--- a/src/HOL/Number_Theory/Residues.thy Fri Jun 19 21:41:33 2015 +0200
+++ b/src/HOL/Number_Theory/Residues.thy Fri Jun 19 23:40:46 2015 +0200
@@ -11,26 +11,21 @@
imports UniqueFactorization MiscAlgebra
begin
-(*
-
- A locale for residue rings
-
-*)
+subsection \<open>A locale for residue rings\<close>
-definition residue_ring :: "int => int ring" where
- "residue_ring m == (|
- carrier = {0..m - 1},
- mult = (%x y. (x * y) mod m),
- one = 1,
- zero = 0,
- add = (%x y. (x + y) mod m) |)"
+definition residue_ring :: "int \<Rightarrow> int ring"
+ where
+ "residue_ring m =
+ \<lparr>carrier = {0..m - 1},
+ mult = \<lambda>x y. (x * y) mod m,
+ one = 1,
+ zero = 0,
+ add = \<lambda>x y. (x + y) mod m\<rparr>"
locale residues =
fixes m :: int and R (structure)
assumes m_gt_one: "m > 1"
- defines "R == residue_ring m"
-
-context residues
+ defines "R \<equiv> residue_ring m"
begin
lemma abelian_group: "abelian_group R"
@@ -76,8 +71,10 @@
context residues
begin
-(* These lemmas translate back and forth between internal and
- external concepts *)
+text \<open>
+ These lemmas translate back and forth between internal and
+ external concepts.
+\<close>
lemma res_carrier_eq: "carrier R = {0..m - 1}"
unfolding R_def residue_ring_def by auto
@@ -94,11 +91,11 @@
lemma res_one_eq: "\<one> = 1"
unfolding R_def residue_ring_def units_of_def by auto
-lemma res_units_eq: "Units R = { x. 0 < x & x < m & coprime x m}"
+lemma res_units_eq: "Units R = {x. 0 < x \<and> x < m \<and> coprime x m}"
apply (insert m_gt_one)
apply (unfold Units_def R_def residue_ring_def)
apply auto
- apply (subgoal_tac "x ~= 0")
+ apply (subgoal_tac "x \<noteq> 0")
apply auto
apply (metis invertible_coprime_int)
apply (subst (asm) coprime_iff_invertible'_int)
@@ -121,19 +118,20 @@
done
lemma finite [iff]: "finite (carrier R)"
- by (subst res_carrier_eq, auto)
+ by (subst res_carrier_eq) auto
lemma finite_Units [iff]: "finite (Units R)"
by (subst res_units_eq) auto
-(* The function a -> a mod m maps the integers to the
- residue classes. The following lemmas show that this mapping
- respects addition and multiplication on the integers. *)
+text \<open>
+ The function @{text "a \<mapsto> a mod m"} maps the integers to the
+ residue classes. The following lemmas show that this mapping
+ respects addition and multiplication on the integers.
+\<close>
-lemma mod_in_carrier [iff]: "a mod m : carrier R"
- apply (unfold res_carrier_eq)
- apply (insert m_gt_one, auto)
- done
+lemma mod_in_carrier [iff]: "a mod m \<in> carrier R"
+ unfolding res_carrier_eq
+ using insert m_gt_one by auto
lemma add_cong: "(x mod m) \<oplus> (y mod m) = (x + y) mod m"
unfolding R_def residue_ring_def
@@ -151,7 +149,7 @@
lemma one_cong: "\<one> = 1 mod m"
using m_gt_one unfolding R_def residue_ring_def by auto
-(* revise algebra library to use 1? *)
+(* FIXME revise algebra library to use 1? *)
lemma pow_cong: "(x mod m) (^) n = x^n mod m"
apply (insert m_gt_one)
apply (induct n)
@@ -162,19 +160,17 @@
lemma neg_cong: "\<ominus> (x mod m) = (- x) mod m"
by (metis mod_minus_eq res_neg_eq)
-lemma (in residues) prod_cong:
- "finite A \<Longrightarrow> (\<Otimes> i:A. (f i) mod m) = (PROD i:A. f i) mod m"
+lemma (in residues) prod_cong: "finite A \<Longrightarrow> (\<Otimes> i:A. (f i) mod m) = (\<Prod>i\<in>A. f i) mod m"
