we restored the scripts of \lambda\delta version 1

[helm.git] / helm / coq-contribs / LAMBDA-TYPES / pr2_defs.v

diff --git a/helm/coq-contribs/LAMBDA-TYPES/pr2_defs.v b/helm/coq-contribs/LAMBDA-TYPES/pr2_defs.v
deleted file mode 100644 (file)
index 9dab9ca..0000000
--- a/helm/coq-contribs/LAMBDA-TYPES/pr2_defs.v
+++ /dev/null
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-Require Export drop_defs.
-Require Export pr0_defs.
-
-(*#* #caption "current axioms for the relation $\\PrS{}{}{}$",
-   "context-free case", "context-dependent $\\delta$-expansion" 
-*)
-(*#* #cap #cap c, d, t, t1, t2 #alpha u in V *)
-
-      Inductive pr2 [c:C; t1:T] : T -> Prop :=
-(* structural rules *)
-         | pr2_free : (t2:?) (pr0 t1 t2) -> (pr2 c t1 t2)
-(* axiom rules *)
-         | pr2_delta: (d:?; u:?; i:?)
-                      (drop i (0) c (CTail d (Bind Abbr) u)) ->
-                      (t2:?) (pr0 t1 t2) -> (t:?) (subst0 i u t2 t) -> 
-                     (pr2 c t1 t).
-
-(*#* #stop file *)
-
-      Hint pr2 : ltlc := Constructors pr2.
-
-   Section pr2_gen_base. (***************************************************)
-
-      Theorem pr2_gen_sort: (c:?; x:?; n:?) (pr2 c (TSort n) x) ->
-                            x = (TSort n).
-      Intros; Inversion H; Pr0GenBase;
-      [ XAuto | Rewrite H1 in H2; Subst0GenBase ].
-      Qed.
-
-      Theorem pr2_gen_lref: (c:?; x:?; n:?) (pr2 c (TLRef n) x) ->
-                            x = (TLRef n) \/
-                            (EX d u | (drop n (0) c (CTail d (Bind Abbr) u)) &
-                                       x = (lift (S n) (0) u)
-                            ).
-      Intros; Inversion H; Pr0GenBase; 
-      [ XAuto | Rewrite H1 in H2; Subst0GenBase; Rewrite <- H2 in H0; XEAuto ].
-      Qed.
-
-      Theorem pr2_gen_abst: (c:?; u1,t1,x:?)
-                            (pr2 c (TTail (Bind Abst) u1 t1) x) ->
-                            (EX u2 t2 | x = (TTail (Bind Abst) u2 t2) &
-                                        (pr2 c u1 u2) & (b:?; u:?)
-                                        (pr2 (CTail c (Bind b) u) t1 t2)
-                            ).
-      Intros; Inversion H; Pr0GenBase;
-      [ XEAuto | Rewrite H1 in H2; Subst0GenBase; XDEAuto 6 ].
-      Qed.
-
-      Theorem pr2_gen_appl: (c:?; u1,t1,x:?)
-                            (pr2 c (TTail (Flat Appl) u1 t1) x) -> (OR
-                            (EX u2 t2 | x = (TTail (Flat Appl) u2 t2) &
-                                        (pr2 c u1 u2) & (pr2 c t1 t2)
-                            ) |
-                            (EX y1 z1 u2 t2 | t1 = (TTail (Bind Abst) y1 z1) &
-                                              x = (TTail (Bind Abbr) u2 t2) &
-                                              (pr2 c u1 u2) & (b:?; u:?) 
-                                             (pr2 (CTail c (Bind b) u) z1 t2)
-                            ) |
-                            (EX b y1 z1 z2 u2 v2 t2 |
-                               ~b=Abst &
-                               t1 = (TTail (Bind b) y1 z1) &
-                               x = (TTail (Bind b) v2 z2) & 
-                               (pr2 c u1 u2) & (pr2 c y1 v2) & (pr0 z1 t2))
-                            ).
