--- a/src/HOL/NumberTheory/IntPrimes.thy Mon Oct 22 11:01:30 2001 +0200
+++ b/src/HOL/NumberTheory/IntPrimes.thy Mon Oct 22 11:54:22 2001 +0200
@@ -29,7 +29,7 @@
"measure ((\<lambda>(m, n, r', r, s', s, t', t). nat r)
:: int * int * int * int *int * int * int * int => nat)"
"xzgcda (m, n, r', r, s', s, t', t) =
- (if r \<le> Numeral0 then (r', s', t')
+ (if r \<le> 0 then (r', s', t')
else xzgcda (m, n, r, r' mod r, s, s' - (r' div r) * s, t, t' - (r' div r) * t))"
(hints simp: pos_mod_bound)
@@ -38,13 +38,13 @@
"zgcd == \<lambda>(x,y). int (gcd (nat (abs x), nat (abs y)))"
defs
- xzgcd_def: "xzgcd m n == xzgcda (m, n, m, n, Numeral1, Numeral0, Numeral0, Numeral1)"
- zprime_def: "zprime == {p. Numeral1 < p \<and> (\<forall>m. m dvd p --> m = Numeral1 \<or> m = p)}"
+ xzgcd_def: "xzgcd m n == xzgcda (m, n, m, n, 1, 0, 0, 1)"
+ zprime_def: "zprime == {p. 1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p)}"
zcong_def: "[a = b] (mod m) == m dvd (a - b)"
lemma zabs_eq_iff:
- "(abs (z::int) = w) = (z = w \<and> Numeral0 <= z \<or> z = -w \<and> z < Numeral0)"
+ "(abs (z::int) = w) = (z = w \<and> 0 <= z \<or> z = -w \<and> z < 0)"
apply (auto simp add: zabs_def)
done
@@ -64,17 +64,17 @@
subsection {* Divides relation *}
-lemma zdvd_0_right [iff]: "(m::int) dvd Numeral0"
+lemma zdvd_0_right [iff]: "(m::int) dvd 0"
apply (unfold dvd_def)
apply (blast intro: zmult_0_right [symmetric])
done
-lemma zdvd_0_left [iff]: "(Numeral0 dvd (m::int)) = (m = Numeral0)"
+lemma zdvd_0_left [iff]: "(0 dvd (m::int)) = (m = 0)"
apply (unfold dvd_def)
apply auto
done
-lemma zdvd_1_left [iff]: "Numeral1 dvd (m::int)"
+lemma zdvd_1_left [iff]: "1 dvd (m::int)"
apply (unfold dvd_def)
apply simp
done
@@ -104,7 +104,7 @@
done
lemma zdvd_anti_sym:
- "Numeral0 < m ==> Numeral0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"
+ "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"
apply (unfold dvd_def)
apply auto
apply (simp add: zmult_assoc zmult_eq_self_iff int_0_less_mult_iff zmult_eq_1_iff)
@@ -186,19 +186,19 @@
apply (simp add: zdvd_zadd zdvd_zmult2)
done
-lemma zdvd_iff_zmod_eq_0: "(k dvd n) = (n mod (k::int) = Numeral0)"
+lemma zdvd_iff_zmod_eq_0: "(k dvd n) = (n mod (k::int) = 0)"
apply (unfold dvd_def)
apply auto
done
-lemma zdvd_not_zless: "Numeral0 < m ==> m < n ==> \<not> n dvd (m::int)"
+lemma zdvd_not_zless: "0 < m ==> m < n ==> \<not> n dvd (m::int)"
apply (unfold dvd_def)
apply auto
- apply (subgoal_tac "Numeral0 < n")
+ apply (subgoal_tac "0 < n")
prefer 2
apply (blast intro: zless_trans)
apply (simp add: int_0_less_mult_iff)
- apply (subgoal_tac "n * k < n * Numeral1")
+ apply (subgoal_tac "n * k < n * 1")
apply (drule zmult_zless_cancel1 [THEN iffD1])
