some more changes and corrections

[helm.git] / helm / coq-contribs / LAMBDA-TYPES / ty0_gen.v

diff --git a/helm/coq-contribs/LAMBDA-TYPES/ty0_gen.v b/helm/coq-contribs/LAMBDA-TYPES/ty0_gen.v
new file mode 100644 (file)
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+++ b/helm/coq-contribs/LAMBDA-TYPES/ty0_gen.v
@@ -0,0 +1,185 @@
+Require pc3_props.
+Require ty0_defs.
+
+(*#* #caption "generation lemma of typing" #clauses *)
+
+   Section ty0_gen_base. (***************************************************)
+
+(*#* #caption "generation lemma for sorts" *)
+(*#* #cap #cap c #alpha x in T, n in h *)
+
+      Theorem ty0_gen_sort: (g:?; c:?; x:?; n:?)
+                            (ty0 g c (TSort n) x) ->
+                            (pc3 c (TSort (next g n)) x).
+                            
+(*#* #stop proof *) 
+
+      Intros until 1; InsertEq H '(TSort n); XElim H; Intros.
+(* case 1 : ty0_conv *)
+      XEAuto.
+(* case 2 : ty0_sort *)
+      Inversion H0; XAuto.
+(* case 3 : ty0_abbr *)
+      Inversion H3.
+(* case 4 : ty0_abst *)
+      Inversion H3.
+(* case 5 : ty0_bind *)
+      Inversion H5.
+(* case 6 : ty0_appl *)
+      Inversion H3.
+(* case 7 : ty0_cast *)
+      Inversion H3.
+      Qed.
+
+(*#* #start proof *) 
+
+(*#* #caption "generation lemma for bound references" *)
+(*#* #cap #cap c, e #alpha x in T, t in U, u in V, n in i *)
+
+      Theorem ty0_gen_lref: (g:?; c:?; x:?; n:?) (ty0 g c (TLRef n) x) ->
+                            (EX e u t | (pc3 c (lift (S n) (0) t) x) &
+                                        (drop n (0) c (CTail e (Bind Abbr) u)) &
+                                        (ty0 g e u t)
+                            ) \/
+                            (EX e u t | (pc3 c (lift (S n) (0) u) x) &
+                                        (drop n (0) c (CTail e (Bind Abst) u)) &
+                                        (ty0 g e u t)
+                            ).
+
+(*#* #stop proof *) 
+                            
+      Intros until 1; InsertEq H '(TLRef n); XElim H; Intros.
+(* case 1 : ty0_conv *)
+      LApply H2; [ Clear H2; Intros H2 | XAuto ].
+      XElim H2; Intros; XElim H2; XEAuto.
+(* case 2 : ty0_sort *)
+      Inversion H0.
+(* case 3 : ty0_abbr *)
+      Inversion H3 ; Rewrite H5 in H0; XEAuto.
+(* case 4 : ty0_abst *)
+      Inversion H3; Rewrite H5 in H0; XEAuto.
+(* case 5 : ty0_bind *)
+      Inversion H5.
+(* case 6 : ty0_appl *)
+      Inversion H3.
+(* case 7 : ty0_cast *)
+      Inversion H3.
+      Qed.
+
+(*#* #start proof *) 
+
+(*#* #caption "generation lemma for binders" *)
+(*#* #cap #cap c #alpha x in T, t1 in U1, t2 in U2, u in V, t in U, t0 in U3 *)
+
+      Theorem ty0_gen_bind: (g:?; b:?; c:?; u,t1,x:?) (ty0 g c (TTail (Bind b) u t1) x) ->
+                            (EX t2 t t0 | (pc3 c (TTail (Bind b) u t2) x) &
+                                          (ty0 g c u t) &
+                                          (ty0 g (CTail c (Bind b) u) t1 t2) &
+                                          (ty0 g (CTail c (Bind b) u) t2 t0)
+                            ).
+      
+(*#* #stop proof *) 
+      
+      Intros until 1; InsertEq H '(TTail (Bind b) u t1); XElim H; Intros.
+(* case 1 : ty0_conv *)
+      LApply H2; [ Clear H2; Intros H2 | XAuto ].
+      XElim H2; XEAuto.
+(* case 2 : ty0_sort *)
+      Inversion H0.
+(* case 3 : ty0_abbr *)
+      Inversion H3.
+(* case 4 : ty0_abst *)
+      Inversion H3.
+(* case 5 : ty0_bind *)
+      Inversion H5.
+      Rewrite H7 in H1; Rewrite H7 in H3.
