1 files changed, 67 insertions, 15 deletions

diff --git a/contrib/subtac/FixSub.v b/contrib/subtac/FixSub.v
index ded069bf..46121ff1 100644
--- a/contrib/subtac/FixSub.v
+++ b/contrib/subtac/FixSub.v
@@ -1,37 +1,87 @@
 Require Import Wf.
+Require Import Coq.subtac.Utils.
 
 Section Well_founded.
-Variable A : Set.
-Variable R : A -> A -> Prop.
-Hypothesis Rwf : well_founded R.
+  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.
 
-Section FixPoint.
-
-Variable P : A -> Set.
+    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.
 
-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))).
+    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.
 
-Definition Fix_sub (x : A) := Fix_F_sub x (Rwf x).
+    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 FixPoint.
 
 End Well_founded.
 
+Extraction Inline Fix_F_sub Fix_sub.
+
 Require Import Wf_nat.
 Require Import Lt.
 
 Section Well_founded_measure.
-Variable A : Set.
+Variable A : Type.
 Variable f : A -> nat.
 Definition R := fun x y => f x < f y.
 
 Section FixPoint.
 
-Variable P : A -> Set.
+Variable P : A -> Type.
 
 Variable F_sub : forall x:A, (forall y: { y : A | f y < f x }, P (proj1_sig y)) -> P x.
  
@@ -44,3 +94,5 @@ Definition Fix_measure_sub (x : A) := Fix_measure_F_sub x (lt_wf (f x)).
 End FixPoint.
 
 End Well_founded_measure.
+
+Extraction Inline Fix_measure_F_sub Fix_measure_sub.