apply auto
done
@@ -221,7 +221,7 @@
nat_mult_distrib [symmetric] nat_eq_iff2)
done
-lemma nat_dvd_iff: "(nat z dvd m) = (if Numeral0 \<le> z then (z dvd int m) else m = 0)"
+lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
apply (auto simp add: dvd_def zmult_int [symmetric])
apply (rule_tac x = "nat k" in exI)
apply (cut_tac k = m in int_less_0_conv)
@@ -245,11 +245,11 @@
subsection {* Euclid's Algorithm and GCD *}
-lemma zgcd_0 [simp]: "zgcd (m, Numeral0) = abs m"
+lemma zgcd_0 [simp]: "zgcd (m, 0) = abs m"
apply (simp add: zgcd_def zabs_def)
done
-lemma zgcd_0_left [simp]: "zgcd (Numeral0, m) = abs m"
+lemma zgcd_0_left [simp]: "zgcd (0, m) = abs m"
apply (simp add: zgcd_def zabs_def)
done
@@ -261,7 +261,7 @@
apply (simp add: zgcd_def)
done
-lemma zgcd_non_0: "Numeral0 < n ==> zgcd (m, n) = zgcd (n, m mod n)"
+lemma zgcd_non_0: "0 < n ==> zgcd (m, n) = zgcd (n, m mod n)"
apply (frule_tac b = n and a = m in pos_mod_sign)
apply (simp add: zgcd_def zabs_def nat_mod_distrib)
apply (cut_tac a = "-m" and b = n in zmod_zminus1_eq_if)
@@ -273,17 +273,17 @@
done
lemma zgcd_eq: "zgcd (m, n) = zgcd (n, m mod n)"
- apply (tactic {* zdiv_undefined_case_tac "n = Numeral0" 1 *})
+ apply (tactic {* zdiv_undefined_case_tac "n = 0" 1 *})
apply (auto simp add: linorder_neq_iff zgcd_non_0)
apply (cut_tac m = "-m" and n = "-n" in zgcd_non_0)
apply auto
done
-lemma zgcd_1 [simp]: "zgcd (m, Numeral1) = Numeral1"
+lemma zgcd_1 [simp]: "zgcd (m, 1) = 1"
apply (simp add: zgcd_def zabs_def)
done
-lemma zgcd_0_1_iff [simp]: "(zgcd (Numeral0, m) = Numeral1) = (abs m = Numeral1)"
+lemma zgcd_0_1_iff [simp]: "(zgcd (0, m) = 1) = (abs m = 1)"
apply (simp add: zgcd_def zabs_def)
done
@@ -303,7 +303,7 @@
apply (simp add: zgcd_def gcd_commute)
done
-lemma zgcd_1_left [simp]: "zgcd (Numeral1, m) = Numeral1"
+lemma zgcd_1_left [simp]: "zgcd (1, m) = 1"
apply (simp add: zgcd_def gcd_1_left)
done
@@ -320,7 +320,7 @@
lemmas zgcd_ac = zgcd_assoc zgcd_commute zgcd_left_commute
-- {* addition is an AC-operator *}
-lemma zgcd_zmult_distrib2: "Numeral0 \<le> k ==> k * zgcd (m, n) = zgcd (k * m, k * n)"
+lemma zgcd_zmult_distrib2: "0 \<le> k ==> k * zgcd (m, n) = zgcd (k * m, k * n)"
apply (simp del: zmult_zminus_right
add: zmult_zminus_right [symmetric] nat_mult_distrib zgcd_def zabs_def
zmult_less_0_iff gcd_mult_distrib2 [symmetric] zmult_int [symmetric])
@@ -330,29 +330,29 @@
apply (simp add: zabs_def zgcd_zmult_distrib2)
done
-lemma zgcd_self [simp]: "Numeral0 \<le> m ==> zgcd (m, m) = m"
- apply (cut_tac k = m and m = "Numeral1" and n = "Numeral1" in zgcd_zmult_distrib2)
+lemma zgcd_self [simp]: "0 \<le> m ==> zgcd (m, m) = m"
+ apply (cut_tac k = m and m = "1" and n = "1" in zgcd_zmult_distrib2)
