2 N) +
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < apply H4 with N = N.
Subgoal 2.2:
Variables: P Q X P1 P2 P3 Q3 N
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (dn X) P1
H3 : oneb P (dn X) P2
H4 : forall N, refl_t (P2 N) (P3 N) (P1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H5 : refl_t (P2 N) (P3 N) (P1 N) (Q3 N)
H6 : bisim_up_to refl_t (P3 N) (Q3 N)
============================
refl_t (P1 N) (P1 N) (P2 N) (P2 N) /\ bisim_up_to refl_t (P1 N) (P2 N) +
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < case H5.
Subgoal 2.2:
Variables: P Q X P1 P2 P3 Q3 N
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (dn X) P1
H3 : oneb P (dn X) P2
H4 : forall N, refl_t (P2 N) (P3 N) (P1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : bisim_up_to refl_t (P3 N) (Q3 N)
H7 : P1 N = Q3 N
H8 : P2 N = P3 N
============================
refl_t (P1 N) (P1 N) (P2 N) (P2 N) /\ bisim_up_to refl_t (P1 N) (P2 N) +
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < split.
Subgoal 2.2.1:
Variables: P Q X P1 P2 P3 Q3 N
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (dn X) P1
H3 : oneb P (dn X) P2
H4 : forall N, refl_t (P2 N) (P3 N) (P1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : bisim_up_to refl_t (P3 N) (Q3 N)
H7 : P1 N = Q3 N
H8 : P2 N = P3 N
============================
refl_t (P1 N) (P1 N) (P2 N) (P2 N)
Subgoal 2.2.2 is:
bisim_up_to refl_t (P1 N) (P2 N) +
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < search.
Subgoal 2.2.2:
Variables: P Q X P1 P2 P3 Q3 N
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (dn X) P1
H3 : oneb P (dn X) P2
H4 : forall N, refl_t (P2 N) (P3 N) (P1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : bisim_up_to refl_t (P3 N) (Q3 N)
H7 : P1 N = Q3 N
H8 : P2 N = P3 N
============================
bisim_up_to refl_t (P1 N) (P2 N) +
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < backchain CH.
Subgoal 2.2.2:
Variables: P Q X P1 P2 P3 Q3 N
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (dn X) P1
H3 : oneb P (dn X) P2
H4 : forall N, refl_t (P2 N) (P3 N) (P1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : bisim_up_to refl_t (P3 N) (Q3 N)
H7 : P1 N = Q3 N
H8 : P2 N = P3 N
============================
bisim_up_to refl_t (P2 N) (P1 N)
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < apply bisim_eq_1R to H6 H8.
Subgoal 2.2.2:
Variables: P Q X P1 P2 P3 Q3 N
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (dn X) P1
H3 : oneb P (dn X) P2
H4 : forall N, refl_t (P2 N) (P3 N) (P1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : bisim_up_to refl_t (P3 N) (Q3 N)
H7 : P1 N = Q3 N
H8 : P2 N = P3 N
H9 : bisim_up_to refl_t (P2 N) (Q3 N)
============================
bisim_up_to refl_t (P2 N) (P1 N)
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < apply bisim_eq_2R to H9 H7.
Subgoal 2.2.2:
Variables: P Q X P1 P2 P3 Q3 N
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (dn X) P1
H3 : oneb P (dn X) P2
H4 : forall N, refl_t (P2 N) (P3 N) (P1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : bisim_up_to refl_t (P3 N) (Q3 N)
H7 : P1 N = Q3 N
H8 : P2 N = P3 N
H9 : bisim_up_to refl_t (P2 N) (Q3 N)
H10 : bisim_up_to refl_t (P2 N) (P1 N)
============================
bisim_up_to refl_t (P2 N) (P1 N)
Subgoal 3 is:
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < search.
Subgoal 3:
Variables: P Q
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
============================
forall X P1, oneb Q (up X) P1 ->
(exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < intros.
Subgoal 3:
Variables: P Q X P1
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (up X) P1
============================
exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +)
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < apply Bis6 to H2.
Subgoal 3:
Variables: P Q X P1 P2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (up X) P1
H3 : oneb P (up X) P2
H4 : refl_t (P2 n1) (P3 n1) (P1 n1) (Q3 n1)
H5 : bisim_up_to refl_t (P3 n1) (Q3 n1)
============================
exists Q1, oneb P (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +)
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < witness P2.
Subgoal 3:
Variables: P Q X P1 P2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (up X) P1
H3 : oneb P (up X) P2
H4 : refl_t (P2 n1) (P3 n1) (P1 n1) (Q3 n1)
H5 : bisim_up_to refl_t (P3 n1) (Q3 n1)
============================
oneb P (up X) P2 /\
(exists P3 Q2, nabla x, refl_t (P1 x) (P3 x) (P2 x) (Q2 x) /\
bisim_up_to refl_t (P3 x) (Q2 x) +)
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < split.
Subgoal 3.1:
Variables: P Q X P1 P2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (up X) P1
H3 : oneb P (up X) P2
H4 : refl_t (P2 n1) (P3 n1) (P1 n1) (Q3 n1)
H5 : bisim_up_to refl_t (P3 n1) (Q3 n1)
============================
oneb P (up X) P2
Subgoal 3.2 is:
exists P3 Q2, nabla x, refl_t (P1 x) (P3 x) (P2 x) (Q2 x) /\
bisim_up_to refl_t (P3 x) (Q2 x) +
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < search.
Subgoal 3.2:
Variables: P Q X P1 P2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (up X) P1
H3 : oneb P (up X) P2
H4 : refl_t (P2 n1) (P3 n1) (P1 n1) (Q3 n1)
H5 : bisim_up_to refl_t (P3 n1) (Q3 n1)
============================
exists P3 Q2, nabla x, refl_t (P1 x) (P3 x) (P2 x) (Q2 x) /\
bisim_up_to refl_t (P3 x) (Q2 x) +
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < witness P1.
Subgoal 3.2:
Variables: P Q X P1 P2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (up X) P1
H3 : oneb P (up X) P2
H4 : refl_t (P2 n1) (P3 n1) (P1 n1) (Q3 n1)
H5 : bisim_up_to refl_t (P3 n1) (Q3 n1)
============================
exists Q2, nabla x, refl_t (P1 x) (P1 x) (P2 x) (Q2 x) /\
bisim_up_to refl_t (P1 x) (Q2 x) +
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < witness P2.
Subgoal 3.2:
Variables: P Q X P1 P2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (up X) P1
H3 : oneb P (up X) P2
H4 : refl_t (P2 n1) (P3 n1) (P1 n1) (Q3 n1)
H5 : bisim_up_to refl_t (P3 n1) (Q3 n1)
============================
nabla x, refl_t (P1 x) (P1 x) (P2 x) (P2 x) /\
bisim_up_to refl_t (P1 x) (P2 x) +
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < intros.
Subgoal 3.2:
Variables: P Q X P1 P2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (up X) P1
H3 : oneb P (up X) P2
H4 : refl_t (P2 n1) (P3 n1) (P1 n1) (Q3 n1)
H5 : bisim_up_to refl_t (P3 n1) (Q3 n1)
============================
refl_t (P1 n1) (P1 n1) (P2 n1) (P2 n1) /\
bisim_up_to refl_t (P1 n1) (P2 n1) +
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < case H4.
Subgoal 3.2:
Variables: P Q X P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (up X) (z1\Q3 z1)
H3 : oneb P (up X) (z1\P3 z1)
H5 : bisim_up_to refl_t (P3 n1) (Q3 n1)
============================
refl_t (Q3 n1) (Q3 n1) (P3 n1) (P3 n1) /\
bisim_up_to refl_t (Q3 n1) (P3 n1) +
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < split.
Subgoal 3.2.1:
Variables: P Q X P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (up X) (z1\Q3 z1)
H3 : oneb P (up X) (z1\P3 z1)
H5 : bisim_up_to refl_t (P3 n1) (Q3 n1)
============================
refl_t (Q3 n1) (Q3 n1) (P3 n1) (P3 n1)
Subgoal 3.2.2 is:
bisim_up_to refl_t (Q3 n1) (P3 n1) +
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < search.
Subgoal 3.2.2:
Variables: P Q X P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb Q (up X) (z1\Q3 z1)
H3 : oneb P (up X) (z1\P3 z1)
H5 : bisim_up_to refl_t (P3 n1) (Q3 n1)
============================
bisim_up_to refl_t (Q3 n1) (P3 n1) +
Subgoal 4 is:
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < backchain CH.
Subgoal 4:
Variables: P Q
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
============================
forall L Q1, one P L Q1 ->
(exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < intros.
Subgoal 4:
Variables: P Q L Q1
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : one P L Q1
============================
exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < apply Bis1 to H2.
Subgoal 4:
Variables: P Q L Q1 Q2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : one P L Q1
H3 : one Q L Q2
H4 : refl_t Q1 P3 Q2 Q3
H5 : bisim_up_to refl_t P3 Q3
============================
exists P1, one Q L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < witness Q2.
