diff options
Diffstat (limited to 'theories/Arith/Wf_nat.v')
| -rw-r--r-- | theories/Arith/Wf_nat.v | 23 |
1 files changed, 4 insertions, 19 deletions
diff --git a/theories/Arith/Wf_nat.v b/theories/Arith/Wf_nat.v index 23419531..b4468dd1 100644 --- a/theories/Arith/Wf_nat.v +++ b/theories/Arith/Wf_nat.v @@ -1,13 +1,11 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Wf_nat.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - (** Well-founded relations and natural numbers *) Require Import Lt. @@ -260,19 +258,6 @@ Qed. Unset Implicit Arguments. -(** [n]th iteration of the function [f] *) - -Fixpoint iter_nat (n:nat) (A:Type) (f:A -> A) (x:A) : A := - match n with - | O => x - | S n' => f (iter_nat n' A f x) - end. - -Theorem iter_nat_plus : - forall (n m:nat) (A:Type) (f:A -> A) (x:A), - iter_nat (n + m) A f x = iter_nat n A f (iter_nat m A f x). -Proof. - simple induction n; - [ simpl in |- *; auto with arith - | intros; simpl in |- *; apply f_equal with (f := f); apply H ]. -Qed. +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). |