contribution about \lambda-\delta

[helm.git] / helm / coq-contribs / LAMBDA-TYPES / pr0_subst1.v

diff --git a/helm/coq-contribs/LAMBDA-TYPES/pr0_subst1.v b/helm/coq-contribs/LAMBDA-TYPES/pr0_subst1.v
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+(*#* #stop file *)
+
+Require subst1_defs.
+Require pr0_defs.
+Require pr0_subst0.
+
+   Section pr0_subst1_props. (***********************************************)
+
+      Theorem pr0_delta1: (u1,u2:?) (pr0 u1 u2) -> (t1,t2:?) (pr0 t1 t2) ->
+                          (w:?) (subst1 (0) u2 t2 w) ->
+                          (pr0 (TTail (Bind Abbr) u1 t1) (TTail (Bind Abbr) u2 w)).
+      Intros until 3; XElim H1; Clear w; XEAuto.
+      Qed.
+
+      Theorem pr0_subst1_back: (u2,t1,t2:?; i:?) (subst1 i u2 t1 t2) ->
+                               (u1:?) (pr0 u1 u2) ->
+                               (EX t | (subst1 i u1 t1 t) & (pr0 t t2)).
+      Intros until 1; XElim H; Intros;
+      Try Pr0Subst0; XEAuto.
+      Qed.
+
+      Theorem pr0_subst1_fwd: (u2,t1,t2:?; i:?) (subst1 i u2 t1 t2) ->
+                              (u1:?) (pr0 u2 u1) ->
+                              (EX t | (subst1 i u1 t1 t) & (pr0 t2 t)).
+      Intros until 1; XElim H; Intros;
+      Try Pr0Subst0; XEAuto.
+      Qed.
+
+      Theorem pr0_subst1: (t1,t2:?) (pr0 t1 t2) ->
+                          (v1,w1:?; i:?) (subst1 i v1 t1 w1) ->
+                          (v2:?) (pr0 v1 v2) ->
+                          (EX w2 | (pr0 w1 w2) & (subst1 i v2 t2 w2)).
+      Intros until 2; XElim H0; Intros;
+      Try Pr0Subst0; XEAuto.
+      Qed.
+
+   End pr0_subst1_props.
+
+      Hints Resolve pr0_delta1 : ltlc.
+
+      Tactic Definition Pr0Subst1 :=
+         Match Context With
+            | [ H1: (pr0 ?1 ?2); H2: (subst1 ?3 ?4 ?1 ?5);
+                H3: (pr0 ?4 ?6) |- ? ] ->
+               LApply (pr0_subst1 ?1 ?2); [ Clear H1; Intros H1 | XAuto ];
+               LApply (H1 ?4 ?5 ?3); [ Clear H1 H2; Intros H1 | XAuto ];
+               LApply (H1 ?6); [ Clear H1; Intros H1 | XAuto ];
+               XElim H1; Intros
+            | [ H1: (pr0 ?1 ?2); H2: (subst1 ?3 ?4 ?1 ?5) |- ? ] ->
+               LApply (pr0_subst1 ?1 ?2); [ Clear H1; Intros H1 | XAuto ];
+               LApply (H1 ?4 ?5 ?3); [ Clear H1 H2; Intros H1 | XAuto ];
+               LApply (H1 ?4); [ Clear H1; Intros H1 | XAuto ];
+               XElim H1; Intros
+            | [ H1: (subst1 ?0 ?1 ?2 ?3); H2: (pr0 ?4 ?1) |- ? ] ->
+               LApply (pr0_subst1_back ?1 ?2 ?3 ?0); [ Clear H1; Intros H1 | XAuto ];
+               LApply (H1 ?4); [ Clear H1 H2; Intros H1 | XAuto ];
+               XElim H1; Intros
+            | [ H1: (subst1 ?0 ?1 ?2 ?3); H2: (pr0 ?1 ?4) |- ? ] ->
+               LApply (pr0_subst1_fwd ?1 ?2 ?3 ?0); [ Clear H1; Intros H1 | XAuto ];
+               LApply (H1 ?4); [ Clear H1 H2; Intros H1 | XAuto ];
+               XElim H1; Intros
+            | _ -> Pr0Subst0.
+