Subgoal 4:
Variables: P Q L Q1 Q2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : one P L Q1
H3 : one Q L Q2
H4 : refl_t Q1 P3 Q2 Q3
H5 : bisim_up_to refl_t P3 Q3
============================
one Q L Q2 /\
(exists P2 Q3, refl_t Q2 P2 Q1 Q3 /\ bisim_up_to refl_t P2 Q3 +)
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < split.
Subgoal 4.1:
Variables: P Q L Q1 Q2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : one P L Q1
H3 : one Q L Q2
H4 : refl_t Q1 P3 Q2 Q3
H5 : bisim_up_to refl_t P3 Q3
============================
one Q L Q2
Subgoal 4.2 is:
exists P2 Q3, refl_t Q2 P2 Q1 Q3 /\ bisim_up_to refl_t P2 Q3 +
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < search.
Subgoal 4.2:
Variables: P Q L Q1 Q2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : one P L Q1
H3 : one Q L Q2
H4 : refl_t Q1 P3 Q2 Q3
H5 : bisim_up_to refl_t P3 Q3
============================
exists P2 Q3, refl_t Q2 P2 Q1 Q3 /\ bisim_up_to refl_t P2 Q3 +
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < witness Q2.
Subgoal 4.2:
Variables: P Q L Q1 Q2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : one P L Q1
H3 : one Q L Q2
H4 : refl_t Q1 P3 Q2 Q3
H5 : bisim_up_to refl_t P3 Q3
============================
exists Q3, refl_t Q2 Q2 Q1 Q3 /\ bisim_up_to refl_t Q2 Q3 +
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < witness Q1.
Subgoal 4.2:
Variables: P Q L Q1 Q2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : one P L Q1
H3 : one Q L Q2
H4 : refl_t Q1 P3 Q2 Q3
H5 : bisim_up_to refl_t P3 Q3
============================
refl_t Q2 Q2 Q1 Q1 /\ bisim_up_to refl_t Q2 Q1 +
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < case H4.
Subgoal 4.2:
Variables: P Q L P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : one P L P3
H3 : one Q L Q3
H5 : bisim_up_to refl_t P3 Q3
============================
refl_t Q3 Q3 P3 P3 /\ bisim_up_to refl_t Q3 P3 +
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < split.
Subgoal 4.2.1:
Variables: P Q L P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : one P L P3
H3 : one Q L Q3
H5 : bisim_up_to refl_t P3 Q3
============================
refl_t Q3 Q3 P3 P3
Subgoal 4.2.2 is:
bisim_up_to refl_t Q3 P3 +
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < search.
Subgoal 4.2.2:
Variables: P Q L P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : one P L P3
H3 : one Q L Q3
H5 : bisim_up_to refl_t P3 Q3
============================
bisim_up_to refl_t Q3 P3 +
Subgoal 5 is:
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < backchain CH.
Subgoal 5:
Variables: P Q
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
============================
forall X Q1, oneb P (dn X) Q1 ->
(exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < intros.
Subgoal 5:
Variables: P Q X Q1
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (dn X) Q1
============================
exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +)
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < apply Bis2 to H2.
Subgoal 5:
Variables: P Q X Q1 Q2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (dn X) Q1
H3 : oneb Q (dn X) Q2
H4 : forall N, refl_t (Q1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
============================
exists P1, oneb Q (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +)
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < witness Q2.
Subgoal 5:
Variables: P Q X Q1 Q2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (dn X) Q1
H3 : oneb Q (dn X) Q2
H4 : forall N, refl_t (Q1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
============================
oneb Q (dn X) Q2 /\
(exists P2 Q3, forall N, refl_t (Q2 N) (P2 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (P2 N) (Q3 N) +)
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < split.
Subgoal 5.1:
Variables: P Q X Q1 Q2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (dn X) Q1
H3 : oneb Q (dn X) Q2
H4 : forall N, refl_t (Q1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
============================
oneb Q (dn X) Q2
Subgoal 5.2 is:
exists P2 Q3, forall N, refl_t (Q2 N) (P2 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (P2 N) (Q3 N) +
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < search.
Subgoal 5.2:
Variables: P Q X Q1 Q2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (dn X) Q1
H3 : oneb Q (dn X) Q2
H4 : forall N, refl_t (Q1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
============================
exists P2 Q3, forall N, refl_t (Q2 N) (P2 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (P2 N) (Q3 N) +
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < witness Q2.
Subgoal 5.2:
Variables: P Q X Q1 Q2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (dn X) Q1
H3 : oneb Q (dn X) Q2
H4 : forall N, refl_t (Q1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
============================
exists Q3, forall N, refl_t (Q2 N) (Q2 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (Q2 N) (Q3 N) +
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < witness Q1.
Subgoal 5.2:
Variables: P Q X Q1 Q2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (dn X) Q1
H3 : oneb Q (dn X) Q2
H4 : forall N, refl_t (Q1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
============================
forall N, refl_t (Q2 N) (Q2 N) (Q1 N) (Q1 N) /\
bisim_up_to refl_t (Q2 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < intros.
Subgoal 5.2:
Variables: P Q X Q1 Q2 P3 Q3 N
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (dn X) Q1
H3 : oneb Q (dn X) Q2
H4 : forall N, refl_t (Q1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
============================
refl_t (Q2 N) (Q2 N) (Q1 N) (Q1 N) /\ bisim_up_to refl_t (Q2 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < apply H4 with N = N.
Subgoal 5.2:
Variables: P Q X Q1 Q2 P3 Q3 N
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (dn X) Q1
H3 : oneb Q (dn X) Q2
H4 : forall N, refl_t (Q1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H5 : refl_t (Q1 N) (P3 N) (Q2 N) (Q3 N)
H6 : bisim_up_to refl_t (P3 N) (Q3 N)
============================
refl_t (Q2 N) (Q2 N) (Q1 N) (Q1 N) /\ bisim_up_to refl_t (Q2 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < case H5.
Subgoal 5.2:
Variables: P Q X Q1 Q2 P3 Q3 N
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (dn X) Q1
H3 : oneb Q (dn X) Q2
H4 : forall N, refl_t (Q1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : bisim_up_to refl_t (P3 N) (Q3 N)
H7 : Q2 N = Q3 N
H8 : Q1 N = P3 N
============================
refl_t (Q2 N) (Q2 N) (Q1 N) (Q1 N) /\ bisim_up_to refl_t (Q2 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < split.
Subgoal 5.2.1:
Variables: P Q X Q1 Q2 P3 Q3 N
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (dn X) Q1
H3 : oneb Q (dn X) Q2
H4 : forall N, refl_t (Q1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : bisim_up_to refl_t (P3 N) (Q3 N)
H7 : Q2 N = Q3 N
H8 : Q1 N = P3 N
============================
refl_t (Q2 N) (Q2 N) (Q1 N) (Q1 N)
Subgoal 5.2.2 is:
bisim_up_to refl_t (Q2 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < search.
Subgoal 5.2.2:
Variables: P Q X Q1 Q2 P3 Q3 N
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (dn X) Q1
H3 : oneb Q (dn X) Q2
H4 : forall N, refl_t (Q1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : bisim_up_to refl_t (P3 N) (Q3 N)
H7 : Q2 N = Q3 N
H8 : Q1 N = P3 N
============================
bisim_up_to refl_t (Q2 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < backchain CH.
Subgoal 5.2.2:
Variables: P Q X Q1 Q2 P3 Q3 N
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (dn X) Q1
H3 : oneb Q (dn X) Q2
H4 : forall N, refl_t (Q1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : bisim_up_to refl_t (P3 N) (Q3 N)
H7 : Q2 N = Q3 N
H8 : Q1 N = P3 N
============================
bisim_up_to refl_t (Q1 N) (Q2 N)
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < apply bisim_eq_1R to H6 H8.
Subgoal 5.2.2:
Variables: P Q X Q1 Q2 P3 Q3 N
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (dn X) Q1
H3 : oneb Q (dn X) Q2
H4 : forall N, refl_t (Q1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : bisim_up_to refl_t (P3 N) (Q3 N)
H7 : Q2 N = Q3 N
H8 : Q1 N = P3 N
H9 : bisim_up_to refl_t (Q1 N) (Q3 N)
============================
bisim_up_to refl_t (Q1 N) (Q2 N)
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < apply bisim_eq_2R to H9 H7.
Subgoal 5.2.2:
Variables: P Q X Q1 Q2 P3 Q3 N
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (dn X) Q1
H3 : oneb Q (dn X) Q2
H4 : forall N, refl_t (Q1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : bisim_up_to refl_t (P3 N) (Q3 N)
H7 : Q2 N = Q3 N
H8 : Q1 N = P3 N
H9 : bisim_up_to refl_t (Q1 N) (Q3 N)
H10 : bisim_up_to refl_t (Q1 N) (Q2 N)
============================
bisim_up_to refl_t (Q1 N) (Q2 N)
Subgoal 6 is:
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < search.
Subgoal 6:
Variables: P Q
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
============================
forall X Q1, oneb P (up X) Q1 ->
(exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_symmetric < intros.
