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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import PeanoNat Plus Mult Lt.
Local Open Scope nat_scope.
(** Factorial *)
Fixpoint fact (n:nat) : nat :=
match n with
| O => 1
| S n => S n * fact n
end.
Arguments fact n%nat.
Lemma lt_O_fact n : 0 < fact n.
Proof.
induction n; simpl; auto with arith.
Qed.
Lemma fact_neq_0 n : fact n <> 0.
Proof.
apply Nat.neq_0_lt_0, lt_O_fact.
Qed.
Lemma fact_le n m : n <= m -> fact n <= fact m.
Proof.
induction 1.
- apply le_n.
- simpl. transitivity (fact m). trivial. apply Nat.le_add_r.
Qed.
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