src/HOL/NumberTheory/WilsonRuss.thy

HOL-NumberTheory: converted to new-style format and proper document setup;

(*  Title:      HOL/NumberTheory/WilsonRuss.thy
    ID:         $Id$
    Author:     Thomas M. Rasmussen
    Copyright   2000  University of Cambridge
*)

header {* Wilson's Theorem according to Russinoff *}

theory WilsonRuss = EulerFermat:

text {*
  Wilson's Theorem following quite closely Russinoff's approach
  using Boyer-Moore (using finite sets instead of lists, though).
*}

subsection {* Definitions and lemmas *}

consts
  inv :: "int => int => int"
  wset :: "int * int => int set"

defs
  inv_def: "inv p a == (a^(nat (p - #2))) mod p"

recdef wset
  "measure ((\<lambda>(a, p). nat a) :: int * int => nat)"
  "wset (a, p) =
    (if #1 < a then
      let ws = wset (a - #1, p)
      in (if a \<in> ws then ws else insert a (insert (inv p a) ws)) else {})"


text {* \medskip @{term [source] inv} *}

lemma aux: "#1 < m ==> Suc (nat (m - #2)) = nat (m - #1)"
  apply (subst int_int_eq [symmetric])
  apply auto
  done

lemma inv_is_inv:
    "p \<in> zprime \<Longrightarrow> #0 < a \<Longrightarrow> a < p ==> [a * inv p a = #1] (mod p)"
  apply (unfold inv_def)
  apply (subst zcong_zmod)
  apply (subst zmod_zmult1_eq [symmetric])
  apply (subst zcong_zmod [symmetric])
  apply (subst power_Suc [symmetric])
  apply (subst aux)
   apply (erule_tac [2] Little_Fermat)
   apply (erule_tac [2] zdvd_not_zless)
   apply (unfold zprime_def)
   apply auto
  done

lemma inv_distinct:
    "p \<in> zprime \<Longrightarrow> #1 < a \<Longrightarrow> a < p - #1 ==> a \<noteq> inv p a"
  apply safe
  apply (cut_tac a = a and p = p in zcong_square)
     apply (cut_tac [3] a = a and p = p in inv_is_inv)
        apply auto
   apply (subgoal_tac "a = #1")
    apply (rule_tac [2] m = p in zcong_zless_imp_eq)
        apply (subgoal_tac [7] "a = p - #1")
         apply (rule_tac [8] m = p in zcong_zless_imp_eq)
             apply auto
  done

lemma inv_not_0:
    "p \<in> zprime \<Longrightarrow> #1 < a \<Longrightarrow> a < p - #1 ==> inv p a \<noteq> #0"
  apply safe
  apply (cut_tac a = a and p = p in inv_is_inv)
     apply (unfold zcong_def)
     apply auto
  apply (subgoal_tac "\<not> p dvd #1")
   apply (rule_tac [2] zdvd_not_zless)
    apply (subgoal_tac "p dvd #1")
     prefer 2
     apply (subst zdvd_zminus_iff [symmetric])
     apply auto
  done

lemma inv_not_1:
    "p \<in> zprime \<Longrightarrow> #1 < a \<Longrightarrow> a < p - #1 ==> inv p a \<noteq> #1"
  apply safe
  apply (cut_tac a = a and p = p in inv_is_inv)
     prefer 4
     apply simp
     apply (subgoal_tac "a = #1")
      apply (rule_tac [2] zcong_zless_imp_eq)
          apply auto
  done

lemma aux: "[a * (p - #1) = #1] (mod p) = [a = p - #1] (mod p)"
  apply (unfold zcong_def)
  apply (simp add: zdiff_zdiff_eq zdiff_zdiff_eq2 zdiff_zmult_distrib2)
  apply (rule_tac s = "p dvd -((a + #1) + (p * -a))" in trans)
   apply (simp add: zmult_commute zminus_zdiff_eq)
  apply (subst zdvd_zminus_iff)
  apply (subst zdvd_reduce)
  apply (rule_tac s = "p dvd (a + #1) + (p * -#1)" in trans)
   apply (subst zdvd_reduce)
   apply auto
  done

lemma inv_not_p_minus_1:
    "p \<in> zprime \<Longrightarrow> #1 < a \<Longrightarrow> a < p - #1 ==> inv p a \<noteq> p - #1"
  apply safe
  apply (cut_tac a = a and p = p in inv_is_inv)
     apply auto
  apply (simp add: aux)
  apply (subgoal_tac "a = p - #1")
   apply (rule_tac [2] zcong_zless_imp_eq)
       apply auto
  done

lemma inv_g_1:
    "p \<in> zprime \<Longrightarrow> #1 < a \<Longrightarrow> a < p - #1 ==> #1 < inv p a"
  apply (case_tac "#0\<le> inv p a")
   apply (subgoal_tac "inv p a \<noteq> #1")
    apply (subgoal_tac "inv p a \<noteq> #0")
     apply (subst order_less_le)
     apply (subst zle_add1_eq_le [symmetric])
     apply (subst order_less_le)
     apply (rule_tac [2] inv_not_0)
       apply (rule_tac [5] inv_not_1)
         apply auto
  apply (unfold inv_def zprime_def)
  apply (simp add: pos_mod_sign)
  done