Subgoal 6:
Variables: P Q X Q1
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (up X) Q1
============================
exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +)
bisim_symmetric < apply Bis3 to H2.
Subgoal 6:
Variables: P Q X Q1 Q2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (up X) Q1
H3 : oneb Q (up X) Q2
H4 : refl_t (Q1 n1) (P3 n1) (Q2 n1) (Q3 n1)
H5 : bisim_up_to refl_t (P3 n1) (Q3 n1)
============================
exists P1, oneb Q (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +)
bisim_symmetric < witness Q2.
Subgoal 6:
Variables: P Q X Q1 Q2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (up X) Q1
H3 : oneb Q (up X) Q2
H4 : refl_t (Q1 n1) (P3 n1) (Q2 n1) (Q3 n1)
H5 : bisim_up_to refl_t (P3 n1) (Q3 n1)
============================
oneb Q (up X) Q2 /\
(exists P2 Q3, nabla x, refl_t (Q2 x) (P2 x) (Q1 x) (Q3 x) /\
bisim_up_to refl_t (P2 x) (Q3 x) +)
bisim_symmetric < split.
Subgoal 6.1:
Variables: P Q X Q1 Q2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (up X) Q1
H3 : oneb Q (up X) Q2
H4 : refl_t (Q1 n1) (P3 n1) (Q2 n1) (Q3 n1)
H5 : bisim_up_to refl_t (P3 n1) (Q3 n1)
============================
oneb Q (up X) Q2
Subgoal 6.2 is:
exists P2 Q3, nabla x, refl_t (Q2 x) (P2 x) (Q1 x) (Q3 x) /\
bisim_up_to refl_t (P2 x) (Q3 x) +
bisim_symmetric < search.
Subgoal 6.2:
Variables: P Q X Q1 Q2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (up X) Q1
H3 : oneb Q (up X) Q2
H4 : refl_t (Q1 n1) (P3 n1) (Q2 n1) (Q3 n1)
H5 : bisim_up_to refl_t (P3 n1) (Q3 n1)
============================
exists P2 Q3, nabla x, refl_t (Q2 x) (P2 x) (Q1 x) (Q3 x) /\
bisim_up_to refl_t (P2 x) (Q3 x) +
bisim_symmetric < witness Q2.
Subgoal 6.2:
Variables: P Q X Q1 Q2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (up X) Q1
H3 : oneb Q (up X) Q2
H4 : refl_t (Q1 n1) (P3 n1) (Q2 n1) (Q3 n1)
H5 : bisim_up_to refl_t (P3 n1) (Q3 n1)
============================
exists Q3, nabla x, refl_t (Q2 x) (Q2 x) (Q1 x) (Q3 x) /\
bisim_up_to refl_t (Q2 x) (Q3 x) +
bisim_symmetric < witness Q1.
Subgoal 6.2:
Variables: P Q X Q1 Q2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (up X) Q1
H3 : oneb Q (up X) Q2
H4 : refl_t (Q1 n1) (P3 n1) (Q2 n1) (Q3 n1)
H5 : bisim_up_to refl_t (P3 n1) (Q3 n1)
============================
nabla x, refl_t (Q2 x) (Q2 x) (Q1 x) (Q1 x) /\
bisim_up_to refl_t (Q2 x) (Q1 x) +
bisim_symmetric < intros.
Subgoal 6.2:
Variables: P Q X Q1 Q2 P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (up X) Q1
H3 : oneb Q (up X) Q2
H4 : refl_t (Q1 n1) (P3 n1) (Q2 n1) (Q3 n1)
H5 : bisim_up_to refl_t (P3 n1) (Q3 n1)
============================
refl_t (Q2 n1) (Q2 n1) (Q1 n1) (Q1 n1) /\
bisim_up_to refl_t (Q2 n1) (Q1 n1) +
bisim_symmetric < case H4.
Subgoal 6.2:
Variables: P Q X P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (up X) (z1\P3 z1)
H3 : oneb Q (up X) (z1\Q3 z1)
H5 : bisim_up_to refl_t (P3 n1) (Q3 n1)
============================
refl_t (Q3 n1) (Q3 n1) (P3 n1) (P3 n1) /\
bisim_up_to refl_t (Q3 n1) (P3 n1) +
bisim_symmetric < split.
Subgoal 6.2.1:
Variables: P Q X P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (up X) (z1\P3 z1)
H3 : oneb Q (up X) (z1\Q3 z1)
H5 : bisim_up_to refl_t (P3 n1) (Q3 n1)
============================
refl_t (Q3 n1) (Q3 n1) (P3 n1) (P3 n1)
Subgoal 6.2.2 is:
bisim_up_to refl_t (Q3 n1) (P3 n1) +
bisim_symmetric < search.
Subgoal 6.2.2:
Variables: P Q X P3 Q3
CH : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P +
Bis1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Bis4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Bis5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Bis6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H2 : oneb P (up X) (z1\P3 z1)
H3 : oneb Q (up X) (z1\Q3 z1)
H5 : bisim_up_to refl_t (P3 n1) (Q3 n1)
============================
bisim_up_to refl_t (Q3 n1) (P3 n1) +
bisim_symmetric < backchain CH.
Proof completed.
Abella < Theorem bisim_transitive :
forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R.
============================
forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R
bisim_transitive < coinduction.
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
============================
forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R #
bisim_transitive < intros.
Variables: P Q R
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
H1 : bisim_up_to refl_t P Q
H2 : bisim_up_to refl_t Q R
============================
bisim_up_to refl_t P R #
bisim_transitive < Left1 : case H1.
Variables: P Q R
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
H2 : bisim_up_to refl_t Q R
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
============================
bisim_up_to refl_t P R #
bisim_transitive < Right1 : case H2.
Variables: P Q R
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
============================
bisim_up_to refl_t P R #
bisim_transitive < unfold.
Subgoal 1:
Variables: P Q R
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
============================
forall L P1, one P L P1 ->
(exists Q1, one R L Q1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 2 is:
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb R (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < intros.
Subgoal 1:
Variables: P Q R L P1
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one P L P1
============================
exists Q1, one R L Q1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)
Subgoal 2 is:
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb R (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < apply Left1 to H3.
Subgoal 1:
Variables: P Q R L P1 Q2 P3 Q3
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one P L P1
H4 : one Q L Q2
H5 : refl_t P1 P3 Q2 Q3
H6 : bisim_up_to refl_t P3 Q3
============================
exists Q1, one R L Q1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)
Subgoal 2 is:
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb R (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < apply Right1 to H4.
Subgoal 1:
Variables: P Q R L P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one P L P1
H4 : one Q L Q2
H5 : refl_t P1 P3 Q2 Q3
H6 : bisim_up_to refl_t P3 Q3
H7 : one R L Q1
H8 : refl_t Q2 P2 Q1 Q4
H9 : bisim_up_to refl_t P2 Q4
============================
exists Q1, one R L Q1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)
Subgoal 2 is:
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb R (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < witness Q1.
Subgoal 1:
Variables: P Q R L P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one P L P1
H4 : one Q L Q2
H5 : refl_t P1 P3 Q2 Q3
H6 : bisim_up_to refl_t P3 Q3
H7 : one R L Q1
H8 : refl_t Q2 P2 Q1 Q4
H9 : bisim_up_to refl_t P2 Q4
============================
one R L Q1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)
Subgoal 2 is:
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb R (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < split.
Subgoal 1.1:
Variables: P Q R L P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one P L P1
H4 : one Q L Q2
H5 : refl_t P1 P3 Q2 Q3
H6 : bisim_up_to refl_t P3 Q3
H7 : one R L Q1
H8 : refl_t Q2 P2 Q1 Q4
H9 : bisim_up_to refl_t P2 Q4
============================
one R L Q1
Subgoal 1.2 is:
exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +
Subgoal 2 is:
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb R (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < search.
Subgoal 1.2:
Variables: P Q R L P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one P L P1
H4 : one Q L Q2
H5 : refl_t P1 P3 Q2 Q3
H6 : bisim_up_to refl_t P3 Q3
H7 : one R L Q1
H8 : refl_t Q2 P2 Q1 Q4
H9 : bisim_up_to refl_t P2 Q4
============================
exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +
Subgoal 2 is:
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb R (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < witness P1.
Subgoal 1.2:
Variables: P Q R L P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one P L P1
H4 : one Q L Q2
H5 : refl_t P1 P3 Q2 Q3
H6 : bisim_up_to refl_t P3 Q3
H7 : one R L Q1
H8 : refl_t Q2 P2 Q1 Q4
H9 : bisim_up_to refl_t P2 Q4
============================
exists Q2, refl_t P1 P1 Q1 Q2 /\ bisim_up_to refl_t P1 Q2 +
Subgoal 2 is:
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb R (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < witness Q1.
Subgoal 1.2:
Variables: P Q R L P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one P L P1
H4 : one Q L Q2
H5 : refl_t P1 P3 Q2 Q3
H6 : bisim_up_to refl_t P3 Q3
H7 : one R L Q1
H8 : refl_t Q2 P2 Q1 Q4
H9 : bisim_up_to refl_t P2 Q4
============================
refl_t P1 P1 Q1 Q1 /\ bisim_up_to refl_t P1 Q1 +
Subgoal 2 is:
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb R (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < split.
