Move Program tactics into a proper theories/ directory as they are general purpose and can be used directly be the user. Document them. Change Prelude to disambiguate an import of a Tactics module.

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@10060 85f007b7-540e-0410-9357-904b9bb8a0f7

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+Require Import Wf.
+Require Import Coq.Program.Utils.
+Require Import ProofIrrelevance.
+
+(** Reformulation of the Wellfounded module using subsets where possible. *)
+
+Section Well_founded.
+  Variable A : Type.
+  Variable R : A -> A -> Prop.
+  Hypothesis Rwf : well_founded R.
+  
+  Section Acc.
+    
+    Variable P : A -> Type.
+    
+    Variable F_sub : forall x:A, (forall y: { y : A | R y x }, P (proj1_sig y)) -> P x.
+    
+    Fixpoint Fix_F_sub (x : A) (r : Acc R x) {struct r} : P x :=
+      F_sub x (fun y: { y : A | R y x}  => Fix_F_sub (proj1_sig y) 
+        (Acc_inv r (proj1_sig y) (proj2_sig y))).
+    
+    Definition Fix_sub (x : A) := Fix_F_sub x (Rwf x).
+  End Acc.
+  
+  Section FixPoint.
+    Variable P : A -> Type.
+    
+    Variable F_sub : forall x:A, (forall y: { y : A | R y x }, P (proj1_sig y)) -> P x.
+    
+    Notation Fix_F := (Fix_F_sub P F_sub) (only parsing). (* alias *)
+    
+    Definition Fix (x:A) := Fix_F_sub P F_sub x (Rwf x).
+    
+    Hypothesis
+      F_ext :
+      forall (x:A) (f g:forall y:{y:A | R y x}, P (`y)),
+        (forall y:{ y:A | R y x}, f y = g y) -> F_sub x f = F_sub x g.
+
+    Lemma Fix_F_eq :
+      forall (x:A) (r:Acc R x),
+        F_sub x (fun (y:{y:A|R y x}) => Fix_F (`y) (Acc_inv r (proj1_sig y) (proj2_sig y))) = Fix_F x r.
+    Proof. 
+      destruct r using Acc_inv_dep; auto.
+    Qed.
+    
+    Lemma Fix_F_inv : forall (x:A) (r s:Acc R x), Fix_F x r = Fix_F x s.
+    Proof.
+      intro x; induction (Rwf x); intros.
+      rewrite <- (Fix_F_eq x r); rewrite <- (Fix_F_eq x s); intros.
+      apply F_ext; auto.
+      intros.
+      rewrite (proof_irrelevance (Acc R x) r s) ; auto.
+    Qed.
+
+    Lemma Fix_eq : forall x:A, Fix x = F_sub x (fun (y:{y:A|R y x}) => Fix (proj1_sig y)).
+    Proof.
+      intro x; unfold Fix in |- *.
+      rewrite <- (Fix_F_eq ).
+      apply F_ext; intros.
+      apply Fix_F_inv.
+    Qed.
+
+    Lemma fix_sub_eq :
+        forall x : A,
+          Fix_sub P F_sub x =
+          let f_sub := F_sub in
+            f_sub x (fun  {y : A | R y x}=> Fix (`y)).
+      exact Fix_eq.
+    Qed.
+
+ End FixPoint.
+
+End Well_founded.
+
+Extraction Inline Fix_F_sub Fix_sub.
+
+Require Import Wf_nat.
+Require Import Lt.
+
+Section Well_founded_measure.
+  Variable A : Type.
+  Variable m : A -> nat.
+  
+  Section Acc.
+    
+    Variable P : A -> Type.
+    
+    Variable F_sub : forall x:A, (forall y: { y : A | m y < m x }, P (proj1_sig y)) -> P x.
+    
+    Fixpoint Fix_measure_F_sub (x : A) (r : Acc lt (m x)) {struct r} : P x :=
+      F_sub x (fun y: { y : A | m y < m x}  => Fix_measure_F_sub (proj1_sig y) 
+        (Acc_inv r (m (proj1_sig y)) (proj2_sig y))).
+    
+    Definition Fix_measure_sub (x : A) := Fix_measure_F_sub x (lt_wf (m x)).
+  
+  End Acc.
+
+  Section FixPoint.
+    Variable P : A -> Type.
+    
+    Variable F_sub : forall x:A, (forall y: { y : A | m y < m x }, P (proj1_sig y)) -> P x.
+    
+    Notation Fix_F := (Fix_measure_F_sub P F_sub) (only parsing). (* alias *)
+    
+    Definition Fix_measure (x:A) := Fix_measure_F_sub P F_sub x (lt_wf (m x)).
+    
+    Hypothesis
+      F_ext :
+      forall (x:A) (f g:forall y:{y:A | m y < m x}, P (`y)),
+        (forall y:{ y:A | m y < m x}, f y = g y) -> F_sub x f = F_sub x g.
+
+    Lemma Fix_measure_F_eq :
+      forall (x:A) (r:Acc lt (m x)),
+        F_sub x (fun (y:{y:A|m y < m x}) => Fix_F (`y) (Acc_inv r (m (proj1_sig y)) (proj2_sig y))) = Fix_F x r.
+    Proof.
+      intros x.
+      set (y := m x).
+      unfold Fix_measure_F_sub.
+      intros r ; case r ; auto.
+    Qed.
+    
+    Lemma Fix_measure_F_inv : forall (x:A) (r s:Acc lt (m x)), Fix_F x r = Fix_F x s.
+    Proof.
+      intros x r s.
+      rewrite (proof_irrelevance (Acc lt (m x)) r s) ; auto.
+    Qed.
+
+    Lemma Fix_measure_eq : forall x:A, Fix_measure x = F_sub x (fun (y:{y:A| m y < m x}) => Fix_measure (proj1_sig y)).
+    Proof.
+      intro x; unfold Fix_measure in |- *.
+      rewrite <- (Fix_measure_F_eq ).
+      apply F_ext; intros.
+      apply Fix_measure_F_inv.
+    Qed.
+
+    Lemma fix_measure_sub_eq :
+        forall x : A,
+          Fix_measure_sub P F_sub x =
+          let f_sub := F_sub in
+            f_sub x (fun  {y : A | m y < m x}=> Fix_measure (`y)).
+      exact Fix_measure_eq.
+    Qed.
+
+ End FixPoint.
+
+End Well_founded_measure.
+
+Extraction Inline Fix_measure_F_sub Fix_measure_sub.