Specification "debruijn_ho". %% Proof of the determinism of the translation of lambda terms into %% the debruijn form %% General property of member Theorem member_prune : forall E (L:olist), nabla (x:hterm), member (E x) L -> exists F, E = y\F. induction on 1. intros. case H1. search. apply IH to H2. search. %% Properties of addition Define nat : nat -> prop by nat z ; nat (s X) := nat X. Define le : nat -> nat -> prop by le X X ; le X (s Y) := le X Y. Theorem le_dec : forall X Y, le (s X) Y -> le X Y. induction on 1. intros. case H1. search. apply IH to H2. search. Theorem le_absurd : forall X, nat X -> le (s X) X -> false. induction on 1. intros. case H1. case H2. case H2. apply le_dec to H4. apply IH to H3 H5. Theorem add_le : forall A B C, {add A B C} -> le B C. induction on 1. intros. case H1. search. apply IH to H2. search. Theorem add_absurd : forall A C, nat C -> {add A (s C) C} -> false. intros. apply add_le to H2. apply le_absurd to H1 H3. Theorem add_zero : forall A C, nat C -> {add A C C} -> A = z. intros. case H2. search. case H1. apply add_absurd to H4 H3. % add is deterministic in its first argument Theorem add_det1 : forall A1 A2 B C, nat C -> {add A1 B C} -> {add A2 B C} -> A1 = A2. induction on 2. intros. case H2. apply add_zero to H1 H3. search. case H3. case H1. apply add_absurd to H5 H4. case H1. apply IH to H6 H4 H5. search. % add is deterministic in its second argument Theorem add_det2 : forall A B1 B2 C, {add A B1 C} -> {add A B2 C} -> B1 = B2. induction on 1. intros. case H1. case H2. search. case H2. apply IH to H3 H4. search. %% Theorems specific to our translation Define ctx : olist -> nat -> prop by ctx nil z ; nabla x, ctx ((pi H'\ pi DX\ add H DX H' => ho2db x H' (dvar DX)) :: L) (s H) := ctx L H. Define name : hterm -> prop by nabla x, name x. Theorem ctx_nat : forall L H, ctx L H -> nat H. induction on 1. intros. case H1. search. apply IH to H2. search. Theorem ctx_inv : forall E L H, ctx L H -> member E L -> exists X HX, E = pi H'\ pi DX\ add HX DX H' => ho2db X H' (dvar DX) /\ name X /\ le (s HX) H. induction on 1. intros. case H1. case H2. case H2. search. apply member_prune to H4. apply IH to H3 H4. search. Theorem ctx_unique1 : forall L H X H1 H2, ctx L H -> member (pi H'\ pi DX\ add H1 DX H' => ho2db X H' (dvar DX)) L -> member (pi H'\ pi DX\ add H2 DX H' => ho2db X H' (dvar DX)) L -> H1 = H2. induction on 2. intros. case H2. case H3. search. case H1. apply member_prune to H4. case H3. case H1. apply member_prune to H4. case H1. apply IH to H6 H4 H5. search. Theorem ctx_unique2 : forall L H X1 X2 HX, ctx L H -> member (pi H'\ pi DX\ add HX DX H' => ho2db X1 H' (dvar DX)) L -> member (pi H'\ pi DX\ add HX DX H' => ho2db X2 H' (dvar DX)) L -> X1 = X2. induction on 2. intros. case H2. case H3. search. case H1. apply ctx_inv to H5 H4. apply ctx_nat to H5. apply le_absurd to H8 H7. case H3. case H1. apply ctx_inv to H5 H4. apply ctx_nat to H5. apply le_absurd to H8 H7. case H1. apply IH to H6 H4 H5. search. Theorem add_ignores_ctx : forall L H A B C, ctx L H -> {L |- add A B C} -> {add A B C}. induction on 2. intros. case H2. search. apply IH to H1 H3. search. apply ctx_inv to H1 H4. case H3. %% ho2db is deterministic in its third argument %% ie, higher-order --> debruijn is unique Theorem ho2db_det3 : forall L M D1 D2 H, ctx L H -> {L |- ho2db M H D1} -> {L |- ho2db M H D2} -> D1 = D2. induction on 2. intros. case H2. case H3. apply IH to H1 H4 H6. apply IH to H1 H5 H7. search. apply ctx_inv to H1 H7. case H6. case H8. case H3. apply IH to _ H4 H5. search. apply ctx_inv to H1 H6. case H5. case H7. apply ctx_inv to H1 H5. case H4. case H3. case H6. case H6. apply ctx_inv to H1 H10. case H9. apply ctx_unique1 to H1 H5 H10. apply add_ignores_ctx to H1 H8. apply add_ignores_ctx to H1 H13. apply add_det2 to H14 H15. search. Theorem ho2db_det3_simple : forall M D1 D2, {ho2db M z D1} -> {ho2db M z D2} -> D1 = D2. intros. apply ho2db_det3 to _ H1 H2. search. %% ho2db is deterministic in its first argument %% ie, debruijn --> higher-order is unique %% proof is mostly the same as ho2db_det3 except with fewer cases Theorem ho2db_det1 : forall L M1 M2 D H, ctx L H -> {L |- ho2db M1 H D} -> {L |- ho2db M2 H D} -> M1 = M2. induction on 2. intros. case H2. case H3. apply IH to H1 H4 H6. apply IH to H1 H5 H7. search. apply ctx_inv to H1 H7. case H6. case H3. apply IH to _ H4 H5. search. apply ctx_inv to H1 H6. case H5. apply ctx_inv to H1 H5. case H4. case H3. apply ctx_inv to H1 H10. case H9. apply add_ignores_ctx to H1 H8. apply add_ignores_ctx to H1 H13. apply ctx_nat to H1. apply add_det1 to H16 H14 H15. apply ctx_unique2 to H1 H5 H10. search. Theorem ho2db_det1_simple : forall M1 M2 D, {ho2db M1 z D} -> {ho2db M2 z D} -> M1 = M2. intros. apply ho2db_det1 to _ H1 H2. search.