Subgoal 1.2.1:
Variables: P Q R L P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one P L P1
H4 : one Q L Q2
H5 : refl_t P1 P3 Q2 Q3
H6 : bisim_up_to refl_t P3 Q3
H7 : one R L Q1
H8 : refl_t Q2 P2 Q1 Q4
H9 : bisim_up_to refl_t P2 Q4
============================
refl_t P1 P1 Q1 Q1
Subgoal 1.2.2 is:
bisim_up_to refl_t P1 Q1 +
Subgoal 2 is:
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb R (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < search.
Subgoal 1.2.2:
Variables: P Q R L P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one P L P1
H4 : one Q L Q2
H5 : refl_t P1 P3 Q2 Q3
H6 : bisim_up_to refl_t P3 Q3
H7 : one R L Q1
H8 : refl_t Q2 P2 Q1 Q4
H9 : bisim_up_to refl_t P2 Q4
============================
bisim_up_to refl_t P1 Q1 +
Subgoal 2 is:
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb R (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < case H5.
Subgoal 1.2.2:
Variables: P Q R L P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one P L P3
H4 : one Q L Q3
H6 : bisim_up_to refl_t P3 Q3
H7 : one R L Q1
H8 : refl_t Q3 P2 Q1 Q4
H9 : bisim_up_to refl_t P2 Q4
============================
bisim_up_to refl_t P3 Q1 +
Subgoal 2 is:
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb R (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < case H8.
Subgoal 1.2.2:
Variables: P Q R L P3 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one P L P3
H4 : one Q L P2
H6 : bisim_up_to refl_t P3 P2
H7 : one R L Q4
H9 : bisim_up_to refl_t P2 Q4
============================
bisim_up_to refl_t P3 Q4 +
Subgoal 2 is:
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb R (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < backchain CH.
Subgoal 2:
Variables: P Q R
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
============================
forall X P1, oneb P (dn X) P1 ->
(exists Q1, oneb R (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < intros.
Subgoal 2:
Variables: P Q R X P1
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (dn X) P1
============================
exists Q1, oneb R (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +)
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < apply Left2 to H3.
Subgoal 2:
Variables: P Q R X P1 Q2 P3 Q3
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (dn X) P1
H4 : oneb Q (dn X) Q2
H5 : forall N, refl_t (P1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
============================
exists Q1, oneb R (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +)
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < apply Right2 to H4.
Subgoal 2:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (dn X) P1
H4 : oneb Q (dn X) Q2
H5 : forall N, refl_t (P1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb R (dn X) Q1
H7 : forall N, refl_t (Q2 N) (P2 N) (Q1 N) (Q4 N) /\
bisim_up_to refl_t (P2 N) (Q4 N)
============================
exists Q1, oneb R (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +)
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < witness Q1.
Subgoal 2:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (dn X) P1
H4 : oneb Q (dn X) Q2
H5 : forall N, refl_t (P1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb R (dn X) Q1
H7 : forall N, refl_t (Q2 N) (P2 N) (Q1 N) (Q4 N) /\
bisim_up_to refl_t (P2 N) (Q4 N)
============================
oneb R (dn X) Q1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +)
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < split.
Subgoal 2.1:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (dn X) P1
H4 : oneb Q (dn X) Q2
H5 : forall N, refl_t (P1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb R (dn X) Q1
H7 : forall N, refl_t (Q2 N) (P2 N) (Q1 N) (Q4 N) /\
bisim_up_to refl_t (P2 N) (Q4 N)
============================
oneb R (dn X) Q1
Subgoal 2.2 is:
exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < search.
Subgoal 2.2:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (dn X) P1
H4 : oneb Q (dn X) Q2
H5 : forall N, refl_t (P1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb R (dn X) Q1
H7 : forall N, refl_t (Q2 N) (P2 N) (Q1 N) (Q4 N) /\
bisim_up_to refl_t (P2 N) (Q4 N)
============================
exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < witness P1.
Subgoal 2.2:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (dn X) P1
H4 : oneb Q (dn X) Q2
H5 : forall N, refl_t (P1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb R (dn X) Q1
H7 : forall N, refl_t (Q2 N) (P2 N) (Q1 N) (Q4 N) /\
bisim_up_to refl_t (P2 N) (Q4 N)
============================
exists Q2, forall N, refl_t (P1 N) (P1 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P1 N) (Q2 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < witness Q1.
Subgoal 2.2:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (dn X) P1
H4 : oneb Q (dn X) Q2
H5 : forall N, refl_t (P1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb R (dn X) Q1
H7 : forall N, refl_t (Q2 N) (P2 N) (Q1 N) (Q4 N) /\
bisim_up_to refl_t (P2 N) (Q4 N)
============================
forall N, refl_t (P1 N) (P1 N) (Q1 N) (Q1 N) /\
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < intros.
Subgoal 2.2:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (dn X) P1
H4 : oneb Q (dn X) Q2
H5 : forall N, refl_t (P1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb R (dn X) Q1
H7 : forall N, refl_t (Q2 N) (P2 N) (Q1 N) (Q4 N) /\
bisim_up_to refl_t (P2 N) (Q4 N)
============================
refl_t (P1 N) (P1 N) (Q1 N) (Q1 N) /\ bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < split.
Subgoal 2.2.1:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (dn X) P1
H4 : oneb Q (dn X) Q2
H5 : forall N, refl_t (P1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb R (dn X) Q1
H7 : forall N, refl_t (Q2 N) (P2 N) (Q1 N) (Q4 N) /\
bisim_up_to refl_t (P2 N) (Q4 N)
============================
refl_t (P1 N) (P1 N) (Q1 N) (Q1 N)
Subgoal 2.2.2 is:
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < search.
Subgoal 2.2.2:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (dn X) P1
H4 : oneb Q (dn X) Q2
H5 : forall N, refl_t (P1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb R (dn X) Q1
H7 : forall N, refl_t (Q2 N) (P2 N) (Q1 N) (Q4 N) /\
bisim_up_to refl_t (P2 N) (Q4 N)
============================
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < apply H5 with N = N.
Subgoal 2.2.2:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (dn X) P1
H4 : oneb Q (dn X) Q2
H5 : forall N, refl_t (P1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb R (dn X) Q1
H7 : forall N, refl_t (Q2 N) (P2 N) (Q1 N) (Q4 N) /\
bisim_up_to refl_t (P2 N) (Q4 N)
H8 : refl_t (P1 N) (P3 N) (Q2 N) (Q3 N)
H9 : bisim_up_to refl_t (P3 N) (Q3 N)
============================
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < apply H7 with N = N.
Subgoal 2.2.2:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (dn X) P1
H4 : oneb Q (dn X) Q2
H5 : forall N, refl_t (P1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb R (dn X) Q1
H7 : forall N, refl_t (Q2 N) (P2 N) (Q1 N) (Q4 N) /\
bisim_up_to refl_t (P2 N) (Q4 N)
H8 : refl_t (P1 N) (P3 N) (Q2 N) (Q3 N)
H9 : bisim_up_to refl_t (P3 N) (Q3 N)
H10 : refl_t (Q2 N) (P2 N) (Q1 N) (Q4 N)
H11 : bisim_up_to refl_t (P2 N) (Q4 N)
============================
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < case H8.
Subgoal 2.2.2:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (dn X) P1
H4 : oneb Q (dn X) Q2
H5 : forall N, refl_t (P1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb R (dn X) Q1
H7 : forall N, refl_t (Q2 N) (P2 N) (Q1 N) (Q4 N) /\
bisim_up_to refl_t (P2 N) (Q4 N)
H9 : bisim_up_to refl_t (P3 N) (Q3 N)
H10 : refl_t (Q2 N) (P2 N) (Q1 N) (Q4 N)
H11 : bisim_up_to refl_t (P2 N) (Q4 N)
H12 : Q2 N = Q3 N
H13 : P1 N = P3 N
============================
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < case H10.
Subgoal 2.2.2:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (dn X) P1
H4 : oneb Q (dn X) Q2
H5 : forall N, refl_t (P1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb R (dn X) Q1
H7 : forall N, refl_t (Q2 N) (P2 N) (Q1 N) (Q4 N) /\
bisim_up_to refl_t (P2 N) (Q4 N)
H9 : bisim_up_to refl_t (P3 N) (Q3 N)
H11 : bisim_up_to refl_t (P2 N) (Q4 N)
H12 : Q2 N = Q3 N
H13 : P1 N = P3 N
H14 : Q1 N = Q4 N
H15 : Q2 N = P2 N
============================
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < apply bisim_eq_1R to H9 H13.
