--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Old_Number_Theory/WilsonBij.thy Tue Sep 01 15:39:33 2009 +0200
@@ -0,0 +1,261 @@
+(* 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 = (\<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 {* \medskip Inverse *}
+
+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, 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 \<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 dvd_minus_iff [symmetric])
+ apply auto
+ done
+
+lemma inv_not_1:
+ "zprime p ==> 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 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"
+ -- {* 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 \<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"
+ -- {* 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: "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)) \<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 {* 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) \<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:
+ "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 setprod_insert)
+ prefer 3
+ apply (subst setprod_insert)
+ apply (auto simp add: fin_bijER)
+ apply (subgoal_tac "zcong ((a * b) * \<Prod>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