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-Require Import Wf.
-Require Import Coq.subtac.Utils.
-
-(** 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.