# Theory IntFact

theory IntFact
imports IntPrimes
```(*  Title:      HOL/Old_Number_Theory/IntFact.thy
Author:     Thomas M. Rasmussen
*)

section ‹Factorial on integers›

theory IntFact
imports IntPrimes
begin

text ‹
Factorial on integers and recursively defined set including all
Integers from @{text 2} up to @{text a}.  Plus definition of product
of finite set.

\bigskip
›

fun zfact :: "int => int"
where "zfact n = (if n ≤ 0 then 1 else n * zfact (n - 1))"

fun d22set :: "int => int set"
where "d22set a = (if 1 < a then insert a (d22set (a - 1)) else {})"

text ‹
\medskip @{term d22set} --- recursively defined set including all
integers from @{text 2} up to @{text a}
›

declare d22set.simps [simp del]

lemma d22set_induct:
assumes "!!a. P {} a"
and "!!a. 1 < (a::int) ==> P (d22set (a - 1)) (a - 1) ==> P (d22set a) a"
shows "P (d22set u) u"
apply (rule d22set.induct)
apply (case_tac "1 < a")
apply (rule_tac assms)
apply (simp_all (no_asm_simp))
apply (simp_all (no_asm_simp) add: d22set.simps assms)
done

lemma d22set_g_1 [rule_format]: "b ∈ d22set a --> 1 < b"
apply (induct a rule: d22set_induct)
apply simp
apply (subst d22set.simps)
apply auto
done

lemma d22set_le [rule_format]: "b ∈ d22set a --> b ≤ a"
apply (induct a rule: d22set_induct)
apply simp
apply (subst d22set.simps)
apply auto
done

lemma d22set_le_swap: "a < b ==> b ∉ d22set a"
by (auto dest: d22set_le)

lemma d22set_mem: "1 < b ⟹ b ≤ a ⟹ b ∈ d22set a"
apply (induct a rule: d22set.induct)
apply auto
apply (subst d22set.simps)
apply (case_tac "b < a", auto)
done

lemma d22set_fin: "finite (d22set a)"
apply (induct a rule: d22set_induct)
prefer 2
apply (subst d22set.simps)
apply auto
done

declare zfact.simps [simp del]

lemma d22set_prod_zfact: "∏(d22set a) = zfact a"
apply (induct a rule: d22set.induct)
apply (subst d22set.simps)
apply (subst zfact.simps)
apply (case_tac "1 < a")
prefer 2