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Require Import Coq.Init.Wf.
Require Import Coq.Program.Utils.
Require Import ProofIrrelevance.
Open Local Scope program_scope.
Implicit Arguments Acc_inv [A R x y].
(** 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 (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 : 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:A|R y x) => Fix_F (`y) (Acc_inv r (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 (proof_irrelevance (Acc R x) r s) ; auto.
Qed.
Lemma Fix_eq : forall x:A, Fix x = F_sub x (fun (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.
Program Fixpoint Fix_measure_F_sub (x : A) (r : Acc lt (m x)) {struct r} : P x :=
F_sub x (fun (y : A | m y < m x) => Fix_measure_F_sub y
(@Acc_inv _ _ _ r (m 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.
Program Variable F_sub : forall x:A, (forall (y : A | m y < m x), P 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.
Program Lemma Fix_measure_F_eq :
forall (x:A) (r:Acc lt (m x)),
F_sub x (fun (y:A | m y < m x) => Fix_F y (Acc_inv r (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.
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