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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import Arith.
Require Import Compare_dec.
Require Import Wf_nat.
Declare ML Module "recdef".
Section Iter.
Variable A : Type.
Fixpoint iter (n : nat) : (A -> A) -> A -> A :=
fun (fl : A -> A) (def : A) =>
match n with
| O => def
| S m => fl (iter m fl def)
end.
End Iter.
Theorem SSplus_lt : forall p p' : nat, p < S (S (p + p')).
auto with arith.
Qed.
Theorem Splus_lt : forall p p' : nat, p' < S (p + p').
auto with arith.
Qed.
Theorem le_lt_SS : forall x y, x <= y -> x < S (S y).
auto with arith.
Qed.
Inductive max_type (m n:nat) : Set :=
cmt : forall v, m <= v -> n <= v -> max_type m n.
Definition max : forall m n:nat, max_type m n.
intros m n; case (le_gt_dec m n).
intros h; exists n; [exact h | apply le_n].
intros h; exists m; [apply le_n | apply lt_le_weak; exact h].
Defined.
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