by (induct set: finite) (auto simp: one_cong mult_cong)
-lemma (in residues) sum_cong:
- "finite A \<Longrightarrow> (\<Oplus> i:A. (f i) mod m) = (SUM i: A. f i) mod m"
+lemma (in residues) sum_cong: "finite A \<Longrightarrow> (\<Oplus> i:A. (f i) mod m) = (\<Sum>i\<in>A. f i) mod m"
by (induct set: finite) (auto simp: zero_cong add_cong)
-lemma mod_in_res_units [simp]: "1 < m \<Longrightarrow> coprime a m \<Longrightarrow>
- a mod m : Units R"
- apply (subst res_units_eq, auto)
+lemma mod_in_res_units [simp]: "1 < m \<Longrightarrow> coprime a m \<Longrightarrow> a mod m \<in> Units R"
+ apply (subst res_units_eq)
+ apply auto
apply (insert pos_mod_sign [of m a])
- apply (subgoal_tac "a mod m ~= 0")
+ apply (subgoal_tac "a mod m \<noteq> 0")
apply arith
apply auto
apply (metis gcd_int.commute gcd_red_int)
@@ -183,13 +179,13 @@
lemma res_eq_to_cong: "((a mod m) = (b mod m)) = [a = b] (mod (m::int))"
unfolding cong_int_def by auto
-(* Simplifying with these will translate a ring equation in R to a
- congruence. *)
+text \<open>Simplifying with these will translate a ring equation in R to a
+ congruence.\<close>
lemmas res_to_cong_simps = add_cong mult_cong pow_cong one_cong
prod_cong sum_cong neg_cong res_eq_to_cong
-(* Other useful facts about the residue ring *)
+text \<open>Other useful facts about the residue ring.\<close>
lemma one_eq_neg_one: "\<one> = \<ominus> \<one> \<Longrightarrow> m = 2"
apply (simp add: res_one_eq res_neg_eq)
@@ -200,12 +196,12 @@
end
-(* prime residues *)
+subsection \<open>Prime residues\<close>
locale residues_prime =
fixes p and R (structure)
assumes p_prime [intro]: "prime p"
- defines "R == residue_ring p"
+ defines "R \<equiv> residue_ring p"
sublocale residues_prime < residues p
apply (unfold R_def residues_def)
@@ -243,40 +239,42 @@
by (rule is_field)
-(*
- Test cases: Euler's theorem and Wilson's theorem.
-*)
+section \<open>Test cases: Euler's theorem and Wilson's theorem\<close>
-
-subsection\<open>Euler's theorem\<close>
+subsection \<open>Euler's theorem\<close>
-(* the definition of the phi function *)
+text \<open>The definition of the phi function.\<close>
-definition phi :: "int => nat"
- where "phi m = card({ x. 0 < x & x < m & gcd x m = 1})"
+definition phi :: "int \<Rightarrow> nat"
+ where "phi m = card {x. 0 < x \<and> x < m \<and> gcd x m = 1}"
-lemma phi_def_nat: "phi m = card({ x. 0 < x & x < nat m & gcd x (nat m) = 1})"
+lemma phi_def_nat: "phi m = card {x. 0 < x \<and> x < nat m \<and> gcd x (nat m) = 1}"
apply (simp add: phi_def)
apply (rule bij_betw_same_card [of nat])
apply (auto simp add: inj_on_def bij_betw_def image_def)
apply (metis dual_order.irrefl dual_order.strict_trans leI nat_1 transfer_nat_int_gcd(1))
- apply (metis One_nat_def int_0 int_1 int_less_0_conv int_nat_eq nat_int transfer_int_nat_gcd(1) zless_int)
+ apply (metis One_nat_def int_0 int_1 int_less_0_conv int_nat_eq nat_int
+ transfer_int_nat_gcd(1) zless_int)
done
lemma prime_phi:
- assumes "2 \<le> p" "phi p = p - 1" shows "prime p"
+ assumes "2 \<le> p" "phi p = p - 1"
+ shows "prime p"
proof -
have "{x. 0 < x \<and> x < p \<and> coprime x p} = {1..p - 1}"
using assms unfolding phi_def_nat
by (intro card_seteq) fastforce+
- then have cop: "\<And>x. x \<in> {1::nat..p - 1} \<Longrightarrow> coprime x p"