Subgoal 2.2.2:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (dn X) P1
H4 : oneb Q (dn X) Q2
H5 : forall N, refl_t (P1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb R (dn X) Q1
H7 : forall N, refl_t (Q2 N) (P2 N) (Q1 N) (Q4 N) /\
bisim_up_to refl_t (P2 N) (Q4 N)
H9 : bisim_up_to refl_t (P3 N) (Q3 N)
H11 : bisim_up_to refl_t (P2 N) (Q4 N)
H12 : Q2 N = Q3 N
H13 : P1 N = P3 N
H14 : Q1 N = Q4 N
H15 : Q2 N = P2 N
H16 : bisim_up_to refl_t (P1 N) (Q3 N)
============================
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < apply bisim_eq_2R to H16 H12.
Subgoal 2.2.2:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (dn X) P1
H4 : oneb Q (dn X) Q2
H5 : forall N, refl_t (P1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb R (dn X) Q1
H7 : forall N, refl_t (Q2 N) (P2 N) (Q1 N) (Q4 N) /\
bisim_up_to refl_t (P2 N) (Q4 N)
H9 : bisim_up_to refl_t (P3 N) (Q3 N)
H11 : bisim_up_to refl_t (P2 N) (Q4 N)
H12 : Q2 N = Q3 N
H13 : P1 N = P3 N
H14 : Q1 N = Q4 N
H15 : Q2 N = P2 N
H16 : bisim_up_to refl_t (P1 N) (Q3 N)
H17 : bisim_up_to refl_t (P1 N) (Q2 N)
============================
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < apply bisim_eq_2L to H17 H15.
Subgoal 2.2.2:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (dn X) P1
H4 : oneb Q (dn X) Q2
H5 : forall N, refl_t (P1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb R (dn X) Q1
H7 : forall N, refl_t (Q2 N) (P2 N) (Q1 N) (Q4 N) /\
bisim_up_to refl_t (P2 N) (Q4 N)
H9 : bisim_up_to refl_t (P3 N) (Q3 N)
H11 : bisim_up_to refl_t (P2 N) (Q4 N)
H12 : Q2 N = Q3 N
H13 : P1 N = P3 N
H14 : Q1 N = Q4 N
H15 : Q2 N = P2 N
H16 : bisim_up_to refl_t (P1 N) (Q3 N)
H17 : bisim_up_to refl_t (P1 N) (Q2 N)
H18 : bisim_up_to refl_t (P1 N) (P2 N)
============================
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < apply bisim_eq_2R to H11 H14.
Subgoal 2.2.2:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (dn X) P1
H4 : oneb Q (dn X) Q2
H5 : forall N, refl_t (P1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb R (dn X) Q1
H7 : forall N, refl_t (Q2 N) (P2 N) (Q1 N) (Q4 N) /\
bisim_up_to refl_t (P2 N) (Q4 N)
H9 : bisim_up_to refl_t (P3 N) (Q3 N)
H11 : bisim_up_to refl_t (P2 N) (Q4 N)
H12 : Q2 N = Q3 N
H13 : P1 N = P3 N
H14 : Q1 N = Q4 N
H15 : Q2 N = P2 N
H16 : bisim_up_to refl_t (P1 N) (Q3 N)
H17 : bisim_up_to refl_t (P1 N) (Q2 N)
H18 : bisim_up_to refl_t (P1 N) (P2 N)
H19 : bisim_up_to refl_t (P2 N) (Q1 N)
============================
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < clear H9 H11 H16 H17.
Subgoal 2.2.2:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (dn X) P1
H4 : oneb Q (dn X) Q2
H5 : forall N, refl_t (P1 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb R (dn X) Q1
H7 : forall N, refl_t (Q2 N) (P2 N) (Q1 N) (Q4 N) /\
bisim_up_to refl_t (P2 N) (Q4 N)
H12 : Q2 N = Q3 N
H13 : P1 N = P3 N
H14 : Q1 N = Q4 N
H15 : Q2 N = P2 N
H18 : bisim_up_to refl_t (P1 N) (P2 N)
H19 : bisim_up_to refl_t (P2 N) (Q1 N)
============================
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 3 is:
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < backchain CH.
Subgoal 3:
Variables: P Q R
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
============================
forall X P1, oneb P (up X) P1 ->
(exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < intros.
Subgoal 3:
Variables: P Q R X P1
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (up X) P1
============================
exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +)
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < apply Left3 to H3.
Subgoal 3:
Variables: P Q R X P1 Q2 P3 Q3
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (up X) P1
H4 : oneb Q (up X) Q2
H5 : refl_t (P1 n1) (P3 n1) (Q2 n1) (Q3 n1)
H6 : bisim_up_to refl_t (P3 n1) (Q3 n1)
============================
exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +)
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < apply Right3 to H4.
Subgoal 3:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (up X) P1
H4 : oneb Q (up X) Q2
H5 : refl_t (P1 n1) (P3 n1) (Q2 n1) (Q3 n1)
H6 : bisim_up_to refl_t (P3 n1) (Q3 n1)
H7 : oneb R (up X) Q1
H8 : refl_t (Q2 n1) (P2 n1) (Q1 n1) (Q4 n1)
H9 : bisim_up_to refl_t (P2 n1) (Q4 n1)
============================
exists Q1, oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +)
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < witness Q1.
Subgoal 3:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (up X) P1
H4 : oneb Q (up X) Q2
H5 : refl_t (P1 n1) (P3 n1) (Q2 n1) (Q3 n1)
H6 : bisim_up_to refl_t (P3 n1) (Q3 n1)
H7 : oneb R (up X) Q1
H8 : refl_t (Q2 n1) (P2 n1) (Q1 n1) (Q4 n1)
H9 : bisim_up_to refl_t (P2 n1) (Q4 n1)
============================
oneb R (up X) Q1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +)
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < split.
Subgoal 3.1:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (up X) P1
H4 : oneb Q (up X) Q2
H5 : refl_t (P1 n1) (P3 n1) (Q2 n1) (Q3 n1)
H6 : bisim_up_to refl_t (P3 n1) (Q3 n1)
H7 : oneb R (up X) Q1
H8 : refl_t (Q2 n1) (P2 n1) (Q1 n1) (Q4 n1)
H9 : bisim_up_to refl_t (P2 n1) (Q4 n1)
============================
oneb R (up X) Q1
Subgoal 3.2 is:
exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < search.
Subgoal 3.2:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (up X) P1
H4 : oneb Q (up X) Q2
H5 : refl_t (P1 n1) (P3 n1) (Q2 n1) (Q3 n1)
H6 : bisim_up_to refl_t (P3 n1) (Q3 n1)
H7 : oneb R (up X) Q1
H8 : refl_t (Q2 n1) (P2 n1) (Q1 n1) (Q4 n1)
H9 : bisim_up_to refl_t (P2 n1) (Q4 n1)
============================
exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < witness P1.
Subgoal 3.2:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (up X) P1
H4 : oneb Q (up X) Q2
H5 : refl_t (P1 n1) (P3 n1) (Q2 n1) (Q3 n1)
H6 : bisim_up_to refl_t (P3 n1) (Q3 n1)
H7 : oneb R (up X) Q1
H8 : refl_t (Q2 n1) (P2 n1) (Q1 n1) (Q4 n1)
H9 : bisim_up_to refl_t (P2 n1) (Q4 n1)
============================
exists Q2, nabla x, refl_t (P1 x) (P1 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P1 x) (Q2 x) +
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < witness Q1.
Subgoal 3.2:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (up X) P1
H4 : oneb Q (up X) Q2
H5 : refl_t (P1 n1) (P3 n1) (Q2 n1) (Q3 n1)
H6 : bisim_up_to refl_t (P3 n1) (Q3 n1)
H7 : oneb R (up X) Q1
H8 : refl_t (Q2 n1) (P2 n1) (Q1 n1) (Q4 n1)
H9 : bisim_up_to refl_t (P2 n1) (Q4 n1)
============================
nabla x, refl_t (P1 x) (P1 x) (Q1 x) (Q1 x) /\
bisim_up_to refl_t (P1 x) (Q1 x) +
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < intros.
Subgoal 3.2:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (up X) P1
H4 : oneb Q (up X) Q2
H5 : refl_t (P1 n1) (P3 n1) (Q2 n1) (Q3 n1)
H6 : bisim_up_to refl_t (P3 n1) (Q3 n1)
H7 : oneb R (up X) Q1
H8 : refl_t (Q2 n1) (P2 n1) (Q1 n1) (Q4 n1)
H9 : bisim_up_to refl_t (P2 n1) (Q4 n1)
============================
refl_t (P1 n1) (P1 n1) (Q1 n1) (Q1 n1) /\
bisim_up_to refl_t (P1 n1) (Q1 n1) +
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < split.
Subgoal 3.2.1:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (up X) P1
H4 : oneb Q (up X) Q2
H5 : refl_t (P1 n1) (P3 n1) (Q2 n1) (Q3 n1)
H6 : bisim_up_to refl_t (P3 n1) (Q3 n1)
H7 : oneb R (up X) Q1
H8 : refl_t (Q2 n1) (P2 n1) (Q1 n1) (Q4 n1)
H9 : bisim_up_to refl_t (P2 n1) (Q4 n1)
============================
refl_t (P1 n1) (P1 n1) (Q1 n1) (Q1 n1)
Subgoal 3.2.2 is:
bisim_up_to refl_t (P1 n1) (Q1 n1) +
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < search.
