(*  Title:      HOL/NumberTheory/WilsonRuss.thy

    ID:         $Id$

    Author:     Thomas M. Rasmussen

    Copyright   2000  University of Cambridge

Changes by Jeremy Avigad, 2003/02/21:

    repaired proof of prime_g_5

*)

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 inv_is_inv_aux: "1 < m ==> Suc (nat (m - 2)) = nat (m - 1)"

by (subst int_int_eq [symmetric], auto)

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 inv_is_inv_aux)

   apply (erule_tac [2] Little_Fermat)

   apply (erule_tac [2] zdvd_not_zless)

   apply (unfold zprime_def, 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, 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, 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, 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], 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, auto)

  done

lemma inv_not_p_minus_1_aux: "[a * (p - 1) = 1] (mod p) = [a = p - 1] (mod p)"

  apply (unfold zcong_def)

  apply (simp add: OrderedGroup.diff_diff_eq diff_diff_eq2 zdiff_zmult_distrib2)

  apply (rule_tac s = "p dvd -((a + 1) + (p * -a))" in trans)

   apply (simp add: mult_commute)

  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, 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, auto)

  apply (simp add: inv_not_p_minus_1_aux)

  apply (subgoal_tac "a = p - 1")

   apply (rule_tac [2] zcong_zless_imp_eq, 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, auto)

  apply (unfold inv_def zprime_def, simp)

  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, auto)

  apply (unfold inv_def zprime_def, simp)

  done

lemma inv_inv_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, 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 inv_inv_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, simp_all)

  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 rule_context

  show ?thesis

    apply (rule wset.induct, safe)

     apply (case_tac [2] "1 < a")

      apply (rule_tac [2] rule_context, simp_all)

      apply (simp_all add: wset.simps rule_context)

    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, simp)

  done

lemma wset_mem_mem [simp]: "1 < a ==> a \<in> wset (a, p)"

  apply (subst wset.simps)

  apply (unfold Let_def, 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, 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, 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, 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, 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, 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, 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, auto)

   apply (case_tac "b = a")

    apply (subst wset.simps)

    apply (unfold Let_def)

    apply (rule_tac [3] wset_subset, 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, 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, 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, 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, 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, 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, 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", auto)

   apply (rule notE)

    prefer 2

    apply assumption

   apply (simp (no_asm))

   apply (rule_tac x = 2 in exI)

   apply (safe, arith)

     apply (rule_tac x = 2 in exI, auto)

  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, 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, auto)

  done

end