Theory IntFact

(*  Title:      HOL/NumberTheory/IntFact.thy
    ID:         $Id: IntFact.thy,v 1.10 2003/12/03 09:49:35 paulson Exp $
    Author:     Thomas M. Rasmussen
    Copyright   2000  University of Cambridge
*)

header {* Factorial on integers *}

theory IntFact = IntPrimes:

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
*}

consts
  zfact :: "int => int"
  ssetprod :: "int set => int"
  d22set :: "int => int set"

recdef zfact  "measure ((λn. nat n) :: int => nat)"
  "zfact n = (if n ≤ 0 then 1 else n * zfact (n - 1))"

defs
  ssetprod_def: "ssetprod A == (if finite A then fold (op *) 1 A else 1)"

recdef d22set  "measure ((λa. nat a) :: int => nat)"
  "d22set a = (if 1 < a then insert a (d22set (a - 1)) else {})"


text {* \medskip @{term ssetprod} --- product of finite set *}

lemma ssetprod_empty [simp]: "ssetprod {} = 1"
  apply (simp add: ssetprod_def)
  done

lemma ssetprod_insert [simp]:
    "finite A ==> a ∉ A ==> ssetprod (insert a A) = a * ssetprod A"
  by (simp add: ssetprod_def mult_left_commute LC.fold_insert [OF LC.intro])

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:
  "(!!a. P {} a) ==>
    (!!a. 1 < (a::int) ==> P (d22set (a - 1)) (a - 1)
      ==> P (d22set a) a)
    ==> P (d22set u) u"
proof -
  case rule_context
  show ?thesis
    apply (rule d22set.induct)
    apply safe
     apply (case_tac [2] "1 < a")
      apply (rule_tac [2] rule_context)
       apply (simp_all (no_asm_simp))
     apply (simp_all (no_asm_simp) add: d22set.simps rule_context)
    done
qed

lemma d22set_g_1 [rule_format]: "b ∈ d22set a --> 1 < b"
  apply (induct a rule: d22set_induct)
   prefer 2
   apply (subst d22set.simps)
   apply auto
  done

lemma d22set_le [rule_format]: "b ∈ d22set a --> b ≤ a"
  apply (induct a rule: d22set_induct)
   prefer 2
   apply (subst d22set.simps)
   apply auto
  done

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

lemma d22set_mem [rule_format]: "1 < b --> b ≤ a --> b ∈ d22set a"
  apply (induct a rule: d22set.induct)
  apply auto
   apply (simp_all add: d22set.simps)
  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: "ssetprod (d22set a) = zfact a"
  apply (induct a rule: d22set.induct)
  apply safe
   apply (simp add: d22set.simps zfact.simps)
  apply (subst d22set.simps)
  apply (subst zfact.simps)
  apply (case_tac "1 < a")
   prefer 2
   apply (simp add: d22set.simps zfact.simps)
  apply (simp add: d22set_fin d22set_le_swap)
  done

end