(*  Title:      HOL/NumberTheory/WilsonBij.thy

     ID:         $Id$

     Author:     Thomas M. Rasmussen

     Copyright   2000  University of Cambridge

 *)

 header {* Wilson's Theorem using a more abstract approach *}

 theory WilsonBij = BijectionRel + IntFact:

 text {*

   Wilson's Theorem using a more ``abstract'' approach based on

   bijections between sets.  Does not use Fermat's Little Theorem

   (unlike Russinoff).

 *}

 subsection {* Definitions and lemmas *}

 constdefs

   reciR :: "int => int => int => bool"

   "reciR p ==

     \<lambda>a b. zcong (a * b) 1 p \<and> 1 < a \<and> a < p - 1 \<and> 1 < b \<and> b < p - 1"

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

   "inv p a ==

     if p \<in> zprime \<and> 0 < a \<and> a < p then

       (SOME x. 0 \<le> x \<and> x < p \<and> zcong (a * x) 1 p)

     else 0"

 text {* \medskip Inverse *}

 lemma inv_correct:

   "p \<in> zprime ==> 0 < a ==> a < p

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

   apply (unfold inv_def)

   apply (simp (no_asm_simp))

   apply (rule zcong_lineq_unique [THEN ex1_implies_ex, THEN someI_ex])

    apply (erule_tac [2] zless_zprime_imp_zrelprime)

     apply (unfold zprime_def)

     apply auto

   done

 lemmas inv_ge = inv_correct [THEN conjunct1, standard]

 lemmas inv_less = inv_correct [THEN conjunct2, THEN conjunct1, standard]

 lemmas inv_is_inv = inv_correct [THEN conjunct2, THEN conjunct2, standard]

 lemma inv_not_0:

   "p \<in> zprime ==> 1 < a ==> a < p - 1 ==> inv p a \<noteq> 0"

   -- {* same as @{text WilsonRuss} *}

   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 ==> 1 < a ==> a < p - 1 ==> inv p a \<noteq> 1"

   -- {* same as @{text WilsonRuss} *}

   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)"

   -- {* same as @{text WilsonRuss} *}

   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)

    apply auto

   done

 lemma inv_not_p_minus_1:

   "p \<in> zprime ==> 1 < a ==> a < p - 1 ==> inv p a \<noteq> p - 1"

   -- {* same as @{text WilsonRuss} *}

   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

 text {*

   Below is slightly different as we don't expand @{term [source] inv}

   but use ``@{text correct}'' theorems.

 *}

 lemma inv_g_1: "p \<in> zprime ==> 1 < a ==> a < p - 1 ==> 1 < 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 (rule inv_ge)

     apply auto

   done

 lemma inv_less_p_minus_1:

   "p \<in> zprime ==> 1 < a ==> a < p - 1 ==> inv p a < p - 1"

   -- {* ditto *}

   apply (subst order_less_le)

   apply (simp add: inv_not_p_minus_1 inv_less)

   done

 text {* \medskip Bijection *}

 lemma aux1: "1 < x ==> 0 \<le> (x::int)"

   apply auto

   done

 lemma aux2: "1 < x ==> 0 < (x::int)"

   apply auto

   done

 lemma aux3: "x \<le> p - 2 ==> x < (p::int)"

   apply auto

   done

 lemma aux4: "x \<le> p - 2 ==> x < (p::int) - 1"

   apply auto

   done

 lemma inv_inj: "p \<in> zprime ==> inj_on (inv p) (d22set (p - 2))"

   apply (unfold inj_on_def)

   apply auto

   apply (rule zcong_zless_imp_eq)

       apply (tactic {* stac (thm "zcong_cancel" RS sym) 5 *})

         apply (rule_tac [7] zcong_trans)

          apply (tactic {* stac (thm "zcong_sym") 8 *})

          apply (erule_tac [7] inv_is_inv)

           apply (tactic "Asm_simp_tac 9")

           apply (erule_tac [9] inv_is_inv)

            apply (rule_tac [6] zless_zprime_imp_zrelprime)

              apply (rule_tac [8] inv_less)

                apply (rule_tac [7] inv_g_1 [THEN aux2])

                  apply (unfold zprime_def)

                  apply (auto intro: d22set_g_1 d22set_le

 		   aux1 aux2 aux3 aux4)

   done

 lemma inv_d22set_d22set:

