--- a/src/HOL/NumberTheory/WilsonRuss.thy Sat Feb 03 17:43:34 2001 +0100
+++ b/src/HOL/NumberTheory/WilsonRuss.thy Sun Feb 04 19:31:13 2001 +0100
@@ -1,21 +1,372 @@
-(* Title: WilsonRuss.thy
+(* Title: HOL/NumberTheory/WilsonRuss.thy
ID: $Id$
- Author: Thomas M. Rasmussen
- Copyright 2000 University of Cambridge
+ Author: Thomas M. Rasmussen
+ Copyright 2000 University of Cambridge
*)
-WilsonRuss = EulerFermat +
+header {* Wilson's Theorem according to Russinoff *}
+
+theory WilsonRuss = EulerFermat:
+
+text {*
+ Wilson's Theorem following quite closely Russinoff's approach
+ using Boyer-Moore (using finite sets instead of lists, though).
+*}
+
+subsection {* Definitions and lemmas *}
consts
- inv :: "[int,int] => int"
- wset :: "int*int=>int set"
+ inv :: "int => int => int"
+ wset :: "int * int => int set"
defs
- inv_def "inv p a == (a ^ (nat (p - #2))) mod p"
+ inv_def: "inv p a == (a^(nat (p - #2))) mod p"
+
+recdef wset
+ "measure ((\<lambda>(a, p). nat a) :: int * int => nat)"
+ "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 aux: "#1 < m ==> Suc (nat (m - #2)) = nat (m - #1)"
+ apply (subst int_int_eq [symmetric])
+ apply auto
+ done
+
+lemma inv_is_inv:
+ "p \<in> zprime \<Longrightarrow> #0 < a \<Longrightarrow> a < p ==> [a * inv p a = #1] (mod p)"
+ apply (unfold inv_def)
+ apply (subst zcong_zmod)
+ apply (subst zmod_zmult1_eq [symmetric])
+ apply (subst zcong_zmod [symmetric])
+ apply (subst power_Suc [symmetric])
+ apply (subst aux)
+ apply (erule_tac [2] Little_Fermat)
+ apply (erule_tac [2] zdvd_not_zless)
+ apply (unfold zprime_def)
+ apply auto
+ done
+
+lemma inv_distinct:
+ "p \<in> zprime \<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)
+ apply 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)
+ apply auto
+ done
+
+lemma inv_not_0:
+ "p \<in> zprime \<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)
+ 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 \<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)
+ apply auto
+ done
+
+lemma aux: "[a * (p - #1) = #1] (mod p) = [a = p - #1] (mod p)"
+ apply (unfold zcong_def)
+ apply (simp add: zdiff_zdiff_eq zdiff_zdiff_eq2 zdiff_zmult_distrib2)
+ apply (rule_tac s = "p dvd -((a + #1) + (p * -a))" in trans)
+ apply (simp add: zmult_commute zminus_zdiff_eq)
+ 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 \<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)
+ apply auto
+ apply (simp add: aux)
+ apply (subgoal_tac "a = p - #1")
+ apply (rule_tac [2] zcong_zless_imp_eq)
+ apply auto
+ done
+
+lemma inv_g_1:
+ "p \<in> zprime \<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)
+ apply auto
+ apply (unfold inv_def zprime_def)
+ apply (simp add: pos_mod_sign)
+ done
+
+lemma inv_less_p_minus_1:
+ "p \<in> zprime \<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)
+ apply auto
+ apply (unfold inv_def zprime_def)
+ apply (simp add: pos_mod_bound)
+ done
+
+lemma 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: zmult_int [symmetric])
+ apply (simp add: zdiff_zmult_distrib zdiff_zmult_distrib2)
+ done
+
+lemma zcong_zpower_zmult:
+ "[x^y = #1] (mod p) \<Longrightarrow> [x^(y * z) = #1] (mod p)"
+ apply (induct z)
+ apply (auto simp add: zpower_zadd_distrib)
+ apply (subgoal_tac "zcong (x^y * x^(y * n)) (#1 * #1) p")
+ apply (rule_tac [2] zcong_zmult)
+ apply simp_all
+ done
+
+lemma inv_inv: "p \<in> zprime \<Longrightarrow>
+ #5 \<le> p \<Longrightarrow> #0 < a \<Longrightarrow> a < p ==> inv p (inv p a) = a"
+ apply (unfold inv_def)
+ apply (subst zpower_zmod)
+ 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 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)
