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[Thery Laurent <thery AT ns.di.univaq.it>]

Hi,

> And I want to proove the theorem:
> "
> inductive A R /\ increasing A R -> well_founded ( RInv R ).
> "
> I guess it should be very easy. I'm not giving the definitions of inductive

Here is something somewhat similar to what you try to do:

Require Import Inclusion.
Require Import Transitive_Closure.
Require Import Relation_Operators.
Require Import Relation_Definitions.
Require Import List.

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

Inductive Group_L : nat -> A -> A ->  Prop :=
  Group_L_0: forall (x y : A), R x y ->  Group_L 0 x y
 | Group_L_n: forall n (x y z : A), Group_L n x y -> R y z ->  Group_L (S 
n) x z .
Hint Resolve Group_L_0 .

Theorem Group_L_inclusion_clos_trans:
 forall n,  inclusion _ (Group_L n) (clos_trans A R).
intros n; elim n; simpl; auto.
intros x y H; inversion_clear H.
apply t_step; auto.
intros n0 H x y H0.
inversion_clear H0 as [|t1 t2 y0].
apply t_trans with y0; auto.
apply t_step; auto.
Qed.

Theorem Acc_R_Group_L: forall n x, Acc R x ->  Acc (Group_L n) x.
intros n x H; apply Acc_incl with (clos_trans A R); auto.
apply Group_L_inclusion_clos_trans; auto.
apply Acc_clos_trans; auto.
Qed.

Theorem wf_R_Group_L: forall n, well_founded R ->  well_founded (Group_L 
n).
intros n; unfold well_founded; intros H y; apply Acc_R_Group_L; auto.
Qed.

--
Laurent Théry