Subgoal 3.2.2:
Variables: P Q R X P1 Q2 P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (up X) P1
H4 : oneb Q (up X) Q2
H5 : refl_t (P1 n1) (P3 n1) (Q2 n1) (Q3 n1)
H6 : bisim_up_to refl_t (P3 n1) (Q3 n1)
H7 : oneb R (up X) Q1
H8 : refl_t (Q2 n1) (P2 n1) (Q1 n1) (Q4 n1)
H9 : bisim_up_to refl_t (P2 n1) (Q4 n1)
============================
bisim_up_to refl_t (P1 n1) (Q1 n1) +
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < case H5.
Subgoal 3.2.2:
Variables: P Q R X P3 Q3 Q1 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (up X) (z1\P3 z1)
H4 : oneb Q (up X) (z1\Q3 z1)
H6 : bisim_up_to refl_t (P3 n1) (Q3 n1)
H7 : oneb R (up X) Q1
H8 : refl_t (Q3 n1) (P2 n1) (Q1 n1) (Q4 n1)
H9 : bisim_up_to refl_t (P2 n1) (Q4 n1)
============================
bisim_up_to refl_t (P3 n1) (Q1 n1) +
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < case H8.
Subgoal 3.2.2:
Variables: P Q R X P3 P2 Q4
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb P (up X) (z1\P3 z1)
H4 : oneb Q (up X) (z1\P2 z1)
H6 : bisim_up_to refl_t (P3 n1) (P2 n1)
H7 : oneb R (up X) (z1\Q4 z1)
H9 : bisim_up_to refl_t (P2 n1) (Q4 n1)
============================
bisim_up_to refl_t (P3 n1) (Q4 n1) +
Subgoal 4 is:
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < backchain CH.
Subgoal 4:
Variables: P Q R
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
============================
forall L Q1, one R L Q1 ->
(exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +))
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < intros.
Subgoal 4:
Variables: P Q R L Q1
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one R L Q1
============================
exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < apply Right4 to H3.
Subgoal 4:
Variables: P Q R L Q1 P2 P3 Q3
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one R L Q1
H4 : one Q L P2
H5 : refl_t P2 P3 Q1 Q3
H6 : bisim_up_to refl_t P3 Q3
============================
exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < apply Left4 to H4.
Subgoal 4:
Variables: P Q R L Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one R L Q1
H4 : one Q L P2
H5 : refl_t P2 P3 Q1 Q3
H6 : bisim_up_to refl_t P3 Q3
H7 : one P L P1
H8 : refl_t P1 P4 P2 Q2
H9 : bisim_up_to refl_t P4 Q2
============================
exists P1, one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < witness P1.
Subgoal 4:
Variables: P Q R L Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one R L Q1
H4 : one Q L P2
H5 : refl_t P2 P3 Q1 Q3
H6 : bisim_up_to refl_t P3 Q3
H7 : one P L P1
H8 : refl_t P1 P4 P2 Q2
H9 : bisim_up_to refl_t P4 Q2
============================
one P L P1 /\
(exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +)
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < split.
Subgoal 4.1:
Variables: P Q R L Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one R L Q1
H4 : one Q L P2
H5 : refl_t P2 P3 Q1 Q3
H6 : bisim_up_to refl_t P3 Q3
H7 : one P L P1
H8 : refl_t P1 P4 P2 Q2
H9 : bisim_up_to refl_t P4 Q2
============================
one P L P1
Subgoal 4.2 is:
exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < search.
Subgoal 4.2:
Variables: P Q R L Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one R L Q1
H4 : one Q L P2
H5 : refl_t P2 P3 Q1 Q3
H6 : bisim_up_to refl_t P3 Q3
H7 : one P L P1
H8 : refl_t P1 P4 P2 Q2
H9 : bisim_up_to refl_t P4 Q2
============================
exists P2 Q2, refl_t P1 P2 Q1 Q2 /\ bisim_up_to refl_t P2 Q2 +
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < witness P1.
Subgoal 4.2:
Variables: P Q R L Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one R L Q1
H4 : one Q L P2
H5 : refl_t P2 P3 Q1 Q3
H6 : bisim_up_to refl_t P3 Q3
H7 : one P L P1
H8 : refl_t P1 P4 P2 Q2
H9 : bisim_up_to refl_t P4 Q2
============================
exists Q2, refl_t P1 P1 Q1 Q2 /\ bisim_up_to refl_t P1 Q2 +
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < witness Q1.
Subgoal 4.2:
Variables: P Q R L Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one R L Q1
H4 : one Q L P2
H5 : refl_t P2 P3 Q1 Q3
H6 : bisim_up_to refl_t P3 Q3
H7 : one P L P1
H8 : refl_t P1 P4 P2 Q2
H9 : bisim_up_to refl_t P4 Q2
============================
refl_t P1 P1 Q1 Q1 /\ bisim_up_to refl_t P1 Q1 +
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < split.
Subgoal 4.2.1:
Variables: P Q R L Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one R L Q1
H4 : one Q L P2
H5 : refl_t P2 P3 Q1 Q3
H6 : bisim_up_to refl_t P3 Q3
H7 : one P L P1
H8 : refl_t P1 P4 P2 Q2
H9 : bisim_up_to refl_t P4 Q2
============================
refl_t P1 P1 Q1 Q1
Subgoal 4.2.2 is:
bisim_up_to refl_t P1 Q1 +
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < search.
Subgoal 4.2.2:
Variables: P Q R L Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one R L Q1
H4 : one Q L P2
H5 : refl_t P2 P3 Q1 Q3
H6 : bisim_up_to refl_t P3 Q3
H7 : one P L P1
H8 : refl_t P1 P4 P2 Q2
H9 : bisim_up_to refl_t P4 Q2
============================
bisim_up_to refl_t P1 Q1 +
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < case H5.
Subgoal 4.2.2:
Variables: P Q R L P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one R L Q3
H4 : one Q L P3
H6 : bisim_up_to refl_t P3 Q3
H7 : one P L P1
H8 : refl_t P1 P4 P3 Q2
H9 : bisim_up_to refl_t P4 Q2
============================
bisim_up_to refl_t P1 Q3 +
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < case H8.
Subgoal 4.2.2:
Variables: P Q R L Q3 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : one R L Q3
H4 : one Q L Q2
H6 : bisim_up_to refl_t Q2 Q3
H7 : one P L P4
H9 : bisim_up_to refl_t P4 Q2
============================
bisim_up_to refl_t P4 Q3 +
Subgoal 5 is:
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < backchain CH.
Subgoal 5:
Variables: P Q R
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
============================
forall X Q1, oneb R (dn X) Q1 ->
(exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +))
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < intros.
Subgoal 5:
Variables: P Q R X Q1
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (dn X) Q1
============================
exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +)
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < apply Right5 to H3.
Subgoal 5:
Variables: P Q R X Q1 P2 P3 Q3
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (dn X) Q1
H4 : oneb Q (dn X) P2
H5 : forall N, refl_t (P2 N) (P3 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
============================
exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +)
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < apply Left5 to H4.
Subgoal 5:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (dn X) Q1
H4 : oneb Q (dn X) P2
H5 : forall N, refl_t (P2 N) (P3 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb P (dn X) P1
H7 : forall N, refl_t (P1 N) (P4 N) (P2 N) (Q2 N) /\
bisim_up_to refl_t (P4 N) (Q2 N)
============================
exists P1, oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +)
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < witness P1.
Subgoal 5:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (dn X) Q1
H4 : oneb Q (dn X) P2
H5 : forall N, refl_t (P2 N) (P3 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb P (dn X) P1
H7 : forall N, refl_t (P1 N) (P4 N) (P2 N) (Q2 N) /\
bisim_up_to refl_t (P4 N) (Q2 N)
============================
oneb P (dn X) P1 /\
(exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +)
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < split.
Subgoal 5.1:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (dn X) Q1
H4 : oneb Q (dn X) P2
H5 : forall N, refl_t (P2 N) (P3 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb P (dn X) P1
H7 : forall N, refl_t (P1 N) (P4 N) (P2 N) (Q2 N) /\
bisim_up_to refl_t (P4 N) (Q2 N)
============================
oneb P (dn X) P1
Subgoal 5.2 is:
exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < search.
Subgoal 5.2:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (dn X) Q1
H4 : oneb Q (dn X) P2
H5 : forall N, refl_t (P2 N) (P3 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb P (dn X) P1
H7 : forall N, refl_t (P1 N) (P4 N) (P2 N) (Q2 N) /\
bisim_up_to refl_t (P4 N) (Q2 N)
============================
exists P2 Q2, forall N, refl_t (P1 N) (P2 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P2 N) (Q2 N) +
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < witness P1.
