Factorial.v

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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.n:nat
0 < fact n
Proof.n:nat
0 < fact n
  induction n; simpl; auto with arith.
Qed.

Lemma fact_neq_0 n : fact n <> 0.n:nat
fact n <> 0
Proof.n:nat
fact n <> 0
 apply Nat.neq_0_lt_0, lt_O_fact.
Qed.

Lemma fact_le n m : n <= m -> fact n <= fact m.n, m:nat
n <= m -> fact n <= fact m
Proof.n, m:nat
n <= m -> fact n <= fact m
  induction 1.n:nat
fact n <= fact n
n, m:nat
H:n <= m
IHle:fact n <= fact m
fact n <= fact (S m)
  -n:nat
fact n <= fact n apply le_n.
  -n, m:nat
H:n <= m
IHle:fact n <= fact m
fact n <= fact (S m) simpl.n, m:nat
H:n <= m
IHle:fact n <= fact m
fact n <= fact m + m * fact m transitivity (fact m).n, m:nat
H:n <= m
IHle:fact n <= fact m
fact n <= fact m
n, m:nat
H:n <= m
IHle:fact n <= fact m
fact m <= fact m + m * fact m trivial.n, m:nat
H:n <= m
IHle:fact n <= fact m
fact m <= fact m + m * fact m apply Nat.le_add_r.
Qed.