diff options
Diffstat (limited to 'theories7/Arith/Factorial.v')
-rw-r--r-- | theories7/Arith/Factorial.v | 51 |
1 files changed, 51 insertions, 0 deletions
diff --git a/theories7/Arith/Factorial.v b/theories7/Arith/Factorial.v new file mode 100644 index 00000000..a8a60c98 --- /dev/null +++ b/theories7/Arith/Factorial.v @@ -0,0 +1,51 @@ +(************************************************************************) +(* 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 *) +(************************************************************************) + +(*i $Id: Factorial.v,v 1.1.2.1 2004/07/16 19:31:24 herbelin Exp $ i*) + +Require Plus. +Require Mult. +Require Lt. +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +(** Factorial *) + +Fixpoint fact [n:nat]:nat:= + Cases n of + O => (S O) + |(S n) => (mult (S n) (fact n)) + end. + +Arguments Scope fact [ nat_scope ]. + +Lemma lt_O_fact : (n:nat)(lt O (fact n)). +Proof. +Induction n; Unfold lt; Simpl; Auto with arith. +Qed. + +Lemma fact_neq_0:(n:nat)~(fact n)=O. +Proof. +Intro. +Apply sym_not_eq. +Apply lt_O_neq. +Apply lt_O_fact. +Qed. + +Lemma fact_growing : (n,m:nat) (le n m) -> (le (fact n) (fact m)). +Proof. +NewInduction 1. +Apply le_n. +Assert (le (mult (S O) (fact n)) (mult (S m) (fact m))). +Apply le_mult_mult. +Apply lt_le_S; Apply lt_O_Sn. +Assumption. +Simpl (mult (S O) (fact n)) in H0. +Rewrite <- plus_n_O in H0. +Assumption. +Qed. |