Subgoal 5.2:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (dn X) Q1
H4 : oneb Q (dn X) P2
H5 : forall N, refl_t (P2 N) (P3 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb P (dn X) P1
H7 : forall N, refl_t (P1 N) (P4 N) (P2 N) (Q2 N) /\
bisim_up_to refl_t (P4 N) (Q2 N)
============================
exists Q2, forall N, refl_t (P1 N) (P1 N) (Q1 N) (Q2 N) /\
bisim_up_to refl_t (P1 N) (Q2 N) +
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < witness Q1.
Subgoal 5.2:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (dn X) Q1
H4 : oneb Q (dn X) P2
H5 : forall N, refl_t (P2 N) (P3 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb P (dn X) P1
H7 : forall N, refl_t (P1 N) (P4 N) (P2 N) (Q2 N) /\
bisim_up_to refl_t (P4 N) (Q2 N)
============================
forall N, refl_t (P1 N) (P1 N) (Q1 N) (Q1 N) /\
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < intros.
Subgoal 5.2:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (dn X) Q1
H4 : oneb Q (dn X) P2
H5 : forall N, refl_t (P2 N) (P3 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb P (dn X) P1
H7 : forall N, refl_t (P1 N) (P4 N) (P2 N) (Q2 N) /\
bisim_up_to refl_t (P4 N) (Q2 N)
============================
refl_t (P1 N) (P1 N) (Q1 N) (Q1 N) /\ bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < split.
Subgoal 5.2.1:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (dn X) Q1
H4 : oneb Q (dn X) P2
H5 : forall N, refl_t (P2 N) (P3 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb P (dn X) P1
H7 : forall N, refl_t (P1 N) (P4 N) (P2 N) (Q2 N) /\
bisim_up_to refl_t (P4 N) (Q2 N)
============================
refl_t (P1 N) (P1 N) (Q1 N) (Q1 N)
Subgoal 5.2.2 is:
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < search.
Subgoal 5.2.2:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (dn X) Q1
H4 : oneb Q (dn X) P2
H5 : forall N, refl_t (P2 N) (P3 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb P (dn X) P1
H7 : forall N, refl_t (P1 N) (P4 N) (P2 N) (Q2 N) /\
bisim_up_to refl_t (P4 N) (Q2 N)
============================
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < apply H5 with N = N.
Subgoal 5.2.2:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (dn X) Q1
H4 : oneb Q (dn X) P2
H5 : forall N, refl_t (P2 N) (P3 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb P (dn X) P1
H7 : forall N, refl_t (P1 N) (P4 N) (P2 N) (Q2 N) /\
bisim_up_to refl_t (P4 N) (Q2 N)
H8 : refl_t (P2 N) (P3 N) (Q1 N) (Q3 N)
H9 : bisim_up_to refl_t (P3 N) (Q3 N)
============================
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < apply H7 with N = N.
Subgoal 5.2.2:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (dn X) Q1
H4 : oneb Q (dn X) P2
H5 : forall N, refl_t (P2 N) (P3 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb P (dn X) P1
H7 : forall N, refl_t (P1 N) (P4 N) (P2 N) (Q2 N) /\
bisim_up_to refl_t (P4 N) (Q2 N)
H8 : refl_t (P2 N) (P3 N) (Q1 N) (Q3 N)
H9 : bisim_up_to refl_t (P3 N) (Q3 N)
H10 : refl_t (P1 N) (P4 N) (P2 N) (Q2 N)
H11 : bisim_up_to refl_t (P4 N) (Q2 N)
============================
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < case H8.
Subgoal 5.2.2:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (dn X) Q1
H4 : oneb Q (dn X) P2
H5 : forall N, refl_t (P2 N) (P3 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb P (dn X) P1
H7 : forall N, refl_t (P1 N) (P4 N) (P2 N) (Q2 N) /\
bisim_up_to refl_t (P4 N) (Q2 N)
H9 : bisim_up_to refl_t (P3 N) (Q3 N)
H10 : refl_t (P1 N) (P4 N) (P2 N) (Q2 N)
H11 : bisim_up_to refl_t (P4 N) (Q2 N)
H12 : Q1 N = Q3 N
H13 : P2 N = P3 N
============================
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < case H10.
Subgoal 5.2.2:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (dn X) Q1
H4 : oneb Q (dn X) P2
H5 : forall N, refl_t (P2 N) (P3 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb P (dn X) P1
H7 : forall N, refl_t (P1 N) (P4 N) (P2 N) (Q2 N) /\
bisim_up_to refl_t (P4 N) (Q2 N)
H9 : bisim_up_to refl_t (P3 N) (Q3 N)
H11 : bisim_up_to refl_t (P4 N) (Q2 N)
H12 : Q1 N = Q3 N
H13 : P2 N = P3 N
H14 : P2 N = Q2 N
H15 : P1 N = P4 N
============================
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < apply bisim_eq_1R to H9 H13.
Subgoal 5.2.2:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (dn X) Q1
H4 : oneb Q (dn X) P2
H5 : forall N, refl_t (P2 N) (P3 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb P (dn X) P1
H7 : forall N, refl_t (P1 N) (P4 N) (P2 N) (Q2 N) /\
bisim_up_to refl_t (P4 N) (Q2 N)
H9 : bisim_up_to refl_t (P3 N) (Q3 N)
H11 : bisim_up_to refl_t (P4 N) (Q2 N)
H12 : Q1 N = Q3 N
H13 : P2 N = P3 N
H14 : P2 N = Q2 N
H15 : P1 N = P4 N
H16 : bisim_up_to refl_t (P2 N) (Q3 N)
============================
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < apply bisim_eq_2R to H16 H12.
Subgoal 5.2.2:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (dn X) Q1
H4 : oneb Q (dn X) P2
H5 : forall N, refl_t (P2 N) (P3 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb P (dn X) P1
H7 : forall N, refl_t (P1 N) (P4 N) (P2 N) (Q2 N) /\
bisim_up_to refl_t (P4 N) (Q2 N)
H9 : bisim_up_to refl_t (P3 N) (Q3 N)
H11 : bisim_up_to refl_t (P4 N) (Q2 N)
H12 : Q1 N = Q3 N
H13 : P2 N = P3 N
H14 : P2 N = Q2 N
H15 : P1 N = P4 N
H16 : bisim_up_to refl_t (P2 N) (Q3 N)
H17 : bisim_up_to refl_t (P2 N) (Q1 N)
============================
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < apply bisim_eq_1R to H11 H15.
Subgoal 5.2.2:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (dn X) Q1
H4 : oneb Q (dn X) P2
H5 : forall N, refl_t (P2 N) (P3 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb P (dn X) P1
H7 : forall N, refl_t (P1 N) (P4 N) (P2 N) (Q2 N) /\
bisim_up_to refl_t (P4 N) (Q2 N)
H9 : bisim_up_to refl_t (P3 N) (Q3 N)
H11 : bisim_up_to refl_t (P4 N) (Q2 N)
H12 : Q1 N = Q3 N
H13 : P2 N = P3 N
H14 : P2 N = Q2 N
H15 : P1 N = P4 N
H16 : bisim_up_to refl_t (P2 N) (Q3 N)
H17 : bisim_up_to refl_t (P2 N) (Q1 N)
H18 : bisim_up_to refl_t (P1 N) (Q2 N)
============================
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < apply bisim_eq_2R to H18 H14.
Subgoal 5.2.2:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (dn X) Q1
H4 : oneb Q (dn X) P2
H5 : forall N, refl_t (P2 N) (P3 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb P (dn X) P1
H7 : forall N, refl_t (P1 N) (P4 N) (P2 N) (Q2 N) /\
bisim_up_to refl_t (P4 N) (Q2 N)
H9 : bisim_up_to refl_t (P3 N) (Q3 N)
H11 : bisim_up_to refl_t (P4 N) (Q2 N)
H12 : Q1 N = Q3 N
H13 : P2 N = P3 N
H14 : P2 N = Q2 N
H15 : P1 N = P4 N
H16 : bisim_up_to refl_t (P2 N) (Q3 N)
H17 : bisim_up_to refl_t (P2 N) (Q1 N)
H18 : bisim_up_to refl_t (P1 N) (Q2 N)
H19 : bisim_up_to refl_t (P1 N) (P2 N)
============================
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < clear H9 H11 H16 H18.
Subgoal 5.2.2:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2 N
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (dn X) Q1
H4 : oneb Q (dn X) P2
H5 : forall N, refl_t (P2 N) (P3 N) (Q1 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)
H6 : oneb P (dn X) P1
H7 : forall N, refl_t (P1 N) (P4 N) (P2 N) (Q2 N) /\
bisim_up_to refl_t (P4 N) (Q2 N)
H12 : Q1 N = Q3 N
H13 : P2 N = P3 N
H14 : P2 N = Q2 N
H15 : P1 N = P4 N
H17 : bisim_up_to refl_t (P2 N) (Q1 N)
H19 : bisim_up_to refl_t (P1 N) (P2 N)
============================
bisim_up_to refl_t (P1 N) (Q1 N) +
Subgoal 6 is:
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < backchain CH.
Subgoal 6:
Variables: P Q R
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
============================
forall X Q1, oneb R (up X) Q1 ->
(exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +))
bisim_transitive < intros.
