--- a/src/HOL/Old_Number_Theory/WilsonBij.thy Tue Oct 18 07:04:08 2016 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,256 +0,0 @@
-(* Title: HOL/Old_Number_Theory/WilsonBij.thy
- Author: Thomas M. Rasmussen
- Copyright 2000 University of Cambridge
-*)
-
-section \<open>Wilson's Theorem using a more abstract approach\<close>
-
-theory WilsonBij
-imports BijectionRel IntFact
-begin
-
-text \<open>
- Wilson's Theorem using a more ``abstract'' approach based on
- bijections between sets. Does not use Fermat's Little Theorem
- (unlike Russinoff).
-\<close>
-
-
-subsection \<open>Definitions and lemmas\<close>
-
-definition reciR :: "int => int => int => bool"
- where "reciR p = (\<lambda>a b. zcong (a * b) 1 p \<and> 1 < a \<and> a < p - 1 \<and> 1 < b \<and> b < p - 1)"
-
-definition inv :: "int => int => int" where
- "inv p a =
- (if zprime p \<and> 0 < a \<and> a < p then
- (SOME x. 0 \<le> x \<and> x < p \<and> zcong (a * x) 1 p)
- else 0)"
-
-
-text \<open>\medskip Inverse\<close>
-
-lemma inv_correct:
- "zprime p ==> 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]
-lemmas inv_less = inv_correct [THEN conjunct2, THEN conjunct1]
-lemmas inv_is_inv = inv_correct [THEN conjunct2, THEN conjunct2]
-
-lemma inv_not_0:
- "zprime p ==> 1 < a ==> a < p - 1 ==> inv p a \<noteq> 0"
- \<comment> \<open>same as \<open>WilsonRuss\<close>\<close>
- apply safe
- apply (cut_tac a = a and p = p in inv_is_inv)
- apply (unfold zcong_def)
- apply auto
- done
-
-lemma inv_not_1:
- "zprime p ==> 1 < a ==> a < p - 1 ==> inv p a \<noteq> 1"
- \<comment> \<open>same as \<open>WilsonRuss\<close>\<close>
- 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)"
- \<comment> \<open>same as \<open>WilsonRuss\<close>\<close>
- 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)
- apply auto
- done
-
-lemma inv_not_p_minus_1:
- "zprime p ==> 1 < a ==> a < p - 1 ==> inv p a \<noteq> p - 1"
- \<comment> \<open>same as \<open>WilsonRuss\<close>\<close>
- 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 \<open>
- Below is slightly different as we don't expand @{term [source] inv}
- but use ``\<open>correct\<close>'' theorems.
-\<close>
-
-lemma inv_g_1: "zprime p ==> 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:
- "zprime p ==> 1 < a ==> a < p - 1 ==> inv p a < p - 1"
- \<comment> \<open>ditto\<close>
- apply (subst order_less_le)
- apply (simp add: inv_not_p_minus_1 inv_less)
- done
-
-
-text \<open>\medskip Bijection\<close>
-
-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: "zprime p ==> inj_on (inv p) (d22set (p - 2))"
- apply (unfold inj_on_def)
- apply auto
- apply (rule zcong_zless_imp_eq)
- apply (tactic \<open>stac @{context} (@{thm zcong_cancel} RS sym) 5\<close>)
- apply (rule_tac [7] zcong_trans)
- apply (tactic \<open>stac @{context} @{thm zcong_sym} 8\<close>)
- apply (erule_tac [7] inv_is_inv)
- apply (tactic "asm_simp_tac @{context} 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)) \<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: "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 \<open>stac @{context} (@{thm zcong_cancel2} RS sym) 5\<close>)
- apply (rule_tac [7] zcong_trans)
- apply (tactic \<open>stac @{context} @{thm zcong_sym} 8\<close>)
- apply (rule_tac [6] zless_zprime_imp_zrelprime)
- apply auto
- apply (rule zcong_zless_imp_eq)
- apply (tactic \<open>stac @{context} (@{thm zcong_cancel} RS sym) 5\<close>)
- apply (rule_tac [7] zcong_trans)
- apply (tactic \<open>stac @{context} @{thm zcong_sym} 8\<close>)
- 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: mult.commute)
- apply auto
- done
-
-lemma bijER_d22set: "zprime p ==> 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 \<open>Wilson\<close>
-
-lemma bijER_zcong_prod_1:
- "zprime p ==> A \<in> bijER (reciR p) ==> [\<Prod>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 prod.insert)
- prefer 3
- apply (subst prod.insert)
- apply (auto simp add: fin_bijER)
- apply (subgoal_tac "zcong ((a * b) * \<Prod>A) (1 * 1) p")
- apply (simp add: mult.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