--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Old_Number_Theory/EulerFermat.thy Tue Sep 01 15:39:33 2009 +0200
@@ -0,0 +1,346 @@
+(* Author: Thomas M. Rasmussen
+ Copyright 2000 University of Cambridge
+*)
+
+header {* Fermat's Little Theorem extended to Euler's Totient function *}
+
+theory EulerFermat
+imports BijectionRel IntFact
+begin
+
+text {*
+ Fermat's Little Theorem extended to Euler's Totient function. More
+ abstract approach than Boyer-Moore (which seems necessary to achieve
+ the extended version).
+*}
+
+
+subsection {* Definitions and lemmas *}
+
+inductive_set
+ RsetR :: "int => int set set"
+ for m :: int
+ where
+ empty [simp]: "{} \<in> RsetR m"
+ | insert: "A \<in> RsetR m ==> zgcd a m = 1 ==>
+ \<forall>a'. a' \<in> A --> \<not> zcong a a' m ==> insert a A \<in> RsetR m"
+
+consts
+ BnorRset :: "int * int => int set"
+
+recdef BnorRset
+ "measure ((\<lambda>(a, m). nat a) :: int * int => nat)"
+ "BnorRset (a, m) =
+ (if 0 < a then
+ let na = BnorRset (a - 1, m)
+ in (if zgcd a m = 1 then insert a na else na)
+ else {})"
+
+definition
+ norRRset :: "int => int set" where
+ "norRRset m = BnorRset (m - 1, m)"
+
+definition
+ noXRRset :: "int => int => int set" where
+ "noXRRset m x = (\<lambda>a. a * x) ` norRRset m"
+
+definition
+ phi :: "int => nat" where
+ "phi m = card (norRRset m)"
+
+definition
+ is_RRset :: "int set => int => bool" where
+ "is_RRset A m = (A \<in> RsetR m \<and> card A = phi m)"
+
+definition
+ RRset2norRR :: "int set => int => int => int" where
+ "RRset2norRR A m a =
+ (if 1 < m \<and> is_RRset A m \<and> a \<in> A then
+ SOME b. zcong a b m \<and> b \<in> norRRset m
+ else 0)"
+
+definition
+ zcongm :: "int => int => int => bool" where
+ "zcongm m = (\<lambda>a b. zcong a b m)"
+
+lemma abs_eq_1_iff [iff]: "(abs z = (1::int)) = (z = 1 \<or> z = -1)"
+ -- {* LCP: not sure why this lemma is needed now *}
+ by (auto simp add: abs_if)
+
+
+text {* \medskip @{text norRRset} *}
+
+declare BnorRset.simps [simp del]
+
+lemma BnorRset_induct:
+ assumes "!!a m. P {} a m"
+ and "!!a m. 0 < (a::int) ==> P (BnorRset (a - 1, m::int)) (a - 1) m
+ ==> P (BnorRset(a,m)) a m"
+ shows "P (BnorRset(u,v)) u v"
+ apply (rule BnorRset.induct)
+ apply safe
+ apply (case_tac [2] "0 < a")
+ apply (rule_tac [2] prems)
+ apply simp_all
+ apply (simp_all add: BnorRset.simps prems)
+ done
+
+lemma Bnor_mem_zle [rule_format]: "b \<in> BnorRset (a, m) \<longrightarrow> b \<le> a"
+ apply (induct a m rule: BnorRset_induct)
+ apply simp
+ apply (subst BnorRset.simps)
+ apply (unfold Let_def, auto)
+ done
+
+lemma Bnor_mem_zle_swap: "a < b ==> b \<notin> BnorRset (a, m)"
+ by (auto dest: Bnor_mem_zle)