lemma inv_less_p_minus_1:
    "p \<in> zprime \<Longrightarrow> #1 < a \<Longrightarrow> a < p - #1 ==> inv p a < p - #1"
  apply (case_tac "inv p a < p")
   apply (subst order_less_le)
   apply (simp add: inv_not_p_minus_1)
  apply auto
  apply (unfold inv_def zprime_def)
  apply (simp add: pos_mod_bound)
  done

lemma aux: "#5 \<le> p ==>
    nat (p - #2) * nat (p - #2) = Suc (nat (p - #1) * nat (p - #3))"
  apply (subst int_int_eq [symmetric])
  apply (simp add: zmult_int [symmetric])
  apply (simp add: zdiff_zmult_distrib zdiff_zmult_distrib2)
  done

lemma zcong_zpower_zmult:
    "[x^y = #1] (mod p) \<Longrightarrow> [x^(y * z) = #1] (mod p)"
  apply (induct z)
   apply (auto simp add: zpower_zadd_distrib)
  apply (subgoal_tac "zcong (x^y * x^(y * n)) (#1 * #1) p")
   apply (rule_tac [2] zcong_zmult)
    apply simp_all
  done

lemma inv_inv: "p \<in> zprime \<Longrightarrow>
    #5 \<le> p \<Longrightarrow> #0 < a \<Longrightarrow> a < p ==> inv p (inv p a) = a"
  apply (unfold inv_def)
  apply (subst zpower_zmod)
  apply (subst zpower_zpower)
  apply (rule zcong_zless_imp_eq)
      prefer 5
      apply (subst zcong_zmod)
      apply (subst mod_mod_trivial)
      apply (subst zcong_zmod [symmetric])
      apply (subst aux)
       apply (subgoal_tac [2]
	 "zcong (a * a^(nat (p - #1) * nat (p - #3))) (a * #1) p")
        apply (rule_tac [3] zcong_zmult)
         apply (rule_tac [4] zcong_zpower_zmult)
         apply (erule_tac [4] Little_Fermat)
         apply (rule_tac [4] zdvd_not_zless)
          apply (simp_all add: pos_mod_bound pos_mod_sign)
  done


text {* \medskip @{term wset} *}

declare wset.simps [simp del]

lemma wset_induct:
  "(!!a p. P {} a p) \<Longrightarrow>
    (!!a p. #1 < (a::int) \<Longrightarrow> P (wset (a - #1, p)) (a - #1) p
      ==> P (wset (a, p)) a p)
    ==> P (wset (u, v)) u v"
proof -
  case antecedent
  show ?thesis
    apply (rule wset.induct)
    apply safe
     apply (case_tac [2] "#1 < a")
      apply (rule_tac [2] antecedent)
        apply simp_all
      apply (simp_all add: wset.simps antecedent)
    done
qed

lemma wset_mem_imp_or [rule_format]:
  "#1 < a \<Longrightarrow> b \<notin> wset (a - #1, p)
    ==> b \<in> wset (a, p) --> b = a \<or> b = inv p a"
  apply (subst wset.simps)
  apply (unfold Let_def)
  apply simp
  done

lemma wset_mem_mem [simp]: "#1 < a ==> a \<in> wset (a, p)"
  apply (subst wset.simps)
  apply (unfold Let_def)
  apply simp
  done

lemma wset_subset: "#1 < a \<Longrightarrow> b \<in> wset (a - #1, p) ==> b \<in> wset (a, p)"
  apply (subst wset.simps)
  apply (unfold Let_def)
  apply auto
  done

lemma wset_g_1 [rule_format]:
    "p \<in> zprime --> a < p - #1 --> b \<in> wset (a, p) --> #1 < b"
  apply (induct a p rule: wset_induct)
   apply auto
  apply (case_tac "b = a")
   apply (case_tac [2] "b = inv p a")
    apply (subgoal_tac [3] "b = a \<or> b = inv p a")
     apply (rule_tac [4] wset_mem_imp_or)
       prefer 2
       apply simp
       apply (rule inv_g_1)
         apply auto
  done

lemma wset_less [rule_format]:
    "p \<in> zprime --> a < p - #1 --> b \<in> wset (a, p) --> b < p - #1"
  apply (induct a p rule: wset_induct)
   apply auto
  apply (case_tac "b = a")
   apply (case_tac [2] "b = inv p a")
    apply (subgoal_tac [3] "b = a \<or> b = inv p a")
     apply (rule_tac [4] wset_mem_imp_or)
       prefer 2
       apply simp
       apply (rule inv_less_p_minus_1)
         apply auto
  done

