contribution about \lambda-\delta

[helm.git] / helm / coq-contribs / LAMBDA-TYPES / subst1_confluence.v

diff --git a/helm/coq-contribs/LAMBDA-TYPES/subst1_confluence.v b/helm/coq-contribs/LAMBDA-TYPES/subst1_confluence.v
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+(*#* #stop file *)
+
+Require lift_gen.
+Require subst0_gen.
+Require subst0_confluence.
+Require subst1_defs.
+Require subst1_gen.
+
+   Section subst1_confluence. (**********************************************)
+
+      Theorem subst1_confluence_neq: (t0,t1,u1:?; i1:?) (subst1 i1 u1 t0 t1) ->
+                                     (t2,u2:?; i2:?) (subst1 i2 u2 t0 t2) ->
+                                     ~i1=i2 ->
+                                     (EX t | (subst1 i2 u2 t1 t) &
+                                             (subst1 i1 u1 t2 t)
+                                     ).
+      Intros until 1; XElim H; Clear t1; Intros.
+(* case 1; subst1_refl *)
+      XEAuto.
+(* case 2; subst1_single *)
+      XElim H0; Intros; Try Subst0Confluence; XEAuto.
+      Qed.
+
+      Theorem subst1_confluence_eq : (t0,t1,u:?; i:?) (subst1 i u t0 t1) ->
+                                     (t2:?) (subst1 i u t0 t2) ->
+                                     (EX t | (subst1 i u t1 t) &
+                                             (subst1 i u t2 t)
+                                     ).
+      Intros until 1; XElim H; Intros.
+(* case 1; subst1_refl *)
+      XEAuto.
+(* case 2; subst1_single *)
+      XElim H0; Intros;
+      Try Subst0Confluence; Try Rewrite H0; XEAuto.
+      Qed.
+
+      Theorem subst1_confluence_lift: (t0,t1,u:?; i:?) (subst1 i u t0 (lift (1) i t1)) ->
+                                      (t2:?) (subst1 i u t0 (lift (1) i t2)) ->
+                                      t1 = t2.
+      Intros until 1; InsertEq H '(lift (1) i t1); XElim H; Clear y; Intros.
+(* case 1: subst1_refl *)
+      Rewrite H in H0; Clear H t0.
+      Subst1Gen; SymEqual; LiftGen; XEAuto.
+(* case 2: subst1_single *)
+      Rewrite H0 in H; Clear H0 t2.
+      InsertEq H1 '(lift (1) i t3); XElim H0; Clear y; Intros.
+(* case 2.1: subst1_refl *)
+      Rewrite H0 in H; Clear H0 t0; Subst0Gen.
+(* case 2.2: subst1_single *)
+      Rewrite H1 in H0; Clear H1 t2; Subst0ConfluenceLift; XAuto.
+      Qed.
+
+   End subst1_confluence.
+
+      Tactic Definition Subst1Confluence :=
+         Match Context With
+            | [ H0: (subst1 ?1 ?2 ?3 (lift (1) ?1 ?4));
+                H1: (subst1 ?1 ?2 ?3 (lift (1) ?1 ?5)) |- ? ] ->
+              LApply (subst1_confluence_lift ?3 ?4 ?2 ?1); [ Clear H0; Intros H0 | XAuto ];
+              LApply (H0 ?5); [ Clear H0; Intros | XAuto ]
+            | [ H0: (subst1 ?1 ?2 ?3 ?4);
+                H1: (subst1 ?1 ?2 ?3 ?5) |- ? ] ->
+              LApply (subst1_confluence_eq ?3 ?4 ?2 ?1); [ Clear H0; Intros H0 | XAuto ];
+              LApply (H0 ?5); [ Clear H0; Intros H0 | XAuto ];
+              XElim H0; Intros
+            | [ H0: (subst0 ?1 ?2 ?3 ?4);
+                H1: (subst1 ?1 ?2 ?3 ?5) |- ? ] ->
+              LApply (subst1_confluence_eq ?3 ?4 ?2 ?1); [ Clear H0; Intros H0 | XAuto ];
+              LApply (H0 ?5); [ Clear H0; Intros H0 | XAuto ];
+              XElim H0; Intros
+            | [ H0: (subst1 ?1 ?2 ?3 ?4);
+                H1: (subst1 ?5 ?6 ?3 ?7) |- ? ] ->
+              LApply (subst1_confluence_neq ?3 ?4 ?2 ?1); [ Clear H0; Intros H0 | XAuto ];
+              LApply (H0 ?7 ?6 ?5); [ Clear H0 H1; Intros H0 | XAuto ];
+              LApply H0; [ Clear H0; Intros H0 | XAuto ];
+              XElim H0; Intros
+            | [ H0: (subst0 ?1 ?2 ?3 ?4);
+                H1: (subst1 ?5 ?6 ?3 ?7) |- ? ] ->
+              LApply (subst1_confluence_neq ?3 ?4 ?2 ?1); [ Clear H0; Intros H0 | XAuto ];
+              LApply (H0 ?7 ?6 ?5); [ Clear H0 H1; Intros H0 | XAuto ];
+              LApply H0; [ Clear H0; Intros H0 | XAuto ];
+              XElim H0; Intros.