apply simp_all
done
-lemma zgcd_zmult_eq_self [simp]: "Numeral0 \<le> k ==> zgcd (k, k * n) = k"
- apply (cut_tac k = k and m = "Numeral1" and n = n in zgcd_zmult_distrib2)
+lemma zgcd_zmult_eq_self [simp]: "0 \<le> k ==> zgcd (k, k * n) = k"
+ apply (cut_tac k = k and m = "1" and n = n in zgcd_zmult_distrib2)
apply simp_all
done
-lemma zgcd_zmult_eq_self2 [simp]: "Numeral0 \<le> k ==> zgcd (k * n, k) = k"
- apply (cut_tac k = k and m = n and n = "Numeral1" in zgcd_zmult_distrib2)
+lemma zgcd_zmult_eq_self2 [simp]: "0 \<le> k ==> zgcd (k * n, k) = k"
+ apply (cut_tac k = k and m = n and n = "1" in zgcd_zmult_distrib2)
apply simp_all
done
-lemma aux: "zgcd (n, k) = Numeral1 ==> k dvd m * n ==> Numeral0 \<le> m ==> k dvd m"
+lemma aux: "zgcd (n, k) = 1 ==> k dvd m * n ==> 0 \<le> m ==> k dvd m"
apply (subgoal_tac "m = zgcd (m * n, m * k)")
apply (erule ssubst, rule zgcd_greatest_iff [THEN iffD2])
apply (simp_all add: zgcd_zmult_distrib2 [symmetric] int_0_le_mult_iff)
done
-lemma zrelprime_zdvd_zmult: "zgcd (n, k) = Numeral1 ==> k dvd m * n ==> k dvd m"
- apply (case_tac "Numeral0 \<le> m")
+lemma zrelprime_zdvd_zmult: "zgcd (n, k) = 1 ==> k dvd m * n ==> k dvd m"
+ apply (case_tac "0 \<le> m")
apply (blast intro: aux)
apply (subgoal_tac "k dvd -m")
apply (rule_tac [2] aux)
@@ -360,20 +360,20 @@
done
lemma zprime_imp_zrelprime:
- "p \<in> zprime ==> \<not> p dvd n ==> zgcd (n, p) = Numeral1"
+ "p \<in> zprime ==> \<not> p dvd n ==> zgcd (n, p) = 1"
apply (unfold zprime_def)
apply auto
done
lemma zless_zprime_imp_zrelprime:
- "p \<in> zprime ==> Numeral0 < n ==> n < p ==> zgcd (n, p) = Numeral1"
+ "p \<in> zprime ==> 0 < n ==> n < p ==> zgcd (n, p) = 1"
apply (erule zprime_imp_zrelprime)
apply (erule zdvd_not_zless)
apply assumption
done
lemma zprime_zdvd_zmult:
- "Numeral0 \<le> (m::int) ==> p \<in> zprime ==> p dvd m * n ==> p dvd m \<or> p dvd n"
+ "0 \<le> (m::int) ==> p \<in> zprime ==> p dvd m * n ==> p dvd m \<or> p dvd n"
apply safe
apply (rule zrelprime_zdvd_zmult)
apply (rule zprime_imp_zrelprime)
@@ -392,7 +392,7 @@
done
lemma zgcd_zmult_zdvd_zgcd:
- "zgcd (k, n) = Numeral1 ==> zgcd (k * m, n) dvd zgcd (m, n)"
+ "zgcd (k, n) = 1 ==> zgcd (k * m, n) dvd zgcd (m, n)"
apply (simp add: zgcd_greatest_iff)
apply (rule_tac n = k in zrelprime_zdvd_zmult)
prefer 2
@@ -402,16 +402,16 @@
apply (simp (no_asm) add: zgcd_ac)
done
-lemma zgcd_zmult_cancel: "zgcd (k, n) = Numeral1 ==> zgcd (k * m, n) = zgcd (m, n)"
+lemma zgcd_zmult_cancel: "zgcd (k, n) = 1 ==> zgcd (k * m, n) = zgcd (m, n)"
apply (simp add: zgcd_def nat_abs_mult_distrib gcd_mult_cancel)
done
lemma zgcd_zgcd_zmult:
- "zgcd (k, m) = Numeral1 ==> zgcd (n, m) = Numeral1 ==> zgcd (k * n, m) = Numeral1"
