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Require Import Wf.
Section Well_founded.
Variable A : Set.
Variable R : A -> A -> Prop.
Hypothesis Rwf : well_founded R.
Section FixPoint.
Variable P : A -> Set.
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 FixPoint.
End Well_founded.
Require Import Wf_nat.
Require Import Lt.
Section Well_founded_measure.
Variable A : Set.
Variable f : A -> nat.
Definition R := fun x y => f x < f y.
Section FixPoint.
Variable P : A -> Set.
Variable F_sub : forall x:A, (forall y: { y : A | f y < f x }, P (proj1_sig y)) -> P x.
Fixpoint Fix_measure_F_sub (x : A) (r : Acc lt (f x)) {struct r} : P x :=
F_sub x (fun y: { y : A | f y < f x} => Fix_measure_F_sub (proj1_sig y)
(Acc_inv r (f (proj1_sig y)) (proj2_sig y))).
Definition Fix_measure_sub (x : A) := Fix_measure_F_sub x (lt_wf (f x)).
End FixPoint.
End Well_founded_measure.
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