-      Intros; Inversion H; Pr0GenBase;
-      Try Rewrite H1 in H2; Try Rewrite H4 in H2; Try Rewrite H5 in H2; 
-      Try Subst0GenBase; XDEAuto 7.
-      Qed.
-
-      Theorem pr2_gen_cast: (c:?; u1,t1,x:?)
-                            (pr2 c (TTail (Flat Cast) u1 t1) x) ->
-                            (EX u2 t2 | x = (TTail (Flat Cast) u2 t2) &
-                                        (pr2 c u1 u2) & (pr2 c t1 t2)
-                            ) \/
-                            (pr2 c t1 x).
-      Intros; Inversion H; Pr0GenBase; 
-      Try Rewrite H1 in H2; Try Subst0GenBase; XEAuto. 
-      Qed.
-
-   End pr2_gen_base.
-
-      Tactic Definition Pr2GenBase :=
-         Match Context With
-            | [ H: (pr2 ?1 (TSort ?2) ?3) |- ? ] ->
-               LApply (pr2_gen_sort ?1 ?3 ?2); [ Clear H; Intros | XAuto ]
-            | [ H: (pr2 ?1 (TLRef ?2) ?3) |- ? ] ->
-               LApply (pr2_gen_lref ?1 ?3 ?2); [ Clear H; Intros H | XAuto ];
-              XDecompose H 
-           | [ H: (pr2 ?1 (TTail (Bind Abst) ?2 ?3) ?4) |- ? ] ->
-               LApply (pr2_gen_abst ?1 ?2 ?3 ?4); [ Clear H; Intros H | XAuto ];
-               XDecompose H
-           | [ H: (pr2 ?1 (TTail (Flat Appl) ?2 ?3) ?4) |- ? ] ->
-               LApply (pr2_gen_appl ?1 ?2 ?3 ?4); [ Clear H; Intros H | XAuto ];
-               XDecompose H
-           | [ H: (pr2 ?1 (TTail (Flat Cast) ?2 ?3) ?4) |- ? ] ->
-               LApply (pr2_gen_cast ?1 ?2 ?3 ?4); [ Clear H; Intros H | XAuto ];
-               XDecompose H.
-
-   Section pr2_props. (******************************************************)
-
-      Theorem pr2_thin_dx: (c:?; t1,t2:?) (pr2 c t1 t2) ->
-                           (u:?; f:?) (pr2 c (TTail (Flat f) u t1)
-                                             (TTail (Flat f) u t2)).
-      Intros; XElim H; XEAuto.
-      Qed.
-
-      Theorem pr2_tail_1: (c:?; u1,u2:?) (pr2 c u1 u2) ->
-                          (k:?; t:?) (pr2 c (TTail k u1 t) (TTail k u2 t)).
-      Intros; XElim H; XEAuto.
-      Qed.
-
-      Theorem pr2_tail_2: (c:?; u,t1,t2:?; k:?) (pr2 (CTail c k u) t1 t2) ->
-                          (pr2 c (TTail k u t1) (TTail k u t2)).
-      XElim k; Intros; (
-      XElim H; [ XAuto | XElim i; Intros; DropGenBase; CGenBase; XEAuto ]).
-      Qed.
-      
-      Hints Resolve pr2_tail_2 : ltlc.
-
-      Theorem pr2_shift: (i:?; c,e:?) (drop i (0) c e) ->
-                         (t1,t2:?) (pr2 c t1 t2) ->
-                         (pr2 e (app c i t1) (app c i t2)).
-      XElim i.
-(* case 1: i = 0 *)
-      Intros; DropGenBase; Rewrite H in H0.
-      Repeat Rewrite app_O; XAuto.
-(* case 2: i > 0 *)
-      XElim c.
-(* case 2.1: CSort *)
-      Intros; DropGenBase; Rewrite H0; XAuto.
-(* case 2.2: CTail *)
-      XElim k; Intros; Simpl; DropGenBase; XAuto.
-      Qed.
-      
-   End pr2_props.
-
-      Hints Resolve pr2_thin_dx pr2_tail_1 pr2_tail_2 pr2_shift : ltlc.