contribution about \lambda-\delta

[helm.git] / helm / coq-contribs / LAMBDA-TYPES / pr3_props.v

diff --git a/helm/coq-contribs/LAMBDA-TYPES/pr3_props.v b/helm/coq-contribs/LAMBDA-TYPES/pr3_props.v
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+Require subst0_subst0.
+Require pr0_subst0.
+Require pr2_lift.
+Require pr3_defs.
+
+(*#* #caption "main properties of predicate \\texttt{pr3}" *)
+(*#* #clauses *)
+
+(*#* #stop file *)
+
+   Section pr3_context. (****************************************************)
+
+      Theorem pr3_pr0_pr2_t : (u1,u2:?) (pr0 u1 u2) ->
+                              (c:?; t1,t2:?; k:?) (pr2 (CTail c k u2) t1 t2) ->
+                              (pr3 (CTail c k u1) t1 t2).
+      Intros.
+      Inversion H0; Clear H0; XAuto.
+      NewInduction i.
+(* case 1 : pr2_delta i = 0 *)
+      DropGenBase; Inversion H0; Clear H0 H3 H4 c k.
+      Rewrite H5 in H; Clear H5 u2.
+      Pr0Subst0; XEAuto.
+(* case 2 : pr2_delta i > 0 *)
+      NewInduction k; DropGenBase; XEAuto.
+      Qed.
+
+      Theorem pr3_pr2_pr2_t : (c:?; u1,u2:?) (pr2 c u1 u2) ->
+                              (t1,t2:?; k:?) (pr2 (CTail c k u2) t1 t2) ->
+                              (pr3 (CTail c k u1) t1 t2).
+      Intros until 1; Inversion H; Clear H; Intros.
+(* case 1 : pr2_pr0 *)
+      EApply pr3_pr0_pr2_t; [ Apply H0 | XAuto ].
+(* case 2 : pr2_delta *)
+      Inversion H; [ XAuto | NewInduction i0 ].
+(* case 2.1 : i0 = 0 *)
+      DropGenBase; Inversion H2; Clear H2.
+      Rewrite <- H5; Rewrite H6 in H; Rewrite <- H7 in H3; Clear H5 H6 H7 d0 k u0.
+      Subst0Subst0; Arith9'In H4 i; XDEAuto 7.
+(* case 2.2 : i0 > 0 *)
+      Clear IHi0; NewInduction k; DropGenBase; XEAuto.
+      Qed.
+
+      Theorem pr3_pr2_pr3_t : (c:?; u2,t1,t2:?; k:?)
+                              (pr3 (CTail c k u2) t1 t2) ->
+                              (u1:?) (pr2 c u1 u2) ->
+                              (pr3 (CTail c k u1) t1 t2).
+      Intros until 1; XElim H; Intros.
+(* case 1 : pr3_r *)
+      XAuto.
+(* case 2 : pr3_u *)
+      EApply pr3_t.
+      EApply pr3_pr2_pr2_t; [ Apply H2 | Apply H ].
+      XAuto.
+      Qed.
+
+(*#* #start file *)
+
+(*#* #caption "reduction inside context items" *)
+(*#* #cap #cap t1, t2 #alpha c in E, u1 in V1, u2 in V2, k in z *)
+
+      Theorem pr3_pr3_pr3_t : (c:?; u1,u2:?) (pr3 c u1 u2) ->
+                              (t1,t2:?; k:?) (pr3 (CTail c k u2) t1 t2) ->
+                              (pr3 (CTail c k u1) t1 t2).
+
+(*#* #stop file *)
+
+      Intros until 1; XElim H; Intros.
+(* case 1 : pr3_r *)
+      XAuto.
+(* case 2 : pr3_u *)
+      EApply pr3_pr2_pr3_t; [ Apply H1; XAuto | XAuto ].
+      Qed.
+
+   End pr3_context.
+
+      Tactic Definition Pr3Context :=
+         Match Context With
+            | [ H1: (pr0 ?2 ?3); H2: (pr2 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
+               LApply (pr3_pr0_pr2_t ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
+               LApply (H1 ?1 ?5 ?6 ?4); [ Clear H1 H2; Intros | XAuto ]
+            | [ H1: (pr0 ?2 ?3); H2: (pr3 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
+               LApply (pr3_pr2_pr3_t ?1 ?3 ?5 ?6 ?4); [ Clear H2; Intros H2 | XAuto ];
+               LApply (H2 ?2); [ Clear H1 H2; Intros | XAuto ]
+            | [ H1: (pr2 ?1 ?2 ?3); H2: (pr2 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
+               LApply (pr3_pr2_pr2_t ?1 ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
+               LApply (H1 ?5 ?6 ?4); [ Clear H1 H2; Intros | XAuto ]
+            | [ H1: (pr2 ?1 ?2 ?3); H2: (pr3 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
+               LApply (pr3_pr2_pr3_t ?1 ?3 ?5 ?6 ?4); [ Clear H2; Intros H2 | XAuto ];
+               LApply (H2 ?2); [ Clear H1 H2; Intros | XAuto ]
+            | [ H1: (pr3 ?1 ?2 ?3); H2: (pr3 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
+               LApply (pr3_pr3_pr3_t ?1 ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
+               LApply (H1 ?5 ?6 ?4); [ Clear H1 H2; Intros | XAuto ].
+
+   Section pr3_lift. (*******************************************************)
+
+(*#* #start file *)
+
+(*#* #caption "conguence with lift" *)
+(*#* #cap #cap c, t1, t2 #alpha e in D, d in i *)
+
+      Theorem pr3_lift : (c,e:?; h,d:?) (drop h d c e) ->
+                         (t1,t2:?) (pr3 e t1 t2) ->
+                         (pr3 c (lift h d t1) (lift h d t2)).
+
+(*#* #stop file *)
+
+      Intros until 2; XElim H0; Intros; XEAuto.
+      Qed.
+
+   End pr3_lift.
+
+      Hints Resolve pr3_lift : ltlc.
+