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theory WilsonBij(* Title: HOL/NumberTheory/WilsonBij.thy ID: $Id: WilsonBij.thy,v 1.13 2007/07/21 21:25:04 wenzelm Exp $ Author: Thomas M. Rasmussen Copyright 2000 University of Cambridge *) header {* Wilson's Theorem using a more abstract approach *} theory WilsonBij imports BijectionRel IntFact begin 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 *} definition reciR :: "int => int => int => bool" where "reciR p = (λa b. zcong (a * b) 1 p ∧ 1 < a ∧ a < p - 1 ∧ 1 < b ∧ b < p - 1)" definition inv :: "int => int => int" where "inv p a = (if zprime p ∧ 0 < a ∧ a < p then (SOME x. 0 ≤ x ∧ x < p ∧ zcong (a * x) 1 p) else 0)" text {* \medskip Inverse *} lemma inv_correct: "zprime p ==> 0 < a ==> a < p ==> 0 ≤ inv p a ∧ inv p a < p ∧ [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: "zprime p ==> 1 < a ==> a < p - 1 ==> inv p a ≠ 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 "¬ 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: "zprime p ==> 1 < a ==> a < p - 1 ==> inv p a ≠ 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: "zprime p ==> 1 < a ==> a < p - 1 ==> inv p a ≠ 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: "zprime p ==> 1 < a ==> a < p - 1 ==> 1 < inv p a" apply (subgoal_tac "inv p a ≠ 1") apply (subgoal_tac "inv p a ≠ 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: "zprime p ==> 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 ≤ (x::int)" apply auto done lemma aux2: "1 < x ==> 0 < (x::int)" apply auto done lemma aux3: "x ≤ p - 2 ==> x < (p::int)" apply auto done lemma aux4: "x ≤ p - 2 ==> x < (p::int) - 1" apply auto done lemma inv_inj: "zprime p ==> 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 @{simpset} 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: "zprime p ==> 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: "zprime p ==> (d22set (p - 2), d22set (p - 2)) ∈ 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: "zprime p ==> bijP (reciR p) (d22set (p - 2))" apply (unfold reciR_def bijP_def) apply auto apply (rule d22set_mem) apply auto done lemma reciP_uniq: "zprime p ==> 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: "zprime p ==> symP (reciR p)" apply (unfold reciR_def symP_def) apply (simp add: zmult_commute) apply auto done lemma bijER_d22set: "zprime p ==> d22set (p - 2) ∈ 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: "zprime p ==> A ∈ bijER (reciR p) ==> [∏A = 1] (mod p)" apply (unfold reciR_def) apply (erule bijER.induct) apply (subgoal_tac [2] "a = 1 ∨ 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) * ∏A) (1 * 1) p") apply (simp add: zmult_assoc) apply (rule zcong_zmult) apply auto done theorem Wilson_Bij: "zprime p ==> [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
lemma inv_correct:
[| zprime p; 0 < a; a < p |]
==> 0 ≤ WilsonBij.inv p a ∧
WilsonBij.inv p a < p ∧ [a * WilsonBij.inv p a = 1] (mod p)
lemma inv_ge:
[| zprime p; 0 < a; a < p |] ==> 0 ≤ WilsonBij.inv p a
lemma inv_less:
[| zprime p; 0 < a; a < p |] ==> WilsonBij.inv p a < p
lemma inv_is_inv:
[| zprime p; 0 < a; a < p |] ==> [a * WilsonBij.inv p a = 1] (mod p)
lemma inv_not_0:
[| zprime p; 1 < a; a < p - 1 |] ==> WilsonBij.inv p a ≠ 0
lemma inv_not_1:
[| zprime p; 1 < a; a < p - 1 |] ==> WilsonBij.inv p a ≠ 1
lemma aux:
[a * (p - 1) = 1] (mod p) = [a = p - 1] (mod p)
lemma inv_not_p_minus_1:
[| zprime p; 1 < a; a < p - 1 |] ==> WilsonBij.inv p a ≠ p - 1
lemma inv_g_1:
[| zprime p; 1 < a; a < p - 1 |] ==> 1 < WilsonBij.inv p a
lemma inv_less_p_minus_1:
[| zprime p; 1 < a; a < p - 1 |] ==> WilsonBij.inv p a < p - 1
lemma aux1:
1 < x ==> 0 ≤ x
lemma aux2:
1 < x ==> 0 < x
lemma aux3:
x ≤ p - 2 ==> x < p
lemma aux4:
x ≤ p - 2 ==> x < p - 1
lemma inv_inj:
zprime p ==> inj_on (WilsonBij.inv p) (d22set (p - 2))
lemma inv_d22set_d22set:
zprime p ==> WilsonBij.inv p ` d22set (p - 2) = d22set (p - 2)
lemma d22set_d22set_bij:
zprime p ==> (d22set (p - 2), d22set (p - 2)) ∈ bijR (reciR p)
lemma reciP_bijP:
zprime p ==> bijP (reciR p) (d22set (p - 2))
lemma reciP_uniq:
zprime p ==> uniqP (reciR p)
lemma reciP_sym:
zprime p ==> symP (reciR p)
lemma bijER_d22set:
zprime p ==> d22set (p - 2) ∈ bijER (reciR p)
lemma bijER_zcong_prod_1:
[| zprime p; A ∈ bijER (reciR p) |] ==> [∏A = 1] (mod p)
theorem Wilson_Bij:
zprime p ==> [zfact (p - 1) = -1] (mod p)