+
+lemma Bnor_mem_zg [rule_format]: "b \<in> BnorRset (a, m) --> 0 < b"
+ apply (induct a m rule: BnorRset_induct)
+ prefer 2
+ apply (subst BnorRset.simps)
+ apply (unfold Let_def, auto)
+ done
+
+lemma Bnor_mem_if [rule_format]:
+ "zgcd b m = 1 --> 0 < b --> b \<le> a --> b \<in> BnorRset (a, m)"
+ apply (induct a m rule: BnorRset.induct, auto)
+ apply (subst BnorRset.simps)
+ defer
+ apply (subst BnorRset.simps)
+ apply (unfold Let_def, auto)
+ done
+
+lemma Bnor_in_RsetR [rule_format]: "a < m --> BnorRset (a, m) \<in> RsetR m"
+ apply (induct a m rule: BnorRset_induct, simp)
+ apply (subst BnorRset.simps)
+ apply (unfold Let_def, auto)
+ apply (rule RsetR.insert)
+ apply (rule_tac [3] allI)
+ apply (rule_tac [3] impI)
+ apply (rule_tac [3] zcong_not)
+ apply (subgoal_tac [6] "a' \<le> a - 1")
+ apply (rule_tac [7] Bnor_mem_zle)
+ apply (rule_tac [5] Bnor_mem_zg, auto)
+ done
+
+lemma Bnor_fin: "finite (BnorRset (a, m))"
+ apply (induct a m rule: BnorRset_induct)
+ prefer 2
+ apply (subst BnorRset.simps)
+ apply (unfold Let_def, auto)
+ done
+
+lemma norR_mem_unique_aux: "a \<le> b - 1 ==> a < (b::int)"
+ apply auto
+ done
+
+lemma norR_mem_unique:
+ "1 < m ==>
+ zgcd a m = 1 ==> \<exists>!b. [a = b] (mod m) \<and> b \<in> norRRset m"
+ apply (unfold norRRset_def)
+ apply (cut_tac a = a and m = m in zcong_zless_unique, auto)
+ apply (rule_tac [2] m = m in zcong_zless_imp_eq)
+ apply (auto intro: Bnor_mem_zle Bnor_mem_zg zcong_trans
+ order_less_imp_le norR_mem_unique_aux simp add: zcong_sym)
+ apply (rule_tac x = b in exI, safe)
+ apply (rule Bnor_mem_if)
+ apply (case_tac [2] "b = 0")
+ apply (auto intro: order_less_le [THEN iffD2])
+ prefer 2
+ apply (simp only: zcong_def)
+ apply (subgoal_tac "zgcd a m = m")
+ prefer 2
+ apply (subst zdvd_iff_zgcd [symmetric])
+ apply (rule_tac [4] zgcd_zcong_zgcd)
+ apply (simp_all add: zcong_sym)
+ done
+
+
+text {* \medskip @{term noXRRset} *}
+
+lemma RRset_gcd [rule_format]:
+ "is_RRset A m ==> a \<in> A --> zgcd a m = 1"
+ apply (unfold is_RRset_def)
+ apply (rule RsetR.induct [where P="%A. a \<in> A --> zgcd a m = 1"], auto)
+ done
+
+lemma RsetR_zmult_mono:
+ "A \<in> RsetR m ==>
+ 0 < m ==> zgcd x m = 1 ==> (\<lambda>a. a * x) ` A \<in> RsetR m"
+ apply (erule RsetR.induct, simp_all)
+ apply (rule RsetR.insert, auto)
+ apply (blast intro: zgcd_zgcd_zmult)
+ apply (simp add: zcong_cancel)
+ done
+
+lemma card_nor_eq_noX:
+ "0 < m ==>
+ zgcd x m = 1 ==> card (noXRRset m x) = card (norRRset m)"
+ apply (unfold norRRset_def noXRRset_def)
+ apply (rule card_image)
+ apply (auto simp add: inj_on_def Bnor_fin)
+ apply (simp add: BnorRset.simps)
+ done
+