+      Rewrite H8 in H; Rewrite H8 in H1; Rewrite H8 in H3.
+      Rewrite H9 in H1; XEAuto.
+(* case 6 : ty0_appl *)
+      Inversion H3.
+(* case 7 : ty0_cast *)
+      Inversion H3.
+      Qed.
+
+(*#* #start proof *) 
+
+(*#* #caption "generation lemma for applications" *)
+(*#* #cap #cap c #alpha x in T, v in U1, w in V1, u in V2, t in U2 *)
+
+      Theorem ty0_gen_appl: (g:?; c:?; w,v,x:?) (ty0 g c (TTail (Flat Appl) w v) x) ->
+                            (EX u t | (pc3 c (TTail (Flat Appl) w (TTail (Bind Abst) u t)) x) &
+                                      (ty0 g c v (TTail (Bind Abst) u t)) &
+                                      (ty0 g c w u)
+                            ).
+                            
+(*#* #stop proof *) 
+                            
+      Intros until 1; InsertEq H '(TTail (Flat Appl) w v); XElim H; Intros.
+(* case 1 : ty0_conv *)
+      LApply H2; [ Clear H2; Intros H2 | XAuto ].
+      XElim H2; XEAuto.
+(* case 2 : ty0_sort *)
+      Inversion H0.
+(* case 3 : ty0_abbr *)
+      Inversion H3.
+(* case 4 : ty0_abst *)
+      Inversion H3.
+(* case 5 : ty0_bind *)
+      Inversion H5.
+(* case 6 : ty0_appl *)
+      Inversion H3; Rewrite H5 in H; Rewrite H6 in H1; XEAuto.
+(* case 7 : ty0_cast *)
+      Inversion H3.
+      Qed.
+
+(*#* #start proof *) 
+
+(*#* #caption "generation lemma for type casts" *)
+(*#* #cap #cap c #alpha x in T, t2 in V, t1 in U *)
+
+      Theorem ty0_gen_cast: (g:?; c:?; t1,t2,x:?)
+                            (ty0 g c (TTail (Flat Cast) t2 t1) x) ->
+                            (pc3 c t2 x) /\ (ty0 g c t1 t2).
+      
+(*#* #stop proof *) 
+
+      Intros until 1; InsertEq H '(TTail (Flat Cast) t2 t1); XElim H; Intros.
+(* case 1 : ty0_conv *)
+      LApply H2; [ Clear H2; Intros H2 | XAuto ].
+      XElim H2; XEAuto.
+(* case 2 : ty0_sort *)
+      Inversion H0.
+(* case 3 : ty0_abbr *)
+      Inversion H3.
+(* case 4 : ty0_abst *)
+      Inversion H3.
+(* case 5 : ty0_bind *)
+      Inversion H5.
+(* case 6 : ty0_appl *)
+      Inversion H3.
+(* case 7 : ty0_cast *)
+      Inversion H3; Rewrite H5 in H; Rewrite H5 in H1; Rewrite H6 in H; XAuto.
+      Qed.
+
+   End ty0_gen_base.
+
+      Tactic Definition Ty0GenBase :=
+         Match Context With
+            | [ H: (ty0 ?1 ?2 (TSort ?3) ?4) |- ? ] ->
+               LApply (ty0_gen_sort ?1 ?2 ?4 ?3); [ Clear H; Intros | XAuto ]
+            | [ H: (ty0 ?1 ?2 (TLRef ?3) ?4) |- ? ] ->
+               LApply (ty0_gen_lref ?1 ?2 ?4 ?3); [ Clear H; Intros H | XAuto ];
+               XElim H; Intros H; XElim H; Intros
+            | [ H: (ty0 ?1 ?2 (TTail (Bind ?3) ?4 ?5) ?6) |- ? ] ->
+               LApply (ty0_gen_bind ?1 ?3 ?2 ?4 ?5 ?6); [ Clear H; Intros H | XAuto ];
+               XElim H; Intros
+            | [ H: (ty0 ?1 ?2 (TTail (Flat Appl) ?3 ?4) ?5) |- ? ] ->
+               LApply (ty0_gen_appl ?1 ?2 ?3 ?4 ?5); [ Clear H; Intros H | XAuto ];
+               XElim H; Intros
+            | [ H: (ty0 ?1 ?2 (TTail (Flat Cast) ?3 ?4) ?5) |- ? ] ->
+               LApply (ty0_gen_cast ?1 ?2 ?4 ?3 ?5); [ Clear H; Intros H | XAuto ];
+               XElim H; Intros.
+
+(*#* #start file *)
+
+(*#* #single *)