contribution about \lambda-\delta

[helm.git] / helm / coq-contribs / LAMBDA-TYPES / subst1_subst1.v

diff --git a/helm/coq-contribs/LAMBDA-TYPES/subst1_subst1.v b/helm/coq-contribs/LAMBDA-TYPES/subst1_subst1.v
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+(*#* #stop file *)
+
+Require subst0_subst0.
+Require subst1_defs.
+
+   Section subst1_subst1. (**************************************************)
+
+      Theorem subst1_subst1: (t1,t2,u2:?; j:?) (subst1 j u2 t1 t2) ->
+                             (u1,u:?; i:?) (subst1 i u u1 u2) ->
+                             (EX t | (subst1 j u1 t1 t) & (subst1 (S (plus i j)) u t t2)).
+      Intros until 1; XElim H; Clear t2.
+(* case 1: subst1_refl on first premise *)
+      XEAuto.
+(* case 2: subst1_single on first premise *)
+      Intros until 2; InsertEq H0 u2; XElim H0; Clear y; Intros.
+(* case 2.1: subst1_refl on second premise *)
+      Rewrite H0; Clear H0 u1; XEAuto.
+(* case 2.2: subst1_single on second premise *)
+      Rewrite H1 in H0; Clear H1 t0; Subst0Subst0; XEAuto.
+      Qed.
+
+      Theorem subst1_subst1_back: (t1,t2,u2:?; j:?) (subst1 j u2 t1 t2) ->
+                                  (u1,u:?; i:?) (subst1 i u u2 u1) ->
+                                  (EX t | (subst1 j u1 t1 t) & (subst1 (S (plus i j)) u t2 t)).
+      Intros until 1; XElim H; Clear t2.
+(* case 1: subst1_refl on first premise *)
+      XEAuto.
+(* case 2: subst1_single on first premise *)
+      Intros until 2; XElim H0; Clear u1; Intros;
+      Try Subst0Subst0; XEAuto.
+      Qed.
+
+      Theorem subst1_trans: (t2,t1,v:?; i:?) (subst1 i v t1 t2) ->
+                            (t3:?) (subst1 i v t2 t3) ->
+                            (subst1 i v t1 t3).
+      Intros until 1; XElim H; Clear t2.
+(* case 1: subst1_refl on first premise *)
+      XEAuto.
+(* case 2: subst1_single on first premise *)
+      Intros until 2; XElim H0; Clear t3; XEAuto.
+      Qed.
+
+   End subst1_subst1.
+
+      Hints Resolve subst1_trans : ltlc.
+
+      Tactic Definition Subst1Subst1 :=
+         Match Context With
+            | [ H1: (subst1 ?0 ?1 ?2 ?3); H2: (subst1 ?4 ?5 ?6 ?1) |- ? ] ->
+               LApply (subst1_subst1 ?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: (subst1 ?0 ?1 ?2 ?3); H2: (subst0 ?4 ?5 ?6 ?1) |- ? ] ->
+               LApply (subst1_subst1 ?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: (subst1 ?0 ?1 ?2 ?3); H2: (subst1 ?4 ?5 ?1 ?6) |- ? ] ->
+               LApply (subst1_subst1_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.