+lemma noX_is_RRset:
+ "0 < m ==> zgcd x m = 1 ==> is_RRset (noXRRset m x) m"
+ apply (unfold is_RRset_def phi_def)
+ apply (auto simp add: card_nor_eq_noX)
+ apply (unfold noXRRset_def norRRset_def)
+ apply (rule RsetR_zmult_mono)
+ apply (rule Bnor_in_RsetR, simp_all)
+ done
+
+lemma aux_some:
+ "1 < m ==> is_RRset A m ==> a \<in> A
+ ==> zcong a (SOME b. [a = b] (mod m) \<and> b \<in> norRRset m) m \<and>
+ (SOME b. [a = b] (mod m) \<and> b \<in> norRRset m) \<in> norRRset m"
+ apply (rule norR_mem_unique [THEN ex1_implies_ex, THEN someI_ex])
+ apply (rule_tac [2] RRset_gcd, simp_all)
+ done
+
+lemma RRset2norRR_correct:
+ "1 < m ==> is_RRset A m ==> a \<in> A ==>
+ [a = RRset2norRR A m a] (mod m) \<and> RRset2norRR A m a \<in> norRRset m"
+ apply (unfold RRset2norRR_def, simp)
+ apply (rule aux_some, simp_all)
+ done
+
+lemmas RRset2norRR_correct1 =
+ RRset2norRR_correct [THEN conjunct1, standard]
+lemmas RRset2norRR_correct2 =
+ RRset2norRR_correct [THEN conjunct2, standard]
+
+lemma RsetR_fin: "A \<in> RsetR m ==> finite A"
+ by (induct set: RsetR) auto
+
+lemma RRset_zcong_eq [rule_format]:
+ "1 < m ==>
+ is_RRset A m ==> [a = b] (mod m) ==> a \<in> A --> b \<in> A --> a = b"
+ apply (unfold is_RRset_def)
+ apply (rule RsetR.induct [where P="%A. a \<in> A --> b \<in> A --> a = b"])
+ apply (auto simp add: zcong_sym)
+ done
+
+lemma aux:
+ "P (SOME a. P a) ==> Q (SOME a. Q a) ==>
+ (SOME a. P a) = (SOME a. Q a) ==> \<exists>a. P a \<and> Q a"
+ apply auto
+ done
+
+lemma RRset2norRR_inj:
+ "1 < m ==> is_RRset A m ==> inj_on (RRset2norRR A m) A"
+ apply (unfold RRset2norRR_def inj_on_def, auto)
+ apply (subgoal_tac "\<exists>b. ([x = b] (mod m) \<and> b \<in> norRRset m) \<and>
+ ([y = b] (mod m) \<and> b \<in> norRRset m)")
+ apply (rule_tac [2] aux)
+ apply (rule_tac [3] aux_some)
+ apply (rule_tac [2] aux_some)
+ apply (rule RRset_zcong_eq, auto)
+ apply (rule_tac b = b in zcong_trans)
+ apply (simp_all add: zcong_sym)
+ done
+
+lemma RRset2norRR_eq_norR:
+ "1 < m ==> is_RRset A m ==> RRset2norRR A m ` A = norRRset m"
+ apply (rule card_seteq)
+ prefer 3
+ apply (subst card_image)
+ apply (rule_tac RRset2norRR_inj, auto)
+ apply (rule_tac [3] RRset2norRR_correct2, auto)
+ apply (unfold is_RRset_def phi_def norRRset_def)
+ apply (auto simp add: Bnor_fin)
+ done
+
+
+lemma Bnor_prod_power_aux: "a \<notin> A ==> inj f ==> f a \<notin> f ` A"
+by (unfold inj_on_def, auto)
+
+lemma Bnor_prod_power [rule_format]:
+ "x \<noteq> 0 ==> a < m --> \<Prod>((\<lambda>a. a * x) ` BnorRset (a, m)) =
+ \<Prod>(BnorRset(a, m)) * x^card (BnorRset (a, m))"
+ apply (induct a m rule: BnorRset_induct)
+ prefer 2
