Library Coq.Arith.Wf_nat


Well-founded relations and natural numbers

Require Import Lt.

Local Open Scope nat_scope.

Implicit Types m n p : nat.

Section Well_founded_Nat.

Variable A : Type.

Variable f : A -> nat.
Definition ltof (a b:A) := f a < f b.
Definition gtof (a b:A) := f b > f a.

Theorem well_founded_ltof : well_founded ltof.

Theorem well_founded_gtof : well_founded gtof.

It is possible to directly prove the induction principle going back to primitive recursion on natural numbers (induction_ltof1) or to use the previous lemmas to extract a program with a fixpoint (induction_ltof2)
the ML-like program for induction_ltof1 is :
let induction_ltof1 f F a =
  let rec indrec n k =
    match n with
    | O -> error
    | S m -> F k (indrec m)
  in indrec (f a + 1) a
the ML-like program for induction_ltof2 is :
   let induction_ltof2 F a = indrec a
   where rec indrec a = F a indrec;;

Theorem induction_ltof1 :
  forall P:A -> Set,
    (forall x:A, (forall y:A, ltof y x -> P y) -> P x) -> forall a:A, P a.

Theorem induction_gtof1 :
  forall P:A -> Set,
    (forall x:A, (forall y:A, gtof y x -> P y) -> P x) -> forall a:A, P a.

Theorem induction_ltof2 :
  forall P:A -> Set,
    (forall x:A, (forall y:A, ltof y x -> P y) -> P x) -> forall a:A, P a.

Theorem induction_gtof2 :
  forall P:A -> Set,
    (forall x:A, (forall y:A, gtof y x -> P y) -> P x) -> forall a:A, P a.

If a relation R is compatible with lt i.e. if x R y => f(x) < f(y) then R is well-founded.

Variable R : A -> A -> Prop.

Hypothesis H_compat : forall x y:A, R x y -> f x < f y.

Theorem well_founded_lt_compat : well_founded R.

End Well_founded_Nat.

Lemma lt_wf : well_founded lt.

Lemma lt_wf_rec1 :
  forall n (P:nat -> Set), (forall n, (forall m, m < n -> P m) -> P n) -> P n.

Lemma lt_wf_rec :
  forall n (P:nat -> Set), (forall n, (forall m, m < n -> P m) -> P n) -> P n.

Lemma lt_wf_ind :
  forall n (P:nat -> Prop), (forall n, (forall m, m < n -> P m) -> P n) -> P n.

Lemma gt_wf_rec :
  forall n (P:nat -> Set), (forall n, (forall m, n > m -> P m) -> P n) -> P n.

Lemma gt_wf_ind :
  forall n (P:nat -> Prop), (forall n, (forall m, n > m -> P m) -> P n) -> P n.

Lemma lt_wf_double_rec :
 forall P:nat -> nat -> Set,
   (forall n m,
     (forall p q, p < n -> P p q) ->
     (forall p, p < m -> P n p) -> P n m) -> forall n m, P n m.

Lemma lt_wf_double_ind :
  forall P:nat -> nat -> Prop,
    (forall n m,
      (forall p (q:nat), p < n -> P p q) ->
      (forall p, p < m -> P n p) -> P n m) -> forall n m, P n m.

Hint Resolve lt_wf: arith.
Hint Resolve well_founded_lt_compat: arith.

Section LT_WF_REL.
  Variable A : Set.
  Variable R : A -> A -> Prop.

  Variable F : A -> nat -> Prop.
  Definition inv_lt_rel x y := exists2 n, F x n & (forall m, F y m -> n < m).

  Hypothesis F_compat : forall x y:A, R x y -> inv_lt_rel x y.
  Remark acc_lt_rel : forall x:A, (exists n, F x n) -> Acc R x.

  Theorem well_founded_inv_lt_rel_compat : well_founded R.

End LT_WF_REL.

Lemma well_founded_inv_rel_inv_lt_rel :
  forall (A:Set) (F:A -> nat -> Prop), well_founded (inv_lt_rel A F).

A constructive proof that any non empty decidable subset of natural numbers has a least element

Set Implicit Arguments.

Require Import Le.
Require Import Compare_dec.
Require Import Decidable.

Definition has_unique_least_element (A:Type) (R:A->A->Prop) (P:A->Prop) :=
  exists! x, P x /\ forall x', P x' -> R x x'.

Lemma dec_inh_nat_subset_has_unique_least_element :
  forall P:nat->Prop, (forall n, P n \/ ~ P n) ->
    (exists n, P n) -> has_unique_least_element le P.

Unset Implicit Arguments.

Notation iter_nat := @nat_iter (only parsing).
Notation iter_nat_plus := @nat_iter_plus (only parsing).
Notation iter_nat_invariant := @nat_iter_invariant (only parsing).