Subgoal 6:
Variables: P Q R X Q1
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (up X) Q1
============================
exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +)
bisim_transitive < apply Right6 to H3.
Subgoal 6:
Variables: P Q R X Q1 P2 P3 Q3
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (up X) Q1
H4 : oneb Q (up X) P2
H5 : refl_t (P2 n1) (P3 n1) (Q1 n1) (Q3 n1)
H6 : bisim_up_to refl_t (P3 n1) (Q3 n1)
============================
exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +)
bisim_transitive < apply Left6 to H4.
Subgoal 6:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (up X) Q1
H4 : oneb Q (up X) P2
H5 : refl_t (P2 n1) (P3 n1) (Q1 n1) (Q3 n1)
H6 : bisim_up_to refl_t (P3 n1) (Q3 n1)
H7 : oneb P (up X) P1
H8 : refl_t (P1 n1) (P4 n1) (P2 n1) (Q2 n1)
H9 : bisim_up_to refl_t (P4 n1) (Q2 n1)
============================
exists P1, oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +)
bisim_transitive < witness P1.
Subgoal 6:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (up X) Q1
H4 : oneb Q (up X) P2
H5 : refl_t (P2 n1) (P3 n1) (Q1 n1) (Q3 n1)
H6 : bisim_up_to refl_t (P3 n1) (Q3 n1)
H7 : oneb P (up X) P1
H8 : refl_t (P1 n1) (P4 n1) (P2 n1) (Q2 n1)
H9 : bisim_up_to refl_t (P4 n1) (Q2 n1)
============================
oneb P (up X) P1 /\
(exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +)
bisim_transitive < split.
Subgoal 6.1:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (up X) Q1
H4 : oneb Q (up X) P2
H5 : refl_t (P2 n1) (P3 n1) (Q1 n1) (Q3 n1)
H6 : bisim_up_to refl_t (P3 n1) (Q3 n1)
H7 : oneb P (up X) P1
H8 : refl_t (P1 n1) (P4 n1) (P2 n1) (Q2 n1)
H9 : bisim_up_to refl_t (P4 n1) (Q2 n1)
============================
oneb P (up X) P1
Subgoal 6.2 is:
exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +
bisim_transitive < search.
Subgoal 6.2:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (up X) Q1
H4 : oneb Q (up X) P2
H5 : refl_t (P2 n1) (P3 n1) (Q1 n1) (Q3 n1)
H6 : bisim_up_to refl_t (P3 n1) (Q3 n1)
H7 : oneb P (up X) P1
H8 : refl_t (P1 n1) (P4 n1) (P2 n1) (Q2 n1)
H9 : bisim_up_to refl_t (P4 n1) (Q2 n1)
============================
exists P2 Q2, nabla x, refl_t (P1 x) (P2 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P2 x) (Q2 x) +
bisim_transitive < witness P1.
Subgoal 6.2:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (up X) Q1
H4 : oneb Q (up X) P2
H5 : refl_t (P2 n1) (P3 n1) (Q1 n1) (Q3 n1)
H6 : bisim_up_to refl_t (P3 n1) (Q3 n1)
H7 : oneb P (up X) P1
H8 : refl_t (P1 n1) (P4 n1) (P2 n1) (Q2 n1)
H9 : bisim_up_to refl_t (P4 n1) (Q2 n1)
============================
exists Q2, nabla x, refl_t (P1 x) (P1 x) (Q1 x) (Q2 x) /\
bisim_up_to refl_t (P1 x) (Q2 x) +
bisim_transitive < witness Q1.
Subgoal 6.2:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (up X) Q1
H4 : oneb Q (up X) P2
H5 : refl_t (P2 n1) (P3 n1) (Q1 n1) (Q3 n1)
H6 : bisim_up_to refl_t (P3 n1) (Q3 n1)
H7 : oneb P (up X) P1
H8 : refl_t (P1 n1) (P4 n1) (P2 n1) (Q2 n1)
H9 : bisim_up_to refl_t (P4 n1) (Q2 n1)
============================
nabla x, refl_t (P1 x) (P1 x) (Q1 x) (Q1 x) /\
bisim_up_to refl_t (P1 x) (Q1 x) +
bisim_transitive < intros.
Subgoal 6.2:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (up X) Q1
H4 : oneb Q (up X) P2
H5 : refl_t (P2 n1) (P3 n1) (Q1 n1) (Q3 n1)
H6 : bisim_up_to refl_t (P3 n1) (Q3 n1)
H7 : oneb P (up X) P1
H8 : refl_t (P1 n1) (P4 n1) (P2 n1) (Q2 n1)
H9 : bisim_up_to refl_t (P4 n1) (Q2 n1)
============================
refl_t (P1 n1) (P1 n1) (Q1 n1) (Q1 n1) /\
bisim_up_to refl_t (P1 n1) (Q1 n1) +
bisim_transitive < split.
Subgoal 6.2.1:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (up X) Q1
H4 : oneb Q (up X) P2
H5 : refl_t (P2 n1) (P3 n1) (Q1 n1) (Q3 n1)
H6 : bisim_up_to refl_t (P3 n1) (Q3 n1)
H7 : oneb P (up X) P1
H8 : refl_t (P1 n1) (P4 n1) (P2 n1) (Q2 n1)
H9 : bisim_up_to refl_t (P4 n1) (Q2 n1)
============================
refl_t (P1 n1) (P1 n1) (Q1 n1) (Q1 n1)
Subgoal 6.2.2 is:
bisim_up_to refl_t (P1 n1) (Q1 n1) +
bisim_transitive < search.
Subgoal 6.2.2:
Variables: P Q R X Q1 P2 P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (up X) Q1
H4 : oneb Q (up X) P2
H5 : refl_t (P2 n1) (P3 n1) (Q1 n1) (Q3 n1)
H6 : bisim_up_to refl_t (P3 n1) (Q3 n1)
H7 : oneb P (up X) P1
H8 : refl_t (P1 n1) (P4 n1) (P2 n1) (Q2 n1)
H9 : bisim_up_to refl_t (P4 n1) (Q2 n1)
============================
bisim_up_to refl_t (P1 n1) (Q1 n1) +
bisim_transitive < case H5.
Subgoal 6.2.2:
Variables: P Q R X P3 Q3 P1 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (up X) (z1\Q3 z1)
H4 : oneb Q (up X) (z1\P3 z1)
H6 : bisim_up_to refl_t (P3 n1) (Q3 n1)
H7 : oneb P (up X) P1
H8 : refl_t (P1 n1) (P4 n1) (P3 n1) (Q2 n1)
H9 : bisim_up_to refl_t (P4 n1) (Q2 n1)
============================
bisim_up_to refl_t (P1 n1) (Q3 n1) +
bisim_transitive < case H8.
Subgoal 6.2.2:
Variables: P Q R X Q3 P4 Q2
CH : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R ->
bisim_up_to refl_t P R +
Left1 : forall L P2, one P L P2 ->
(exists Q2, one Q L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left2 : forall X P2, oneb P (dn X) P2 ->
(exists Q2, oneb Q (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left3 : forall X P2, oneb P (up X) P2 ->
(exists Q2, oneb Q (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Left4 : forall L Q2, one Q L Q2 ->
(exists P2, one P L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Left5 : forall X Q2, oneb Q (dn X) Q2 ->
(exists P2, oneb P (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Left6 : forall X Q2, oneb Q (up X) Q2 ->
(exists P2, oneb P (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right1 : forall L P2, one Q L P2 ->
(exists Q2, one R L Q2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right2 : forall X P2, oneb Q (dn X) P2 ->
(exists Q2, oneb R (dn X) Q2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right3 : forall X P2, oneb Q (up X) P2 ->
(exists Q2, oneb R (up X) Q2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
Right4 : forall L Q2, one R L Q2 ->
(exists P2, one Q L P2 /\
(exists P3 Q3, refl_t P2 P3 Q2 Q3 /\ bisim_up_to refl_t P3 Q3))
Right5 : forall X Q2, oneb R (dn X) Q2 ->
(exists P2, oneb Q (dn X) P2 /\
(exists P3 Q3, forall N, refl_t (P2 N) (P3 N) (Q2 N) (Q3 N) /\
bisim_up_to refl_t (P3 N) (Q3 N)))
Right6 : forall X Q2, oneb R (up X) Q2 ->
(exists P2, oneb Q (up X) P2 /\
(exists P3 Q3, nabla x, refl_t (P2 x) (P3 x) (Q2 x) (Q3 x) /\
bisim_up_to refl_t (P3 x) (Q3 x)))
H3 : oneb R (up X) (z1\Q3 z1)
H4 : oneb Q (up X) (z1\Q2 z1)
H6 : bisim_up_to refl_t (Q2 n1) (Q3 n1)
H7 : oneb P (up X) (z1\P4 z1)
H9 : bisim_up_to refl_t (P4 n1) (Q2 n1)
============================
bisim_up_to refl_t (P4 n1) (Q3 n1) +
bisim_transitive < backchain CH.
Proof completed.
Abella < Goodbye.
