contribution about \lambda-\delta

[helm.git] / helm / coq-contribs / LAMBDA-TYPES / subst0_confluence.v

diff --git a/helm/coq-contribs/LAMBDA-TYPES/subst0_confluence.v b/helm/coq-contribs/LAMBDA-TYPES/subst0_confluence.v
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+++ b/helm/coq-contribs/LAMBDA-TYPES/subst0_confluence.v
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+(*#* #stop file *)
+
+Require lift_gen.
+Require subst0_gen.
+Require subst0_defs.
+
+   Section subst0_confluence. (**********************************************)
+
+      Tactic Definition IH :=
+         Match Context With
+            | [ H1: (t2,u2:?; i2:?) (subst0 i2 u2 ?1 t2) -> ~?2=i2 -> ?;
+                H2: (subst0 ?3 ?4 ?1 ?5); H3: ~?2=?3 |- ? ] ->
+                 LApply (H1 ?5 ?4 ?3); [ Clear H1; Intros H1 | XAuto ];
+                 LApply H1; [ Clear H1; Intros H1 | XAuto ];
+                 XElim H1; Intros
+            | [ H1: (t2,u2:?; i2:?) (subst0 i2 u2 ?1 t2) -> ~(s k ?2)=i2 -> ?;
+                H2: (subst0 (s k ?3) ?4 ?1 ?5); H3: ~?2=?3 |- ? ] ->
+                 LApply (H1 ?5 ?4 (s k ?3)); [ Clear H1; Intros H1 | XAuto ];
+                 LApply H1; [ Clear H1; Intros H1 | Unfold not in H3; Unfold not; XEAuto ];
+                 XElim H1; Intros
+            | [ H1: (t2:T) (subst0 ?1 ?2 ?3 t2) -> ?; H2: (subst0 ?1 ?2 ?3 ?4) |- ? ] ->
+                 LApply (H1 ?4); [ Clear H1; Intros H1 | XAuto ];
+                 XElim H1; Intros H1; [ Try Rewrite H1 | XElim H1; Intros | Idtac | Idtac ].
+
+      Theorem subst0_confluence_neq : (t0,t1,u1:?; i1:?) (subst0 i1 u1 t0 t1) ->
+                                      (t2,u2:?; i2:?) (subst0 i2 u2 t0 t2) ->
+                                      ~i1=i2 ->
+                                      (EX t | (subst0 i2 u2 t1 t) &
+                                              (subst0 i1 u1 t2 t)).
+
+      Intros until 1; XElim H; Intros;
+      Subst0GenBase; Try Rewrite H in H0; Try Rewrite H1; Try Rewrite H3;
+      Try EqFalse; Repeat IH; XEAuto.
+      Qed.
+
+      Theorem subst0_confluence_eq : (t0,t1,u:?; i:?) (subst0 i u t0 t1) ->
+                                     (t2:?) (subst0 i u t0 t2) -> (OR
+                                     t1 = t2 |
+                                     (EX t | (subst0 i u t1 t) & (subst0 i u t2 t)) |
+                                     (subst0 i u t1 t2) |
+                                     (subst0 i u t2 t1)).
+      Intros until 1; XElim H; Intros;
+      Subst0GenBase; Try Rewrite H1; Try Rewrite H3;
+      Repeat IH; XEAuto.
+      Qed.
+
+   End subst0_confluence.
+
+      Tactic Definition Subst0Confluence :=
+         Match Context With
+            | [ H0: (subst0 ?1 ?2 ?3 ?4);
+                H1: (subst0 ?1 ?2 ?3 ?5) |- ? ] ->
+              LApply (subst0_confluence_eq ?3 ?4 ?2 ?1); [ Clear H0; Intros H0 | XAuto ];
+              LApply (H0 ?5); [ Clear H0; Intros H0 | XAuto ];
+              XElim H0; [ Intros | Intros H0; XElim H0; Intros | Intros | Intros ]
+            | [ H0: (subst0 ?1 ?2 ?3 ?4);
+                H1: (subst0 ?5 ?6 ?3 ?7) |- ? ] ->
+              LApply (subst0_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 | Simpl; XAuto ];
+              XElim H0; Intros.
+
+   Section subst0_confluence_lift. (*****************************************)
+
+      Theorem subst0_confluence_lift: (t0,t1,u:?; i:?) (subst0 i u t0 (lift (1) i t1)) ->
+                                      (t2:?) (subst0 i u t0 (lift (1) i t2)) ->
+                                      t1 = t2.
+      Intros; Subst0Confluence;
+      Try Subst0Gen; SymEqual; LiftGen; XEAuto.
+      Qed.
+
+   End subst0_confluence_lift.
+
+      Tactic Definition Subst0ConfluenceLift :=
+         Match Context With
+            | [ H0: (subst0 ?1 ?2 ?3 (lift (1) ?1 ?4));
+                H1: (subst0 ?1 ?2 ?3 (lift (1) ?1 ?5)) |- ? ] ->
+              LApply (subst0_confluence_lift ?3 ?4 ?2 ?1); [ Clear H0; Intros H0 | XAuto ];
+              LApply (H0 ?5); [ Clear H0; Intros | XAuto ]
+            | _ -> Subst0Confluence.