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The Coq Proof Assistant

# Library Coq.Wellfounded.Well_Ordering

Author: Cristina Cornes. From: Constructing Recursion Operators in Type Theory L. Paulson JSC (1986) 2, 325-355
Require Import EqdepFacts.
#[universes(template)]
Inductive WO (A : Type) (B : A -> Type) : Type :=
sup : forall (a:A) (f:B a -> WO A B), WO A B.
Section WellOrdering.
Variable A : Type.
Variable B : A -> Type.
Notation WO := (WO A B).
Inductive le_WO : WO -> WO -> Prop :=
le_sup : forall (a:A) (f:B a -> WO) (v:B a), le_WO (f v) (sup _ _ a f).
Lemma le_WO_inv x y : le_WO x y -> exists a f v, f v = x /\ sup _ _ a f = y.
Theorem wf_WO : well_founded le_WO.
End WellOrdering.
Section Characterisation_wf_relations.
Wellfounded relations are the inverse image of wellordering types
Variable A : Type.
Variable leA : A -> A -> Prop.
Definition B (a:A) := {x : A | leA x a}.
Definition wof : well_founded leA -> A -> WO A B.
End Characterisation_wf_relations.