fichiers prelude Coq

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+
+(* $Id$ *)
+
+(* This module proves the validity of
+    - well-founded recursion (also called course of values)
+    - well-founded induction
+   from a well-founded ordering on a given set *)
+
+Require Export Logic.
+Require Export LogicSyntax.
+
+(* Well-founded induction principle on Prop *)
+
+Chapter Well_founded.
+
+ Variable A : Set.
+ Variable R : A -> A -> Prop.
+
+ (* The accessibility predicate is defined to be non-informative *)
+
+ Inductive  Acc : A -> Prop 
+    := Acc_intro : (x:A)((y:A)(R y x)->(Acc y))->(Acc x).
+
+ Lemma Acc_inv : (x:A)(Acc x) -> (y:A)(R y x) -> (Acc y).
+  Destruct 1; Trivial.
+ Defined.
+
+  (* the informative elimination :
+     let Acc_rec F = let rec wf x = F x wf in wf *)
+
+ Section AccRec.
+  Variable P : A -> Set.
+  Variable F : (x:A)((y:A)(R y x)->(Acc y))->((y:A)(R y x)->(P y))->(P x).
+
+  Fixpoint Acc_rec [x:A;a:(Acc x)] : (P x)
+     := (F x (Acc_inv x a) ([y:A][h:(R y x)](Acc_rec y (Acc_inv x a y h)))).
+
+ End AccRec.
+
+ (* A relation is well-founded if every element is accessible *)
+
+ Definition well_founded := (a:A)(Acc a).
+
+ (* well-founded induction on Set and Prop *)
+
+ Hypothesis Rwf : well_founded.
+
+ Theorem well_founded_induction : 
+        (P:A->Set)((x:A)((y:A)(R y x)->(P y))->(P x))->(a:A)(P a).
+ Proof.
+  Intros; Apply (Acc_rec P); Auto.
+ Save.
+
+  Theorem well_founded_ind : 
+         (P:A->Prop)((x:A)((y:A)(R y x)->(P y))->(P x))->(a:A)(P a).
+   Proof.
+    Intros; Apply (Acc_ind P); Auto.
+   Qed.
+
+(* Building fixpoints  *) 
+
+Section FixPoint.
+
+Variable P : A -> Set.
+Variable F : (x:A)((y:A)(R y x)->(P y))->(P x).
+
+Fixpoint Fix_F [x:A;r:(Acc x)] : (P x) := 
+         (F x [y:A][p:(R y x)](Fix_F y (Acc_inv x r y p))).
+
+Definition fix := [x:A](Fix_F x (Rwf x)).
+
+(* Proof that well_founded_induction satisfies the fixpoint equation. 
+   It requires an extra property of the functional *)
+
+Hypothesis F_ext : 
+  (x:A)(f,g:(y:A)(R y x)->(P y))
+  ((y:A)(p:(R y x))((f y p)=(g y p)))->(F x f)=(F x g).
+
+Scheme Acc_inv_dep := Induction for Acc Sort Prop.
+
+Lemma Fix_F_eq
+  : (x:A)(r:(Acc x))
+    (F x [y:A][p:(R y x)](Fix_F y (Acc_inv x r y p)))=(Fix_F x r).
+Intros x r; Elim r using  Acc_inv_dep; Auto.
+Save.
+
+Lemma Fix_F_inv : (x:A)(r,s:(Acc x))(Fix_F x r)=(Fix_F x s).
+Intro x; Elim (Rwf x); Intros.
+Case (Fix_F_eq x0 r); Case (Fix_F_eq x0 s); Intros.
+Apply F_ext; Auto.
+Save.
+
+
+Lemma fix_eq : (x:A)(fix x)=(F x [y:A][p:(R y x)](fix y)).
+Intro; Unfold fix.
+Case (Fix_F_eq x).
+Apply F_ext; Intros.
+Apply Fix_F_inv.
+Save.
+
+End FixPoint.
+
+End Well_founded.