+ "zgcd (k, m) = 1 ==> zgcd (n, m) = 1 ==> zgcd (k * n, m) = 1"
apply (simp (no_asm_simp) add: zgcd_zmult_cancel)
done
-lemma zdvd_iff_zgcd: "Numeral0 < m ==> (m dvd n) = (zgcd (n, m) = m)"
+lemma zdvd_iff_zgcd: "0 < m ==> (m dvd n) = (zgcd (n, m) = m)"
apply safe
apply (rule_tac [2] n = "zgcd (n, m)" in zdvd_trans)
apply (rule_tac [3] zgcd_zdvd1)
@@ -423,7 +423,7 @@
subsection {* Congruences *}
-lemma zcong_1 [simp]: "[a = b] (mod Numeral1)"
+lemma zcong_1 [simp]: "[a = b] (mod 1)"
apply (unfold zcong_def)
apply auto
done
@@ -494,19 +494,19 @@
done
lemma zcong_square:
- "p \<in> zprime ==> Numeral0 < a ==> [a * a = Numeral1] (mod p)
- ==> [a = Numeral1] (mod p) \<or> [a = p - Numeral1] (mod p)"
+ "p \<in> zprime ==> 0 < a ==> [a * a = 1] (mod p)
+ ==> [a = 1] (mod p) \<or> [a = p - 1] (mod p)"
apply (unfold zcong_def)
apply (rule zprime_zdvd_zmult)
- apply (rule_tac [3] s = "a * a - Numeral1 + p * (Numeral1 - a)" in subst)
+ apply (rule_tac [3] s = "a * a - 1 + p * (1 - a)" in subst)
prefer 4
apply (simp add: zdvd_reduce)
apply (simp_all add: zdiff_zmult_distrib zmult_commute zdiff_zmult_distrib2)
done
lemma zcong_cancel:
- "Numeral0 \<le> m ==>
- zgcd (k, m) = Numeral1 ==> [a * k = b * k] (mod m) = [a = b] (mod m)"
+ "0 \<le> m ==>
+ zgcd (k, m) = 1 ==> [a * k = b * k] (mod m) = [a = b] (mod m)"
apply safe
prefer 2
apply (blast intro: zcong_scalar)
@@ -523,19 +523,19 @@
done
lemma zcong_cancel2:
- "Numeral0 \<le> m ==>
- zgcd (k, m) = Numeral1 ==> [k * a = k * b] (mod m) = [a = b] (mod m)"
+ "0 \<le> m ==>
+ zgcd (k, m) = 1 ==> [k * a = k * b] (mod m) = [a = b] (mod m)"
apply (simp add: zmult_commute zcong_cancel)
done
lemma zcong_zgcd_zmult_zmod:
- "[a = b] (mod m) ==> [a = b] (mod n) ==> zgcd (m, n) = Numeral1
+ "[a = b] (mod m) ==> [a = b] (mod n) ==> zgcd (m, n) = 1
==> [a = b] (mod m * n)"
apply (unfold zcong_def dvd_def)
apply auto
apply (subgoal_tac "m dvd n * ka")
apply (subgoal_tac "m dvd ka")
- apply (case_tac [2] "Numeral0 \<le> ka")
+ apply (case_tac [2] "0 \<le> ka")
prefer 3
apply (subst zdvd_zminus_iff [symmetric])
apply (rule_tac n = n in zrelprime_zdvd_zmult)
@@ -550,8 +550,8 @@
done
lemma zcong_zless_imp_eq:
- "Numeral0 \<le> a ==>
- a < m ==> Numeral0 \<le> b ==> b < m ==> [a = b] (mod m) ==> a = b"
+ "0 \<le> a ==>
+ a < m ==> 0 \<le> b ==> b < m ==> [a = b] (mod m) ==> a = b"
apply (unfold zcong_def dvd_def)
apply auto
apply (drule_tac f = "\<lambda>z. z mod m" in arg_cong)
@@ -566,38 +566,38 @@
done
lemma zcong_square_zless:
- "p \<in> zprime ==> Numeral0 < a ==> a < p ==>
- [a * a = Numeral1] (mod p) ==> a = Numeral1 \<or> a = p - Numeral1"
+ "p \<in> zprime ==> 0 < a ==> a < p ==>