+ apply (simplesubst BnorRset.simps) --{*multiple redexes*}
+ apply (unfold Let_def, auto)
+ apply (simp add: Bnor_fin Bnor_mem_zle_swap)
+ apply (subst setprod_insert)
+ apply (rule_tac [2] Bnor_prod_power_aux)
+ apply (unfold inj_on_def)
+ apply (simp_all add: zmult_ac Bnor_fin finite_imageI
+ Bnor_mem_zle_swap)
+ done
+
+
+subsection {* Fermat *}
+
+lemma bijzcong_zcong_prod:
+ "(A, B) \<in> bijR (zcongm m) ==> [\<Prod>A = \<Prod>B] (mod m)"
+ apply (unfold zcongm_def)
+ apply (erule bijR.induct)
+ apply (subgoal_tac [2] "a \<notin> A \<and> b \<notin> B \<and> finite A \<and> finite B")
+ apply (auto intro: fin_bijRl fin_bijRr zcong_zmult)
+ done
+
+lemma Bnor_prod_zgcd [rule_format]:
+ "a < m --> zgcd (\<Prod>(BnorRset(a, m))) m = 1"
+ apply (induct a m rule: BnorRset_induct)
+ prefer 2
+ apply (subst BnorRset.simps)
+ apply (unfold Let_def, auto)
+ apply (simp add: Bnor_fin Bnor_mem_zle_swap)
+ apply (blast intro: zgcd_zgcd_zmult)
+ done
+
+theorem Euler_Fermat:
+ "0 < m ==> zgcd x m = 1 ==> [x^(phi m) = 1] (mod m)"
+ apply (unfold norRRset_def phi_def)
+ apply (case_tac "x = 0")
+ apply (case_tac [2] "m = 1")
+ apply (rule_tac [3] iffD1)
+ apply (rule_tac [3] k = "\<Prod>(BnorRset(m - 1, m))"
+ in zcong_cancel2)
+ prefer 5
+ apply (subst Bnor_prod_power [symmetric])
+ apply (rule_tac [7] Bnor_prod_zgcd, simp_all)
+ apply (rule bijzcong_zcong_prod)
+ apply (fold norRRset_def noXRRset_def)
+ apply (subst RRset2norRR_eq_norR [symmetric])
+ apply (rule_tac [3] inj_func_bijR, auto)
+ apply (unfold zcongm_def)
+ apply (rule_tac [2] RRset2norRR_correct1)
+ apply (rule_tac [5] RRset2norRR_inj)
+ apply (auto intro: order_less_le [THEN iffD2]
+ simp add: noX_is_RRset)
+ apply (unfold noXRRset_def norRRset_def)
+ apply (rule finite_imageI)
+ apply (rule Bnor_fin)
+ done
+
+lemma Bnor_prime:
+ "\<lbrakk> zprime p; a < p \<rbrakk> \<Longrightarrow> card (BnorRset (a, p)) = nat a"
+ apply (induct a p rule: BnorRset.induct)
+ apply (subst BnorRset.simps)
+ apply (unfold Let_def, auto simp add:zless_zprime_imp_zrelprime)
+ apply (subgoal_tac "finite (BnorRset (a - 1,m))")
+ apply (subgoal_tac "a ~: BnorRset (a - 1,m)")
+ apply (auto simp add: card_insert_disjoint Suc_nat_eq_nat_zadd1)
+ apply (frule Bnor_mem_zle, arith)
+ apply (frule Bnor_fin)
+ done
+
+lemma phi_prime: "zprime p ==> phi p = nat (p - 1)"
+ apply (unfold phi_def norRRset_def)
+ apply (rule Bnor_prime, auto)
+ done
+
+theorem Little_Fermat:
+ "zprime p ==> \<not> p dvd x ==> [x^(nat (p - 1)) = 1] (mod p)"
+ apply (subst phi_prime [symmetric])
+ apply (rule_tac [2] Euler_Fermat)
+ apply (erule_tac [3] zprime_imp_zrelprime)
+ apply (unfold zprime_def, auto)
+ done
+
+end