src/HOL/Hyperreal/Fact.thy
 author paulson Sat, 31 Jul 2004 20:54:23 +0200 changeset 15094 a7d1a3fdc30d parent 12196 a3be6b3a9c0b child 15131 c69542757a4d permissions -rw-r--r--
conversion of Hyperreal/{Fact,Filter} to Isar scripts
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(*  Title       : Fact.thy
Author      : Jacques D. Fleuriot
Copyright   : 1998  University of Cambridge
Conversion to Isar and new proofs by Lawrence C Paulson, 2004
*)

theory Fact = Real:

consts fact :: "nat => nat"
primrec
fact_0:     "fact 0 = 1"
fact_Suc:   "fact (Suc n) = (Suc n) * fact n"

lemma fact_gt_zero [simp]: "0 < fact n"
by (induct "n", auto)

lemma fact_not_eq_zero [simp]: "fact n \<noteq> 0"
by simp

lemma real_of_nat_fact_not_zero [simp]: "real (fact n) \<noteq> 0"
by auto

lemma real_of_nat_fact_gt_zero [simp]: "0 < real(fact n)"
by auto

lemma real_of_nat_fact_ge_zero [simp]: "0 \<le> real(fact n)"
by simp

lemma fact_ge_one [simp]: "1 \<le> fact n"
by (induct "n", auto)

lemma fact_mono: "m \<le> n ==> fact m \<le> fact n"
apply (drule le_imp_less_or_eq)
apply (induct_tac "k", auto)
done

text{*Note that @{term "fact 0 = fact 1"}*}
lemma fact_less_mono: "[| 0 < m; m < n |] ==> fact m < fact n"
apply (drule_tac m = m in less_imp_Suc_add, auto)
apply (induct_tac "k", auto)
done

lemma inv_real_of_nat_fact_gt_zero [simp]: "0 < inverse (real (fact n))"

lemma inv_real_of_nat_fact_ge_zero [simp]: "0 \<le> inverse (real (fact n))"
by (auto intro: order_less_imp_le)

lemma fact_diff_Suc [rule_format]:
"\<forall>m. n < Suc m --> fact (Suc m - n) = (Suc m - n) * fact (m - n)"
apply (induct n, auto)
apply (drule_tac x = "m - 1" in spec, auto)
done

lemma fact_num0 [simp]: "fact 0 = 1"
by auto

lemma fact_num_eq_if: "fact m = (if m=0 then 1 else m * fact (m - 1))"
by (case_tac "m", auto)