--- a/src/HOL/NumberTheory/EulerFermat.thy Tue Sep 29 22:15:54 2009 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,348 +0,0 @@
-(* Title: HOL/NumberTheory/EulerFermat.thy
- ID: $Id$
- 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