+ [a * a = 1] (mod p) ==> a = 1 \<or> a = p - 1"
apply (cut_tac p = p and a = a in zcong_square)
apply (simp add: zprime_def)
apply (auto intro: zcong_zless_imp_eq)
done
lemma zcong_not:
- "Numeral0 < a ==> a < m ==> Numeral0 < b ==> b < a ==> \<not> [a = b] (mod m)"
+ "0 < a ==> a < m ==> 0 < b ==> b < a ==> \<not> [a = b] (mod m)"
apply (unfold zcong_def)
apply (rule zdvd_not_zless)
apply auto
done
lemma zcong_zless_0:
- "Numeral0 \<le> a ==> a < m ==> [a = Numeral0] (mod m) ==> a = Numeral0"
+ "0 \<le> a ==> a < m ==> [a = 0] (mod m) ==> a = 0"
apply (unfold zcong_def dvd_def)
apply auto
- apply (subgoal_tac "Numeral0 < m")
+ apply (subgoal_tac "0 < m")
apply (rotate_tac -1)
apply (simp add: int_0_le_mult_iff)
- apply (subgoal_tac "m * k < m * Numeral1")
+ apply (subgoal_tac "m * k < m * 1")
apply (drule zmult_zless_cancel1 [THEN iffD1])
apply (auto simp add: linorder_neq_iff)
done
lemma zcong_zless_unique:
- "Numeral0 < m ==> (\<exists>!b. Numeral0 \<le> b \<and> b < m \<and> [a = b] (mod m))"
+ "0 < m ==> (\<exists>!b. 0 \<le> b \<and> b < m \<and> [a = b] (mod m))"
apply auto
apply (subgoal_tac [2] "[b = y] (mod m)")
- apply (case_tac [2] "b = Numeral0")
- apply (case_tac [3] "y = Numeral0")
+ apply (case_tac [2] "b = 0")
+ apply (case_tac [3] "y = 0")
apply (auto intro: zcong_trans zcong_zless_0 zcong_zless_imp_eq order_less_le
simp add: zcong_sym)
apply (unfold zcong_def dvd_def)
@@ -616,8 +616,8 @@
done
lemma zgcd_zcong_zgcd:
- "Numeral0 < m ==>
- zgcd (a, m) = Numeral1 ==> [a = b] (mod m) ==> zgcd (b, m) = Numeral1"
+ "0 < m ==>
+ zgcd (a, m) = 1 ==> [a = b] (mod m) ==> zgcd (b, m) = 1"
apply (auto simp add: zcong_iff_lin)
done
@@ -643,7 +643,7 @@
apply (simp add: zadd_commute)
done
-lemma zcong_zmod_eq: "Numeral0 < m ==> [a = b] (mod m) = (a mod m = b mod m)"
+lemma zcong_zmod_eq: "0 < m ==> [a = b] (mod m) = (a mod m = b mod m)"
apply auto
apply (rule_tac m = m in zcong_zless_imp_eq)
prefer 5
@@ -659,13 +659,13 @@
apply (auto simp add: zcong_def)
done
-lemma zcong_zero [iff]: "[a = b] (mod Numeral0) = (a = b)"
+lemma zcong_zero [iff]: "[a = b] (mod 0) = (a = b)"
apply (auto simp add: zcong_def)
done
lemma "[a = b] (mod m) = (a mod m = b mod m)"
- apply (tactic {* zdiv_undefined_case_tac "m = Numeral0" 1 *})
- apply (case_tac "Numeral0 < m")
+ apply (tactic {* zdiv_undefined_case_tac "m = 0" 1 *})
+ apply (case_tac "0 < m")
apply (simp add: zcong_zmod_eq)
apply (rule_tac t = m in zminus_zminus [THEN subst])
apply (subst zcong_zminus)
@@ -677,7 +677,7 @@
subsection {* Modulo *}
lemma zmod_zdvd_zmod:
- "Numeral0 < (m::int) ==> m dvd b ==> (a mod b mod m) = (a mod m)"
+ "0 < (m::int) ==> m dvd b ==> (a mod b mod m) = (a mod m)"
apply (unfold dvd_def)
