Library Coq.Init.Wf

This module proves the validity of

  • well-founded recursion (also known as course of values)
  • well-founded induction
from a well-founded ordering on a given set

Set Implicit Arguments.

Require Import Notations.
Require Import Ltac.
Require Import Logic.
Require Import Datatypes.

Well-founded induction principle on Prop

Section Well_founded.

 Variable A : Type.
 Variable R : A -> A -> Prop.

The accessibility predicate is defined to be non-informative (Acc_rect is automatically defined because Acc is a singleton type)

 Inductive Acc (x: A) : Prop :=
     Acc_intro : (forall y:A, R y x -> Acc y) -> Acc x.

 Register Acc as

 Lemma Acc_inv : forall x:A, Acc x -> forall y:A, R y x -> Acc y.

 Global Arguments Acc_inv [x] _ [y] _, [x] _ y _.
 Register Acc_inv as

A relation is well-founded if every element is accessible

 Definition well_founded := forall a:A, Acc a.

 Register well_founded as

Well-founded induction on Set and Prop

 Hypothesis Rwf : well_founded.

 Theorem well_founded_induction_type :
  forall P:A -> Type,
    (forall x:A, (forall y:A, R y x -> P y) -> P x) -> forall a:A, P a.

 Theorem well_founded_induction :
  forall P:A -> Set,
    (forall x:A, (forall y:A, R y x -> P y) -> P x) -> forall a:A, P a.

 Theorem well_founded_ind :
  forall P:A -> Prop,
    (forall x:A, (forall y:A, R y x -> P y) -> P x) -> forall a:A, P a.

Well-founded fixpoints

 Section FixPoint.

  Variable P : A -> Type.
  Variable F : forall x:A, (forall y:A, R y x -> P y) -> P x.

  Fixpoint Fix_F (x:A) (a:Acc x) : P x :=
    F (fun (y:A) (h:R y x) => Fix_F (Acc_inv a h)).

  Scheme Acc_inv_dep := Induction for Acc Sort Prop.

  Lemma Fix_F_eq (x:A) (r:Acc x) :
     F (fun (y:A) (p:R y x) => Fix_F (x:=y) (Acc_inv r p)) = Fix_F (x:=x) r.

  Definition Fix (x:A) := Fix_F (Rwf x).

Proof that well_founded_induction satisfies the fixpoint equation. It requires an extra property of the functional

    F_ext :
      forall (x:A) (f g:forall y:A, R y x -> P y),
        (forall (y:A) (p:R y x), f y p = g y p) -> F f = F g.

  Lemma Fix_F_inv : forall (x:A) (r s:Acc x), Fix_F r = Fix_F s.

  Lemma Fix_eq : forall x:A, Fix x = F (fun (y:A) (p:R y x) => Fix y).

 End FixPoint.

End Well_founded.

Well-founded fixpoints over pairs

Section Well_founded_2.

  Variables A B : Type.
  Variable R : A * B -> A * B -> Prop.

  Variable P : A -> B -> Type.

  Section FixPoint_2.

    F :
      forall (x:A) (x':B),
        (forall (y:A) (y':B), R (y, y') (x, x') -> P y y') -> P x x'.

  Fixpoint Fix_F_2 (x:A) (x':B) (a:Acc R (x, x')) : P x x' :=
      (fun (y:A) (y':B) (h:R (y, y') (x, x')) =>
         Fix_F_2 (x:=y) (x':=y') (Acc_inv a (y,y') h)).

  End FixPoint_2.

  Hypothesis Rwf : well_founded R.

  Theorem well_founded_induction_type_2 :
   (forall (x:A) (x':B),
      (forall (y:A) (y':B), R (y, y') (x, x') -> P y y') -> P x x') ->
   forall (a:A) (b:B), P a b.

End Well_founded_2.

Notation Acc_iter := Fix_F (only parsing). Notation Acc_iter_2 := Fix_F_2 (only parsing).
Section Acc_generator.
  Variable A : Type.
  Variable R : A -> A -> Prop.

  Fixpoint Acc_intro_generator n (wf : well_founded R) :=
    match n with
        | O => wf
        | S n => fun x => Acc_intro x (fun y _ => Acc_intro_generator n (Acc_intro_generator n wf) y)

End Acc_generator.