(*  Title:      HOL/Old_Number_Theory/WilsonRuss.thy

     Author:     Thomas M. Rasmussen

     Copyright   2000  University of Cambridge

 *)

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

 theory WilsonRuss

 imports EulerFermat

 begin

 text {*

   Wilson's Theorem following quite closely Russinoff's approach

   using Boyer-Moore (using finite sets instead of lists, though).

 *}

 subsection {* Definitions and lemmas *}

 definition inv :: "int => int => int"

   where "inv p a = (a^(nat (p - 2))) mod p"

 fun wset :: "int \<Rightarrow> int => int set" where

   "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:

     "zprime p \<Longrightarrow> 0 < a \<Longrightarrow> a < p ==> [a * inv p a = 1] (mod p)"

   apply (unfold inv_def)

   apply (subst zcong_zmod)

   apply (subst mod_mult_right_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:

     "zprime p \<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:

     "zprime p \<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 dvd_minus_iff [symmetric], auto)

   done

 lemma inv_not_1:

     "zprime p \<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: diff_diff_eq diff_diff_eq2 right_diff_distrib)

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

    apply (simp add: algebra_simps)

   apply (subst dvd_minus_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:

     "zprime p \<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:

     "zprime p \<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:

     "zprime p \<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: of_nat_mult)

   apply (simp add: left_diff_distrib right_diff_distrib)

   done

 lemma zcong_zpower_zmult:

     "[x^y = 1] (mod p) \<Longrightarrow> [x^(y * z) = 1] (mod p)"

   apply (induct z)

    apply (auto simp add: power_add)

   apply (subgoal_tac "zcong (x^y * x^(y * z)) (1 * 1) p")

    apply (rule_tac [2] zcong_zmult, simp_all)

   done

 lemma inv_inv: "zprime p \<Longrightarrow>

     5 \<le> p \<Longrightarrow> 0 < a \<Longrightarrow> a < p ==> inv p (inv p a) = a"

   apply (unfold inv_def)

   apply (subst power_mod)

   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:

   assumes "!!a p. P {} a p"

     and "!!a p. 1 < (a::int) \<Longrightarrow>

       P (wset (a - 1) p) (a - 1) p ==> P (wset a p) a p"

   shows "P (wset u v) u v"

   apply (rule wset.induct)

   apply (case_tac "1 < a")

    apply (rule assms)

     apply (simp_all add: wset.simps assms)

   done

 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]:

     "zprime p --> 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]:

     "zprime p --> 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]:

   "zprime p -->

     a < p - 1 --> 1 < b --> b \<le> a --> b \<in> wset a p"

   apply (induct a p rule: wset.induct, auto)

   apply (rule_tac wset_subset)

   apply (simp (no_asm_simp))

   apply auto

   done

 lemma wset_mem_inv_mem [rule_format]:

   "zprime p --> 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:

   "zprime p \<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]:

   "zprime p -->

     5 \<le> p --> a < p - 1 --> [(\<Prod>x\<in>wset a p. x) = 1] (mod p)"

   apply (induct a p rule: wset_induct)

    prefer 2

    apply (subst wset.simps)

    apply (auto, unfold Let_def, auto)

   apply (subst setprod_insert)

     apply (tactic {* stac @{thm setprod_insert} 3 *})

       apply (subgoal_tac [5]

         "zcong (a * inv p a * (\<Prod>x\<in>wset (a - 1) p. x)) (1 * 1) p")

        prefer 5

        apply (simp add: mult_assoc)

       apply (rule_tac [5] zcong_zmult)

        apply (rule_tac [5] inv_is_inv)

          apply (tactic "clarify_tac @{context} 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: "zprime p ==> 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: "zprime p \<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:

     "zprime p ==> [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