lemma wset_mem [rule_format]:
  "p \<in> zprime -->
    a < p - #1 --> #1 < b --> b \<le> a --> b \<in> wset (a, p)"
  apply (induct a p rule: wset.induct)
  apply auto
   apply (subgoal_tac "b = a")
    apply (rule_tac [2] zle_anti_sym)
     apply (rule_tac [4] wset_subset)
      apply (simp (no_asm_simp))
     apply auto
  done

lemma wset_mem_inv_mem [rule_format]:
  "p \<in> zprime --> #5 \<le> p --> a < p - #1 --> b \<in> wset (a, p)
    --> inv p b \<in> wset (a, p)"
  apply (induct a p rule: wset_induct)
   apply auto
   apply (case_tac "b = a")
    apply (subst wset.simps)
    apply (unfold Let_def)
    apply (rule_tac [3] wset_subset)
     apply auto
  apply (case_tac "b = inv p a")
   apply (simp (no_asm_simp))
   apply (subst inv_inv)
       apply (subgoal_tac [6] "b = a \<or> b = inv p a")
        apply (rule_tac [7] wset_mem_imp_or)
          apply auto
  done

lemma wset_inv_mem_mem:
  "p \<in> zprime \<Longrightarrow> #5 \<le> p \<Longrightarrow> a < p - #1 \<Longrightarrow> #1 < b \<Longrightarrow> b < p - #1
    \<Longrightarrow> inv p b \<in> wset (a, p) \<Longrightarrow> b \<in> wset (a, p)"
  apply (rule_tac s = "inv p (inv p b)" and t = b in subst)
   apply (rule_tac [2] wset_mem_inv_mem)
      apply (rule inv_inv)
         apply simp_all
  done

lemma wset_fin: "finite (wset (a, p))"
  apply (induct a p rule: wset_induct)
   prefer 2
   apply (subst wset.simps)
   apply (unfold Let_def)
   apply auto
  done

lemma wset_zcong_prod_1 [rule_format]:
  "p \<in> zprime -->
    #5 \<le> p --> a < p - #1 --> [setprod (wset (a, p)) = #1] (mod p)"
  apply (induct a p rule: wset_induct)
   prefer 2
   apply (subst wset.simps)
   apply (unfold Let_def)
   apply auto
  apply (subst setprod_insert)
    apply (tactic {* stac (thm "setprod_insert") 3 *})
      apply (subgoal_tac [5]
	"zcong (a * inv p a * setprod (wset (a - #1, p))) (#1 * #1) p")
       prefer 5
       apply (simp add: zmult_assoc)
      apply (rule_tac [5] zcong_zmult)
       apply (rule_tac [5] inv_is_inv)
         apply (tactic "Clarify_tac 4")
         apply (subgoal_tac [4] "a \<in> wset (a - #1, p)")
          apply (rule_tac [5] wset_inv_mem_mem)
               apply (simp_all add: wset_fin)
  apply (rule inv_distinct)
    apply auto
  done

lemma d22set_eq_wset: "p \<in> zprime ==> d22set (p - #2) = wset (p - #2, p)"
  apply safe
   apply (erule wset_mem)
     apply (rule_tac [2] d22set_g_1)
     apply (rule_tac [3] d22set_le)
     apply (rule_tac [4] d22set_mem)
      apply (erule_tac [4] wset_g_1)
       prefer 6
       apply (subst zle_add1_eq_le [symmetric])
       apply (subgoal_tac "p - #2 + #1 = p - #1")
        apply (simp (no_asm_simp))
        apply (erule wset_less)
         apply auto
  done


subsection {* Wilson *}

lemma prime_g_5: "p \<in> zprime \<Longrightarrow> p \<noteq> #2 \<Longrightarrow> p \<noteq> #3 ==> #5 \<le> p"
  apply (unfold zprime_def dvd_def)
  apply (case_tac "p = #4")
   apply auto
   apply (rule notE)
    prefer 2
    apply assumption
   apply (simp (no_asm))
   apply (rule_tac x = "#2" in exI)
   apply safe
     apply (rule_tac x = "#2" in exI)
     apply auto
  apply arith
  done

theorem Wilson_Russ:
    "p \<in> zprime ==> [zfact (p - #1) = #-1] (mod p)"
  apply (subgoal_tac "[(p - #1) * zfact (p - #2) = #-1 * #1] (mod p)")
   apply (rule_tac [2] zcong_zmult)
    apply (simp only: zprime_def)
    apply (subst zfact.simps)
    apply (rule_tac t = "p - #1 - #1" and s = "p - #2" in subst)
     apply auto
   apply (simp only: zcong_def)
   apply (simp (no_asm_simp))
  apply (case_tac "p = #2")
   apply (simp add: zfact.simps)
  apply (case_tac "p = #3")
   apply (simp add: zfact.simps)
  apply (subgoal_tac "#5 \<le> p")
   apply (erule_tac [2] prime_g_5)
    apply (subst d22set_prod_zfact [symmetric])
    apply (subst d22set_eq_wset)
     apply (rule_tac [2] wset_zcong_prod_1)
       apply auto
  done

end