contribution about \lambda-\delta

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

diff --git a/helm/coq-contribs/LAMBDA-TYPES/contexts_defs.v b/helm/coq-contribs/LAMBDA-TYPES/contexts_defs.v
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+(*#* #stop file *)
+
+Require Export terms_defs.
+
+      Inductive Set C := CSort : nat -> C
+                      |  CTail : C -> K -> T -> C.
+
+      Hint f3CKT : ltlc := Resolve (f_equal3 C K T).
+
+      Definition r: K -> nat -> nat := [k;i] Cases k of
+         | (Bind _) => i
+         | (Flat _) => (S i)
+      end.
+
+      Fixpoint app [c:C] : nat -> T -> T := [j;t]Cases j c of
+         | (0)   _             => t
+         | _     (CSort _)     => t
+         | (S i) (CTail c k u) => (app c (r k i) (TTail k u t))
+      end.
+
+   Section r_props. (********************************************************)
+
+      Theorem r_S: (k:?; i:?) (r k (S i)) = (S (r k i)).
+      XElim k; XAuto.
+      Qed.
+
+      Theorem r_plus_sym: (k:?; i,j:?) (r k (plus i j)) = (plus i (r k j)).
+      XElim k; Intros; Simpl; XAuto.
+      Qed.
+
+      Theorem r_minus: (i,n:?) (lt n i) ->
+                       (k:?) (minus (r k i) (S n)) = (r k (minus i (S n))).
+      XElim k; Intros; Simpl; XEAuto.
+      Qed.
+
+      Theorem r_dis: (k:?; P:Prop)
+                     (((i:?) (r k i) = i) -> P) ->
+                     (((i:?) (r k i) = (S i)) -> P) -> P.
+      XElim k; XAuto.
+      Qed.
+
+   End r_props.
+
+      Tactic Definition RRw :=
+         Repeat (Rewrite r_S Orelse Rewrite r_plus_sym).
+
+   Section r_arith. (********************************************************)
+
+      Theorem r_arith0: (k:?; i:?) (minus (r k (S i)) (1)) = (r k i).
+      Intros; RRw; Rewrite minus_Sx_SO; XAuto.
+      Qed.
+
+      Theorem r_arith1: (k:?; i,j:?) (minus (r k (S i)) (S j)) = (minus (r k i) j).
+      Intros; RRw; XAuto.
+      Qed.
+
+   End r_arith.
+
+   Section app_props. (******************************************************)
+
+      Theorem app_csort : (t:?; i,n:?) (app (CSort n) i t) = t.
+      XElim i; Intros; Simpl; XAuto.
+      Qed.
+
+      Theorem app_O : (c:?; t:?) (app c (0) t) = t.
+      XElim c; XAuto.
+      Qed.
+
+   End app_props.
+
+      Hints Resolve app_csort app_O : ltlc.