+ then have cop: "\<And>x::nat. x \<in> {1..p - 1} \<Longrightarrow> coprime x p"
by blast
- { fix x::nat assume *: "1 < x" "x < p" and "x dvd p"
+ have False if *: "1 < x" "x < p" and "x dvd p" for x :: nat
+ proof -
have "coprime x p"
apply (rule cop)
using * apply auto
done
- with \<open>x dvd p\<close> \<open>1 < x\<close> have "False" by auto }
+ with \<open>x dvd p\<close> \<open>1 < x\<close> show ?thesis
+ by auto
+ qed
then show ?thesis
using \<open>2 \<le> p\<close>
by (simp add: prime_def)
@@ -285,9 +283,9 @@
qed
lemma phi_zero [simp]: "phi 0 = 0"
- apply (subst phi_def)
+ unfolding phi_def
(* Auto hangs here. Once again, where is the simplification rule
- 1 == Suc 0 coming from? *)
+ 1 \<equiv> Suc 0 coming from? *)
apply (auto simp add: card_eq_0_iff)
(* Add card_eq_0_iff as a simp rule? delete card_empty_imp? *)
done
@@ -295,19 +293,19 @@
lemma phi_one [simp]: "phi 1 = 0"
by (auto simp add: phi_def card_eq_0_iff)
-lemma (in residues) phi_eq: "phi m = card(Units R)"
+lemma (in residues) phi_eq: "phi m = card (Units R)"
by (simp add: phi_def res_units_eq)
lemma (in residues) euler_theorem1:
assumes a: "gcd a m = 1"
shows "[a^phi m = 1] (mod m)"
proof -
- from a m_gt_one have [simp]: "a mod m : Units R"
+ from a m_gt_one have [simp]: "a mod m \<in> Units R"
by (intro mod_in_res_units)
from phi_eq have "(a mod m) (^) (phi m) = (a mod m) (^) (card (Units R))"
by simp
also have "\<dots> = \<one>"
- by (intro units_power_order_eq_one, auto)
+ by (intro units_power_order_eq_one) auto
finally show ?thesis
by (simp add: res_to_cong_simps)
qed
@@ -329,55 +327,57 @@
(* outside the locale, we can relax the restriction m > 1 *)
lemma euler_theorem:
- assumes "m >= 0" and "gcd a m = 1"
+ assumes "m \<ge> 0"
+ and "gcd a m = 1"
shows "[a^phi m = 1] (mod m)"
-proof (cases)
- assume "m = 0 | m = 1"
+proof (cases "m = 0 | m = 1")
+ case True
then show ?thesis by auto
next
- assume "~(m = 0 | m = 1)"
+ case False
with assms show ?thesis
by (intro residues.euler_theorem1, unfold residues_def, auto)
qed
-lemma (in residues_prime) phi_prime: "phi p = (nat p - 1)"
+lemma (in residues_prime) phi_prime: "phi p = nat p - 1"
apply (subst phi_eq)
apply (subst res_prime_units_eq)
apply auto
done
-lemma phi_prime: "prime p \<Longrightarrow> phi p = (nat p - 1)"
+lemma phi_prime: "prime p \<Longrightarrow> phi p = nat p - 1"
apply (rule residues_prime.phi_prime)
apply (erule residues_prime.intro)
done
lemma fermat_theorem:
- fixes a::int
- assumes "prime p" and "~ (p dvd a)"
+ fixes a :: int
+ assumes "prime p"
+ and "\<not> p dvd a"
shows "[a^(p - 1) = 1] (mod p)"
proof -
- from assms have "[a^phi p = 1] (mod p)"
+ from assms have "[a ^ phi p = 1] (mod p)"
apply (intro euler_theorem)
apply (metis of_nat_0_le_iff)
apply (metis gcd_int.commute prime_imp_coprime_int)
done
also have "phi p = nat p - 1"
- by (rule phi_prime, rule assms)
+ by (rule phi_prime) (rule assms)
finally show ?thesis
by (metis nat_int)
qed
lemma fermat_theorem_nat:
- assumes "prime p" and "~ (p dvd a)"
- shows "[a^(p - 1) = 1] (mod p)"
-using fermat_theorem [of p a] assms
-by (metis int_1 of_nat_power transfer_int_nat_cong zdvd_int)
+ assumes "prime p" and "\<not> p dvd a"
+ shows "[a ^ (p - 1) = 1] (mod p)"
+ using fermat_theorem [of p a] assms
+ by (metis int_1 of_nat_power transfer_int_nat_cong zdvd_int)