     "p \<in> zprime ==> inv p ` d22set (p - 2) = d22set (p - 2)"

   apply (rule endo_inj_surj)

     apply (rule d22set_fin)

    apply (erule_tac [2] inv_inj)

   apply auto

   apply (rule d22set_mem)

    apply (erule inv_g_1)

     apply (subgoal_tac [3] "inv p xa < p - 1")

      apply (erule_tac [4] inv_less_p_minus_1)

       apply (auto intro: d22set_g_1 d22set_le aux4)

   done

 lemma d22set_d22set_bij:

     "p \<in> zprime ==> (d22set (p - 2), d22set (p - 2)) \<in> bijR (reciR p)"

   apply (unfold reciR_def)

   apply (rule_tac s = "(d22set (p - 2), inv p ` d22set (p - 2))" in subst)

    apply (simp add: inv_d22set_d22set)

   apply (rule inj_func_bijR)

     apply (rule_tac [3] d22set_fin)

    apply (erule_tac [2] inv_inj)

   apply auto

       apply (erule inv_is_inv)

        apply (erule_tac [5] inv_g_1)

         apply (erule_tac [7] inv_less_p_minus_1)

          apply (auto intro: d22set_g_1 d22set_le aux2 aux3 aux4)

   done

 lemma reciP_bijP: "p \<in> zprime ==> bijP (reciR p) (d22set (p - 2))"

   apply (unfold reciR_def bijP_def)

   apply auto

   apply (rule d22set_mem)

    apply auto

   done

 lemma reciP_uniq: "p \<in> zprime ==> uniqP (reciR p)"

   apply (unfold reciR_def uniqP_def)

   apply auto

    apply (rule zcong_zless_imp_eq)

        apply (tactic {* stac (thm "zcong_cancel2" RS sym) 5 *})

          apply (rule_tac [7] zcong_trans)

           apply (tactic {* stac (thm "zcong_sym") 8 *})

           apply (rule_tac [6] zless_zprime_imp_zrelprime)

             apply auto

   apply (rule zcong_zless_imp_eq)

       apply (tactic {* stac (thm "zcong_cancel" RS sym) 5 *})

         apply (rule_tac [7] zcong_trans)

          apply (tactic {* stac (thm "zcong_sym") 8 *})

          apply (rule_tac [6] zless_zprime_imp_zrelprime)

            apply auto

   done

 lemma reciP_sym: "p \<in> zprime ==> symP (reciR p)"

   apply (unfold reciR_def symP_def)

   apply (simp add: zmult_commute)

   apply auto

   done

 lemma bijER_d22set: "p \<in> zprime ==> d22set (p - 2) \<in> bijER (reciR p)"

   apply (rule bijR_bijER)

      apply (erule d22set_d22set_bij)

     apply (erule reciP_bijP)

    apply (erule reciP_uniq)

   apply (erule reciP_sym)

   done

 subsection {* Wilson *}

 lemma bijER_zcong_prod_1:

     "p \<in> zprime ==> A \<in> bijER (reciR p) ==> [setprod A = 1] (mod p)"

   apply (unfold reciR_def)

   apply (erule bijER.induct)

     apply (subgoal_tac [2] "a = 1 \<or> a = p - 1")

      apply (rule_tac [3] zcong_square_zless)

         apply auto

   apply (subst setprod_insert)

     prefer 3

     apply (subst setprod_insert)

       apply (auto simp add: fin_bijER)

   apply (subgoal_tac "zcong ((a * b) * setprod A) (1 * 1) p")

    apply (simp add: zmult_assoc)

   apply (rule zcong_zmult)

    apply auto

   done

 theorem Wilson_Bij: "p \<in> zprime ==> [zfact (p - 1) = -1] (mod p)"

   apply (subgoal_tac "zcong ((p - 1) * zfact (p - 2)) (-1 * 1) p")

    apply (rule_tac [2] zcong_zmult)

     apply (simp add: zprime_def)

     apply (subst zfact.simps)

     apply (rule_tac t = "p - 1 - 1" and s = "p - 2" in subst)

      apply auto

    apply (simp add: zcong_def)

   apply (subst d22set_prod_zfact [symmetric])

   apply (rule bijER_zcong_prod_1)

    apply (rule_tac [2] bijER_d22set)

    apply auto

   done

 end