X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fcoq-contribs%2FLAMBDA-TYPES%2Fpr3_props.v;h=b5c7df9375c9d586a7c7d5407d176b1767ae893c;hb=4167cea65ca58897d1a3dbb81ff95de5074700cc;hp=106bfe66cab6a2851a247ee4e8b7aa8637e27770;hpb=249d79bebff886846fbab65cc079623d90684baf;p=helm.git

diff --git a/helm/coq-contribs/LAMBDA-TYPES/pr3_props.v b/helm/coq-contribs/LAMBDA-TYPES/pr3_props.v
index 106bfe66c..b5c7df937 100644
--- a/helm/coq-contribs/LAMBDA-TYPES/pr3_props.v
+++ b/helm/coq-contribs/LAMBDA-TYPES/pr3_props.v
@@ -1,73 +1,70 @@
 Require subst0_subst0.
 Require pr0_subst0.
+Require cpr0_defs.
 Require pr2_lift.
+Require pr2_gen.
 Require pr3_defs.
 
-(*#* #caption "main properties of predicate \\texttt{pr3}" *)
+(*#* #caption "main properties of the relation $\\PrT{}{}{}$" *)
 (*#* #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.
+      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.
+(* case 1 : pr2_delta i = 0 *) 
+      DropGenBase; Inversion H0; Clear H0 H4 H5 H6 c k t.
+      Rewrite H7 in H; Clear H7 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).
+      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 *)
+(* case 1 : pr2_free *)
       EApply pr3_pr0_pr2_t; [ Apply H0 | XAuto ].
 (* case 2 : pr2_delta *)
-      Inversion H; [ XAuto | NewInduction i0 ].
+      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.
+      DropGenBase; Inversion H4; Clear H3 H4 H7 t t4.
+      Rewrite <- H9; Rewrite H10 in H; Rewrite <- H11 in H6; Clear H9 H10 H11 d0 k u0.
+      Subst0Subst0; Arith9'In H4 i; Clear H2 H H6 u2.
+      Pr0Subst0; Apply pr3_sing with t2:=x0; XEAuto.
 (* 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).
+      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 *)
+(* case 1 : pr3_refl *)
       XAuto.
-(* case 2 : pr3_u *)
+(* case 2 : pr3_sing *)
       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 *)
-
+      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).
       Intros until 1; XElim H; Intros.
-(* case 1 : pr3_r *)
+(* case 1 : pr3_refl *)
       XAuto.
-(* case 2 : pr3_u *)
+(* case 2 : pr3_sing *)
       EApply pr3_pr2_pr3_t; [ Apply H1; XAuto | XAuto ].
       Qed.
 
@@ -76,34 +73,29 @@ Require pr3_defs.
       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 ]
+               LApply (pr3_pr0_pr2_t ?2 ?3); [ Intros H_x | XAuto ];
+               LApply (H_x ?1 ?5 ?6 ?4); [ Clear H_x 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 ]
+               LApply (H2 ?2); [ Clear 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 ]
+               LApply (pr3_pr2_pr2_t ?1 ?2 ?3); [ Intros H_x | XAuto ];
+               LApply (H_x ?5 ?6 ?4); [ Clear H_x 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 ]
+               LApply (H2 ?2); [ Clear 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 ].
+               LApply (pr3_pr3_pr3_t ?1 ?2 ?3); [ Intros H_x | XAuto ];
+               LApply (H_x ?5 ?6 ?4); [ Clear H_x 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 *)
-
+      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)).
       Intros until 2; XElim H0; Intros; XEAuto.
       Qed.
 
@@ -111,4 +103,25 @@ Require pr3_defs.
 
       Hints Resolve pr3_lift : ltlc.
 
-*)
+   Section pr3_cpr0. (*******************************************************)
+
+      Theorem pr3_cpr0_t: (c1,c2:?) (cpr0 c2 c1) -> (t1,t2:?) (pr3 c1 t1 t2) ->
+                          (pr3 c2 t1 t2).
+      Intros until 1; XElim H; Intros.
+(* case 1 : cpr0_refl *)
+      XAuto.
+(* case 2 : cpr0_comp *)
+      Pr3Context; Clear H1.
+      XElim H2; Intros.
+(* case 2.1 : pr3_refl *)
+      XAuto.
+(* case 2.2 : pr3_sing *)
+      EApply pr3_t; [ Idtac | XEAuto ]. Clear H2 H3 c1 c2 t1 t2 t4 u2.
+      Inversion_clear H1.
+(* case 2.2.1 : pr2_free *)
+      XAuto.
+(* case 2.2.1 : pr2_delta *)
+      Cpr0Drop; Pr0Subst0; Apply pr3_sing with t2:=x; XEAuto.
+      Qed.
+
+   End pr3_cpr0.