subsection \<open>Wilson's theorem\<close>
-lemma (in field) inv_pair_lemma: "x : Units R \<Longrightarrow> y : Units R \<Longrightarrow>
- {x, inv x} ~= {y, inv y} \<Longrightarrow> {x, inv x} Int {y, inv y} = {}"
+lemma (in field) inv_pair_lemma: "x \<in> Units R \<Longrightarrow> y \<in> Units R \<Longrightarrow>
+ {x, inv x} \<noteq> {y, inv y} \<Longrightarrow> {x, inv x} \<inter> {y, inv y} = {}"
apply auto
apply (metis Units_inv_inv)+
done
@@ -386,41 +386,43 @@
assumes a: "p > 2"
shows "[fact (p - 1) = (-1::int)] (mod p)"
proof -
- let ?InversePairs = "{ {x, inv x} | x. x : Units R - {\<one>, \<ominus> \<one>}}"
- have UR: "Units R = {\<one>, \<ominus> \<one>} Un (Union ?InversePairs)"
+ let ?Inverse_Pairs = "{{x, inv x}| x. x \<in> Units R - {\<one>, \<ominus> \<one>}}"
+ have UR: "Units R = {\<one>, \<ominus> \<one>} \<union> \<Union>?Inverse_Pairs"
by auto
- have "(\<Otimes>i: Units R. i) =
- (\<Otimes>i: {\<one>, \<ominus> \<one>}. i) \<otimes> (\<Otimes>i: Union ?InversePairs. i)"
+ have "(\<Otimes>i\<in>Units R. i) = (\<Otimes>i\<in>{\<one>, \<ominus> \<one>}. i) \<otimes> (\<Otimes>i\<in>\<Union>?Inverse_Pairs. i)"
apply (subst UR)
apply (subst finprod_Un_disjoint)
apply (auto intro: funcsetI)
- apply (metis Units_inv_inv inv_one inv_neg_one)+
+ using inv_one apply auto[1]
+ using inv_eq_neg_one_eq apply auto
done
- also have "(\<Otimes>i: {\<one>, \<ominus> \<one>}. i) = \<ominus> \<one>"
+ also have "(\<Otimes>i\<in>{\<one>, \<ominus> \<one>}. i) = \<ominus> \<one>"
apply (subst finprod_insert)
apply auto
apply (frule one_eq_neg_one)
- apply (insert a, force)
+ using a apply force
done
- also have "(\<Otimes>i:(Union ?InversePairs). i) =
- (\<Otimes>A: ?InversePairs. (\<Otimes>y:A. y))"
- apply (subst finprod_Union_disjoint, auto)
+ also have "(\<Otimes>i\<in>(\<Union>?Inverse_Pairs). i) = (\<Otimes>A\<in>?Inverse_Pairs. (\<Otimes>y\<in>A. y))"
+ apply (subst finprod_Union_disjoint)
+ apply auto
apply (metis Units_inv_inv)+
done
also have "\<dots> = \<one>"
- apply (rule finprod_one, auto)
- apply (subst finprod_insert, auto)
+ apply (rule finprod_one)
+ apply auto
+ apply (subst finprod_insert)
+ apply auto
apply (metis inv_eq_self)
done
- finally have "(\<Otimes>i: Units R. i) = \<ominus> \<one>"
+ finally have "(\<Otimes>i\<in>Units R. i) = \<ominus> \<one>"
by simp
- also have "(\<Otimes>i: Units R. i) = (\<Otimes>i: Units R. i mod p)"
+ also have "(\<Otimes>i\<in>Units R. i) = (\<Otimes>i\<in>Units R. i mod p)"
apply (rule finprod_cong')
- apply (auto)
+ apply auto
apply (subst (asm) res_prime_units_eq)
apply auto
done
- also have "\<dots> = (PROD i: Units R. i) mod p"
+ also have "\<dots> = (\<Prod>i\<in>Units R. i) mod p"
apply (rule prod_cong)
apply auto
done
@@ -430,13 +432,14 @@
apply (subst res_prime_units_eq)
apply (simp add: int_setprod zmod_int setprod_int_eq)
done
- finally have "fact (p - 1) mod p = (\<ominus> \<one> :: int)".
- then show ?thesis
+ finally have "fact (p - 1) mod p = \<ominus> \<one>" .
+ then show ?thesis
by (metis of_nat_fact Divides.transfer_int_nat_functions(2) cong_int_def res_neg_eq res_one_eq)
qed
lemma wilson_theorem:
- assumes "prime p" shows "[fact (p - 1) = - 1] (mod p)"
+ assumes "prime p"
+ shows "[fact (p - 1) = - 1] (mod p)"
proof (cases "p = 2")
case True
then show ?thesis