--- a/src/HOL/NumberTheory/WilsonRuss.thy Tue Sep 29 22:15:54 2009 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,329 +0,0 @@
-(* Title: HOL/NumberTheory/WilsonRuss.thy
- ID: $Id$
- Author: Thomas M. Rasmussen
- Copyright 2000 University of Cambridge
-*)
-
-header {* Wilson's Theorem according to Russinoff *}
-
-theory WilsonRuss imports EulerFermat begin
-
-text {*
- Wilson's Theorem following quite closely Russinoff's approach
- using Boyer-Moore (using finite sets instead of lists, though).
-*}
-
-subsection {* Definitions and lemmas *}
-
-definition
- inv :: "int => int => int" where
- "inv p a = (a^(nat (p - 2))) mod p"
-
-consts
- wset :: "int * int => int set"
-
-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 inv_is_inv_aux: "1 < m ==> Suc (nat (m - 2)) = nat (m - 1)"
-by (subst int_int_eq [symmetric], auto)
-
-lemma inv_is_inv:
- "zprime p \<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 inv_is_inv_aux)
- apply (erule_tac [2] Little_Fermat)
- apply (erule_tac [2] zdvd_not_zless)
- apply (unfold zprime_def, auto)
- done
-
-lemma inv_distinct:
- "zprime p \<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, 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, auto)
- done
-
-lemma inv_not_0:
- "zprime p \<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, 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], auto)
- done
-
-lemma inv_not_1:
- "zprime p \<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, auto)
- done
-
-lemma inv_not_p_minus_1_aux:
- "[a * (p - 1) = 1] (mod p) = [a = p - 1] (mod p)"
- 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, auto)
- done
-
-lemma inv_not_p_minus_1:
- "zprime p \<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, auto)
- apply (simp add: inv_not_p_minus_1_aux)
- apply (subgoal_tac "a = p - 1")
- apply (rule_tac [2] zcong_zless_imp_eq, auto)
- done
-
-lemma inv_g_1:
- "zprime p \<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, auto)
- apply (unfold inv_def zprime_def, simp)
- done
-
-lemma inv_less_p_minus_1:
- "zprime p \<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, auto)
- apply (unfold inv_def zprime_def, simp)
- done
-
-lemma inv_inv_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 * z)) (1 * 1) p")
- apply (rule_tac [2] zcong_zmult, simp_all)
- done
-
-lemma inv_inv: "zprime p \<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 inv_inv_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, simp_all)
- done
-
-
-text {* \medskip @{term wset} *}
-
-declare wset.simps [simp del]
-
-lemma wset_induct:
- assumes "!!a p. P {} a p"
- and "!!a p. 1 < (a::int) \<Longrightarrow>
- P (wset (a - 1, p)) (a - 1) p ==> P (wset (a, p)) a p"
- shows "P (wset (u, v)) u v"
- apply (rule wset.induct, safe)
- prefer 2
- apply (case_tac "1 < a")
- apply (rule prems)
- apply simp_all
- apply (simp_all add: wset.simps prems)
- done
-
-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, simp)
- done
-
-lemma wset_mem_mem [simp]: "1 < a ==> a \<in> wset (a, p)"
- apply (subst wset.simps)
- apply (unfold Let_def, 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, auto)
- done
-
-lemma wset_g_1 [rule_format]:
- "zprime p --> a < p - 1 --> b \<in> wset (a, p) --> 1 < b"
- apply (induct a p rule: wset_induct, 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, auto)
- done
-
-lemma wset_less [rule_format]:
- "zprime p --> a < p - 1 --> b \<in> wset (a, p) --> b < p - 1"
- apply (induct a p rule: wset_induct, 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, auto)
- done
-
-lemma wset_mem [rule_format]:
- "zprime p -->
- a < p - 1 --> 1 < b --> b \<le> a --> b \<in> wset (a, p)"
- apply (induct a p rule: wset.induct, auto)
- apply (rule_tac wset_subset)
- apply (simp (no_asm_simp))
- apply auto
- done
-
-lemma wset_mem_inv_mem [rule_format]:
- "zprime p --> 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, auto)
- apply (case_tac "b = a")
- apply (subst wset.simps)
- apply (unfold Let_def)
- apply (rule_tac [3] wset_subset, 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, auto)
- done
-
-lemma wset_inv_mem_mem:
- "zprime p \<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, 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, auto)
- done
-
-lemma wset_zcong_prod_1 [rule_format]:
- "zprime p -->
- 5 \<le> p --> a < p - 1 --> [(\<Prod>x\<in>wset(a, p). x) = 1] (mod p)"
- apply (induct a p rule: wset_induct)
- prefer 2
- apply (subst wset.simps)
- apply (unfold Let_def, auto)
- apply (subst setprod_insert)
- apply (tactic {* stac (thm "setprod_insert") 3 *})
- apply (subgoal_tac [5]
- "zcong (a * inv p a * (\<Prod>x\<in> wset(a - 1, p). x)) (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 @{claset} 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, auto)
- done
-
-lemma d22set_eq_wset: "zprime p ==> 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, auto)
- done
-
-
-subsection {* Wilson *}
-
-lemma prime_g_5: "zprime p \<Longrightarrow> p \<noteq> 2 \<Longrightarrow> p \<noteq> 3 ==> 5 \<le> p"
- apply (unfold zprime_def dvd_def)
- apply (case_tac "p = 4", auto)
- apply (rule notE)
- prefer 2
- apply assumption
- apply (simp (no_asm))
- apply (rule_tac x = 2 in exI)
- apply (safe, arith)
- apply (rule_tac x = 2 in exI, auto)
- done
-
-theorem Wilson_Russ:
- "zprime p ==> [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, 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, auto)
- done
-
-end