feat(library/data/nat/fact): define factorial

library/data/nat/fact.lean

@@ -0,0 +1,63 @@
+/-
+Copyright (c) 2015 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+
+Factorial
+-/
+import data.nat.div
+
+namespace nat
+definition fact : nat → nat
+| 0        := 1
+| (succ n) := (succ n) * fact n
+
+lemma fact_zero : fact 0 = 1 :=
+rfl
+
+lemma fact_one : fact 1 = 1 :=
+rfl
+
+lemma fact_succ (n) : fact (succ n) = succ n * fact n :=
+rfl
+
+lemma fact_ne_zero : ∀ n, fact n ≠ 0
+| 0        := by contradiction
+| (succ n) :=
+  begin
+    intro h,
+    rewrite [fact_succ at h],
+    cases (eq_zero_or_eq_zero_of_mul_eq_zero h) with h₁ h₂,
+      contradiction,
+      exact fact_ne_zero n h₂
+  end
+
+lemma fact_gt_0 (n) : fact n > 0 :=
+pos_of_ne_zero (fact_ne_zero n)
+
+lemma dvd_fact : ∀ {m n}, m > 0 → m ≤ n → m ∣ fact n
+| m 0        h₁ h₂ := absurd h₁ (not_lt_of_ge h₂)
+| m (succ n) h₁ h₂ :=
+  begin
+    rewrite fact_succ,
+    cases (eq_or_lt_of_le h₂) with he hl,
+      {subst m, apply dvd_mul_right},
+      {have aux : m ∣ fact n, from dvd_fact h₁ (le_of_lt_succ hl),
+       apply dvd_mul_of_dvd_right aux}
+  end
+
+lemma fact_le {m n} : m ≤ n → fact m ≤ fact n :=
+begin
+  induction n with n ih,
+   {intro h,
+    have meq0 : m = 0, from eq_zero_of_le_zero h,
+    subst m},
+   {intro m_le_succ_n,
+    cases (eq_or_lt_of_le m_le_succ_n) with h₁ h₂,
+      {subst m},
+      {transitivity (fact n),
+        exact ih (le_of_lt_succ h₂),
+        rewrite [fact_succ, -one_mul at {1}],
+        exact mul_le_mul (succ_le_succ (zero_le n)) !le.refl}}
+end
+end nat