contribution about \lambda-\delta

[helm.git] / helm / coq-contribs / LAMBDA-TYPES / subst0_subst0.v

diff --git a/helm/coq-contribs/LAMBDA-TYPES/subst0_subst0.v b/helm/coq-contribs/LAMBDA-TYPES/subst0_subst0.v
new file mode 100644 (file)
index 0000000..e092e23
--- /dev/null
+++ b/helm/coq-contribs/LAMBDA-TYPES/subst0_subst0.v
@@ -0,0 +1,68 @@
+(*#* #stop file *)
+
+Require subst0_defs.
+Require subst0_gen.
+Require subst0_lift.
+
+   Section subst0_subst0. (**************************************************)
+
+      Tactic Definition IH :=
+         Match Context With
+            | [ H1: (u1,u:?; i:?) (subst0 i u u1 ?1) -> ?;
+                H2: (subst0 ?2 ?3 ?4 ?1) |- ? ] ->
+               LApply (H1 ?4 ?3 ?2); [ Clear H1; Intros H1 | XAuto ];
+               XElim H1; Intros
+            | [ H1: (u1,u:?; i:?) (subst0 i u ?1 u1) -> ?;
+                H2: (subst0 ?2 ?3 ?1 ?4) |- ? ] ->
+               LApply (H1 ?4 ?3 ?2); [ Clear H1; Intros H1 | XAuto ];
+               XElim H1; Intros.
+
+      Theorem subst0_subst0: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
+                             (u1,u:?; i:?) (subst0 i u u1 u2) ->
+                             (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t t2)).
+      Intros until 1; XElim H; Intros.
+(* case 1 : subst0_bref *)
+      Arith5 i0 i; XEAuto.
+(* case 2 : subst0_fst *)
+      IH; XEAuto.
+(* case 3 : subst0_snd *)
+      IH; SRwBackIn H2; XEAuto.
+(* case 4 : subst0_both *)
+      Repeat IH; SRwBackIn H4; XEAuto.
+      Qed.
+
+      Theorem subst0_subst0_back: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
+                                  (u1,u:?; i:?) (subst0 i u u2 u1) ->
+                                  (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t2 t)).
+      Intros until 1; XElim H; Intros.
+(* case 1 : subst0_bref *)
+      Arith5 i0 i; XEAuto.
+(* case 2 : subst0_fst *)
+      IH; XEAuto.
+(* case 3 : subst0_snd *)
+      IH; SRwBackIn H2; XEAuto.
+(* case 4 : subst0_both *)
+      Repeat IH; SRwBackIn H4; XEAuto.
+      Qed.
+
+      Theorem subst0_trans: (t2,t1,v:?; i:?) (subst0 i v t1 t2) ->
+                            (t3:?) (subst0 i v t2 t3) ->
+                            (subst0 i v t1 t3).
+      Intros until 1; XElim H; Intros;
+      Subst0Gen; Try Rewrite H1; Try Rewrite H3; XAuto.
+      Qed.
+
+   End subst0_subst0.
+
+      Hints Resolve subst0_trans : ltlc.
+
+      Tactic Definition Subst0Subst0 :=
+         Match Context With
+            | [ H1: (subst0 ?0 ?1 ?2 ?3); H2: (subst0 ?4 ?5 ?6 ?1) |- ? ] ->
+               LApply (subst0_subst0 ?2 ?3 ?1 ?0); [ Intros H_x | XAuto ];
+               LApply (H_x ?6 ?5 ?4); [ Clear H_x; Intros H_x | XAuto ];
+               XElim H_x; Intros
+            | [ H1: (subst0 ?0 ?1 ?2 ?3); H2: (subst0 ?4 ?5 ?1 ?6) |- ? ] ->
+               LApply (subst0_subst0_back ?2 ?3 ?1 ?0); [ Intros H_x | XAuto ];
+               LApply (H_x ?6 ?5 ?4); [ Clear H_x; Intros H_x | XAuto ];
+               XElim H_x; Intros.