paulson@15094: (*  Title       : Fact.thy
paulson@12196:     Author      : Jacques D. Fleuriot
paulson@12196:     Copyright   : 1998  University of Cambridge
paulson@15094:     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
paulson@12196: *)
paulson@12196: 
paulson@15094: header{*Factorial Function*}
paulson@15094: 
nipkow@15131: theory Fact
huffman@30073: imports Main
nipkow@15131: begin
paulson@15094: 
paulson@15094: consts fact :: "nat => nat"
paulson@15094: primrec
wenzelm@19765:   fact_0:     "fact 0 = 1"
wenzelm@19765:   fact_Suc:   "fact (Suc n) = (Suc n) * fact n"
paulson@15094: 
paulson@15094: 
paulson@15094: lemma fact_gt_zero [simp]: "0 < fact n"
nipkow@25134: by (induct n) auto
paulson@15094: 
paulson@15094: lemma fact_not_eq_zero [simp]: "fact n \<noteq> 0"
nipkow@25162: by simp
paulson@15094: 
chaieb@29693: lemma of_nat_fact_not_zero [simp]: "of_nat (fact n) \<noteq> (0::'a::semiring_char_0)"
nipkow@25134: by auto
paulson@15094: 
chaieb@29693: class ordered_semiring_1 = ordered_semiring + semiring_1
chaieb@29693: class ordered_semiring_1_strict = ordered_semiring_strict + semiring_1
paulson@15094: 
chaieb@29693: lemma of_nat_fact_gt_zero [simp]: "(0::'a::{ordered_semidom}) < of_nat(fact n)" by auto
chaieb@29693: 
chaieb@29693: lemma of_nat_fact_ge_zero [simp]: "(0::'a::ordered_semidom) \<le> of_nat(fact n)"
nipkow@25134: by simp
paulson@15094: 
paulson@15094: lemma fact_ge_one [simp]: "1 \<le> fact n"
nipkow@25134: by (induct n) auto
paulson@12196: 
paulson@15094: lemma fact_mono: "m \<le> n ==> fact m \<le> fact n"
nipkow@25134: apply (drule le_imp_less_or_eq)
nipkow@25134: apply (auto dest!: less_imp_Suc_add)
nipkow@25134: apply (induct_tac k, auto)
nipkow@25134: done
paulson@15094: 
paulson@15094: text{*Note that @{term "fact 0 = fact 1"}*}
paulson@15094: lemma fact_less_mono: "[| 0 < m; m < n |] ==> fact m < fact n"
nipkow@25134: apply (drule_tac m = m in less_imp_Suc_add, auto)
nipkow@25134: apply (induct_tac k, auto)
nipkow@25134: done
paulson@15094: 
chaieb@29693: lemma inv_of_nat_fact_gt_zero [simp]: "(0::'a::ordered_field) < inverse (of_nat (fact n))"
nipkow@25134: by (auto simp add: positive_imp_inverse_positive)
paulson@15094: 
chaieb@29693: lemma inv_of_nat_fact_ge_zero [simp]: "(0::'a::ordered_field) \<le> inverse (of_nat (fact n))"
nipkow@25134: by (auto intro: order_less_imp_le)
paulson@15094: 
paulson@15094: lemma fact_diff_Suc [rule_format]:
nipkow@25134:   "n < Suc m ==> fact (Suc m - n) = (Suc m - n) * fact (m - n)"
nipkow@25134: apply (induct n arbitrary: m)
nipkow@25134: apply auto
huffman@30082: apply (drule_tac x = "m - Suc 0" in meta_spec, auto)
nipkow@25134: done
paulson@15094: 
chaieb@29693: lemma fact_num0: "fact 0 = 1"
nipkow@25134: by auto
paulson@15094: 
paulson@15094: lemma fact_num_eq_if: "fact m = (if m=0 then 1 else m * fact (m - 1))"
nipkow@25134: by (cases m) auto
paulson@15094: 
paulson@15094: lemma fact_add_num_eq_if:
nipkow@25134:   "fact (m + n) = (if m + n = 0 then 1 else (m + n) * fact (m + n - 1))"
nipkow@25134: by (cases "m + n") auto
paulson@15094: 
paulson@15094: lemma fact_add_num_eq_if2:
nipkow@25134:   "fact (m + n) = (if m = 0 then fact n else (m + n) * fact ((m - 1) + n))"
nipkow@25134: by (cases m) auto
paulson@12196: 
nipkow@15131: end