+ apply (simp_all add: pos_mod_bound pos_mod_sign)
+ done
+
+
+text {* \medskip @{term wset} *}
+
+declare wset.simps [simp del]
-recdef wset "measure ((%(a,p).(nat a)) ::int*int=>nat)"
- "wset (a,p) = (if #1<a then let ws = wset (a-#1,p) in
- (if a:ws then ws else insert a (insert (inv p a) ws))
- else {})"
+lemma wset_induct:
+ "(!!a p. P {} a p) \<Longrightarrow>
+ (!!a p. #1 < (a::int) \<Longrightarrow> P (wset (a - #1, p)) (a - #1) p
+ ==> P (wset (a, p)) a p)
+ ==> P (wset (u, v)) u v"
+proof -
+ case antecedent
+ show ?thesis
+ apply (rule wset.induct)
+ apply safe
+ apply (case_tac [2] "#1 < a")
+ apply (rule_tac [2] antecedent)
+ apply simp_all
+ apply (simp_all add: wset.simps antecedent)
+ done
+qed
+
+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)
+ apply simp
+ done
+
+lemma wset_mem_mem [simp]: "#1 < a ==> a \<in> wset (a, p)"
+ apply (subst wset.simps)
+ apply (unfold Let_def)
+ apply 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)
+ apply auto
+ done
+
+lemma wset_g_1 [rule_format]:
+ "p \<in> zprime --> a < p - #1 --> b \<in> wset (a, p) --> #1 < b"
+ apply (induct a p rule: wset_induct)
+ apply 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)
+ apply auto
+ done
+
+lemma wset_less [rule_format]:
+ "p \<in> zprime --> a < p - #1 --> b \<in> wset (a, p) --> b < p - #1"
+ apply (induct a p rule: wset_induct)
+ apply 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)
+ apply auto
+ done
+
+lemma wset_mem [rule_format]:
+ "p \<in> zprime -->
+ a < p - #1 --> #1 < b --> b \<le> a --> b \<in> wset (a, p)"
+ apply (induct a p rule: wset.induct)
+ apply auto
+ apply (subgoal_tac "b = a")
+ apply (rule_tac [2] zle_anti_sym)
+ apply (rule_tac [4] wset_subset)
+ apply (simp (no_asm_simp))
+ apply auto
+ done
+
+lemma wset_mem_inv_mem [rule_format]:
+ "p \<in> zprime --> #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)
+ apply auto
+ apply (case_tac "b = a")
+ apply (subst wset.simps)
+ apply (unfold Let_def)
+ apply (rule_tac [3] wset_subset)
+ apply 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)
+ apply auto
+ done
+
+lemma wset_inv_mem_mem:
+ "p \<in> zprime \<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)
+ apply 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)
+ apply auto
+ done
+
+lemma wset_zcong_prod_1 [rule_format]:
+ "p \<in> zprime -->
+ #5 \<le> p --> a < p - #1 --> [setprod (wset (a, p)) = #1] (mod p)"
+ apply (induct a p rule: wset_induct)
+ prefer 2
+ apply (subst wset.simps)
+ apply (unfold Let_def)
+ apply auto
+ apply (subst setprod_insert)
+ apply (tactic {* stac (thm "setprod_insert") 3 *})
+ apply (subgoal_tac [5]
+ "zcong (a * inv p a * setprod (wset (a - #1, p))) (#1 * #1) p")
+ prefer 5
+ apply (simp add: zmult_assoc)
+ apply (rule_tac [5] zcong_zmult)
+ apply (rule_tac [5] inv_is_inv)
+ apply (tactic "Clarify_tac 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)
+ apply auto
+ done
+
+lemma d22set_eq_wset: "p \<in> zprime ==> 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)
+ apply auto
+ done
+
+
+subsection {* Wilson *}
+
+lemma prime_g_5: "p \<in> zprime \<Longrightarrow> p \<noteq> #2 \<Longrightarrow> p \<noteq> #3 ==> #5 \<le> p"
+ apply (unfold zprime_def dvd_def)
+ apply (case_tac "p = #4")
+ apply auto
+ apply (rule notE)
+ prefer 2
+ apply assumption
+ apply (simp (no_asm))
+ apply (rule_tac x = "#2" in exI)
+ apply safe
+ apply (rule_tac x = "#2" in exI)
+ apply auto
+ apply arith
+ done
+
+theorem Wilson_Russ:
+ "p \<in> zprime ==> [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)
+ apply 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)
+ apply auto
+ done
end