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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import PeanoNat.
Require Compare_dec.
Require Wf_nat.
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 le_lt_SS x y : x <= y -> x < S (S y).
Proof.
intros. now apply Nat.lt_succ_r, Nat.le_le_succ_r.
Qed.
Theorem Splus_lt x y : y < S (x + y).
Proof.
apply Nat.lt_succ_r. rewrite Nat.add_comm. apply Nat.le_add_r.
Qed.
Theorem SSplus_lt x y : x < S (S (x + y)).
Proof.
apply le_lt_SS, Nat.le_add_r.
Qed.
Inductive max_type (m n:nat) : Set :=
cmt : forall v, m <= v -> n <= v -> max_type m n.
Definition max m n : max_type m n.
Proof.
destruct (Compare_dec.le_gt_dec m n) as [h|h].
- exists n; [exact h | apply le_n].
- exists m; [apply le_n | apply Nat.lt_le_incl; exact h].
Defined.
Definition Acc_intro_generator_function := fun A R => @Acc_intro_generator A R 100.
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