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(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
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(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
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(*i $Id: Wf_nat.v 8642 2006-03-17 10:09:02Z notin $ i*)

(** Well-founded relations and natural numbers *)

Require Import Lt.

Open Local Scope nat_scope.

Implicit Types m n p : nat.

Section Well_founded_Nat.

Variable A : Set.

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.
Proof.
red in |- *.
cut (forall n (a:A), f a < n -> Acc ltof a).
intros H a; apply (H (S (f a))); auto with arith.
induction n.
intros; absurd (f a < 0); auto with arith.
intros a ltSma.
apply Acc_intro.
unfold ltof in |- *; intros b ltfafb.
apply IHn.
apply lt_le_trans with (f a); auto with arith.
Defined.

Theorem well_founded_gtof : well_founded gtof.
Proof.
exact well_founded_ltof.
Defined.

(** 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 a = indrec ((f a)+1) a 
   where rec indrec = 
        function 0    -> (function a -> error)
               |(S m) -> (function a -> (F a (function y -> indrec y m)));;
]]

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.
Proof.
intros P F; cut (forall n (a:A), f a < n -> P a).
intros H a; apply (H (S (f a))); auto with arith.
induction n.
intros; absurd (f a < 0); auto with arith.
intros a ltSma.
apply F.
unfold ltof in |- *; intros b ltfafb.
apply IHn.
apply lt_le_trans with (f a); auto with arith.
Defined. 

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.
Proof.
exact induction_ltof1.
Defined.

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.
Proof.
exact (well_founded_induction well_founded_ltof).
Defined.

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.
Proof.
exact induction_ltof2.
Defined.

(** 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.
Proof.
red in |- *.
cut (forall n (a:A), f a < n -> Acc R a).
intros H a; apply (H (S (f a))); auto with arith.
induction n.
intros; absurd (f a < 0); auto with arith.
intros a ltSma.
apply Acc_intro.
intros b ltfafb.
apply IHn.
apply lt_le_trans with (f a); auto with arith.
Defined.

End Well_founded_Nat.

Lemma lt_wf : well_founded lt.
Proof.
exact (well_founded_ltof nat (fun m => m)).
Defined.

Lemma lt_wf_rec1 :
 forall n (P:nat -> Set), (forall n, (forall m, m < n -> P m) -> P n) -> P n.
Proof.
exact (fun p P F => induction_ltof1 nat (fun m => m) P F p).
Defined.

Lemma lt_wf_rec :
 forall n (P:nat -> Set), (forall n, (forall m, m < n -> P m) -> P n) -> P n.
Proof.
exact (fun p P F => induction_ltof2 nat (fun m => m) P F p).
Defined.

Lemma lt_wf_ind :
 forall n (P:nat -> Prop), (forall n, (forall m, m < n -> P m) -> P n) -> P n.
Proof.
intro p; intros; elim (lt_wf p); auto with arith.
Qed.

Lemma gt_wf_rec :
 forall n (P:nat -> Set), (forall n, (forall m, n > m -> P m) -> P n) -> P n.
Proof.
exact lt_wf_rec.
Defined.

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

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.
Proof.
intros P Hrec p; pattern p in |- *; apply lt_wf_rec.
intros n H q; pattern q in |- *; apply lt_wf_rec; auto with arith.
Defined.

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.
Proof.
intros P Hrec p; pattern p in |- *; apply lt_wf_ind.
intros n H q; pattern q in |- *; apply lt_wf_ind; auto with arith.
Qed.

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

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

(* Relational form of inversion *)
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.
Proof.
intros x [n fxn]; generalize dependent x.
pattern n in |- *; apply lt_wf_ind; intros.
constructor; intros.
destruct (F_compat y x) as (x0,H1,H2); trivial.
apply (H x0); auto.
Qed.

Theorem well_founded_inv_lt_rel_compat : well_founded R.
Proof.
constructor; intros.
case (F_compat y a); trivial; intros.
apply acc_lt_rel; trivial.
exists x; trivial.
Qed.

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).
intros; apply (well_founded_inv_lt_rel_compat A (inv_lt_rel A F) F); trivial.
Qed.