contribution about \lambda-\delta

[helm.git] / helm / coq-contribs / LAMBDA-TYPES / pr2_gen_context.v

diff --git a/helm/coq-contribs/LAMBDA-TYPES/pr2_gen_context.v b/helm/coq-contribs/LAMBDA-TYPES/pr2_gen_context.v
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+(*#* #stop file *)
+
+Require drop_props.
+Require subst1_gen.
+Require subst1_subst1.
+Require subst1_confluence.
+Require csubst1_defs.
+Require pr0_gen.
+Require pr0_subst1.
+Require pr2_defs.
+Require pr2_subst1.
+
+   Section pr2_gen_context. (************************************************)
+
+      Theorem pr2_gen_cabbr: (c:?; t1,t2:?) (pr2 c t1 t2) -> (e:?; u:?; d:?)
+                             (drop d (0) c (CTail e (Bind Abbr) u)) ->
+                             (a0:?) (csubst1 d u c a0) ->
+                             (a:?) (drop (1) d a0 a) ->
+                             (x1:?) (subst1 d u t1 (lift (1) d x1)) ->
+                             (EX x2 | (subst1 d u t2 (lift (1) d x2)) &
+                                      (pr2 a x1 x2)
+                             ).
+      Intros until 1; XElim H; Intros.
+(* case 1: pr2_pr0 *)
+      Pr0Subst1; Pr0Gen; Rewrite H in H3; Clear H x; XEAuto.
+(* case 2: pr2_delta *)
+      Apply (lt_eq_gt_e i d0); Intros.
+(* case 2.1: i < d0 *)
+      Subst1Confluence; CSubst1Drop.
+      Rewrite minus_x_Sy in H; [ Idtac | XAuto ].
+      CSubst1GenBase; Rewrite H in H6; Clear H x0.
+      Rewrite (lt_plus_minus i d0) in H3; [ Idtac | XAuto ].
+      DropDis; Rewrite H in H7; Clear H x2.
+      Subst1Subst1; Pattern 2 d0 in H; Rewrite (lt_plus_minus i d0) in H; [ Idtac | XAuto ].
+      Subst1Gen; Rewrite H in H9; Simpl in H9; Clear H x2.
+      Rewrite <- lt_plus_minus in H9; [ Idtac | XAuto ].
+      Rewrite <- lt_plus_minus_r in H9; XEAuto.
+(* case 2.2: i = d0 *)
+      Rewrite H5 in H; Rewrite H5 in H0; Clear H5 i.
+      DropDis; Inversion H; Rewrite <- H7 in H0; Rewrite <- H7 in H1; Rewrite <- H7; Clear H H6 H7 e u.
+      Subst1Confluence; Subst1Gen; Rewrite H0 in H; Clear H0 x; XEAuto.
+(* case 2.3: i > d0 *)
+      Subst1Confluence; Subst1Gen; Rewrite H4 in H0; Clear H1 H4 x.
+      CSubst1Drop; DropDis; XEAuto.
+      Qed.
+
+   End pr2_gen_context.
+
+      Tactic Definition Pr2GenContext :=
+         Match Context With
+            | [ H0: (pr2 ?1 ?2 ?3); H1: (drop ?4 (0) ?1 (CTail ?5 (Bind Abbr) ?6));
+                H2: (csubst1 ?4 ?6 ?1 ?7); H3: (drop (1) ?4 ?7 ?8);
+                H4: (subst1 ?4 ?6 ?2 (lift (1) ?4 ?9)) |- ? ] ->
+               LApply (pr2_gen_cabbr ?1 ?2 ?3); [ Clear H0; Intros H0 | XAuto ];
+               LApply (H0 ?5 ?6 ?4); [ Clear H0; Intros H0 | XAuto ];
+               LApply (H0 ?7); [ Clear H0; Intros H0 | XAuto ];
+               LApply (H0 ?8); [ Clear H0; Intros H0 | XAuto ];
+               LApply (H0 ?9); [ Clear H0 H4; Intros H0 | XAuto ];
+               XElim H0; Intros.