apply auto
apply (subst zcong_zmod_eq [symmetric])
@@ -696,14 +696,14 @@
declare xzgcda.simps [simp del]
lemma aux1:
- "zgcd (r', r) = k --> Numeral0 < r -->
+ "zgcd (r', r) = k --> 0 < r -->
(\<exists>sn tn. xzgcda (m, n, r', r, s', s, t', t) = (k, sn, tn))"
apply (rule_tac u = m and v = n and w = r' and x = r and y = s' and
z = s and aa = t' and ab = t in xzgcda.induct)
apply (subst zgcd_eq)
apply (subst xzgcda.simps)
apply auto
- apply (case_tac "r' mod r = Numeral0")
+ apply (case_tac "r' mod r = 0")
prefer 2
apply (frule_tac a = "r'" in pos_mod_sign)
apply auto
@@ -716,14 +716,14 @@
done
lemma aux2:
- "(\<exists>sn tn. xzgcda (m, n, r', r, s', s, t', t) = (k, sn, tn)) --> Numeral0 < r -->
+ "(\<exists>sn tn. xzgcda (m, n, r', r, s', s, t', t) = (k, sn, tn)) --> 0 < r -->
zgcd (r', r) = k"
apply (rule_tac u = m and v = n and w = r' and x = r and y = s' and
z = s and aa = t' and ab = t in xzgcda.induct)
apply (subst zgcd_eq)
apply (subst xzgcda.simps)
apply (auto simp add: linorder_not_le)
- apply (case_tac "r' mod r = Numeral0")
+ apply (case_tac "r' mod r = 0")
prefer 2
apply (frule_tac a = "r'" in pos_mod_sign)
apply auto
@@ -735,7 +735,7 @@
done
lemma xzgcd_correct:
- "Numeral0 < n ==> (zgcd (m, n) = k) = (\<exists>s t. xzgcd m n = (k, s, t))"
+ "0 < n ==> (zgcd (m, n) = k) = (\<exists>s t. xzgcd m n = (k, s, t))"
apply (unfold xzgcd_def)
apply (rule iffI)
apply (rule_tac [2] aux2 [THEN mp, THEN mp])
@@ -768,17 +768,17 @@
by (rule iffD2 [OF order_less_le conjI])
lemma xzgcda_linear [rule_format]:
- "Numeral0 < r --> xzgcda (m, n, r', r, s', s, t', t) = (rn, sn, tn) -->
+ "0 < r --> xzgcda (m, n, r', r, s', s, t', t) = (rn, sn, tn) -->
r' = s' * m + t' * n --> r = s * m + t * n --> rn = sn * m + tn * n"
apply (rule_tac u = m and v = n and w = r' and x = r and y = s' and
z = s and aa = t' and ab = t in xzgcda.induct)
apply (subst xzgcda.simps)
apply (simp (no_asm))
apply (rule impI)+
- apply (case_tac "r' mod r = Numeral0")
+ apply (case_tac "r' mod r = 0")
apply (simp add: xzgcda.simps)
apply clarify
- apply (subgoal_tac "Numeral0 < r' mod r")
+ apply (subgoal_tac "0 < r' mod r")
apply (rule_tac [2] order_le_neq_implies_less)
apply (rule_tac [2] pos_mod_sign)
apply (cut_tac m = m and n = n and r' = r' and r = r and s' = s' and
@@ -787,7 +787,7 @@
done
lemma xzgcd_linear:
- "Numeral0 < n ==> xzgcd m n = (r, s, t) ==> r = s * m + t * n"
+ "0 < n ==> xzgcd m n = (r, s, t) ==> r = s * m + t * n"
apply (unfold xzgcd_def)
apply (erule xzgcda_linear)
apply assumption
@@ -795,7 +795,7 @@
done
lemma zgcd_ex_linear:
- "Numeral0 < n ==> zgcd (m, n) = k ==> (\<exists>s t. k = s * m + t * n)"
+ "0 < n ==> zgcd (m, n) = k ==> (\<exists>s t. k = s * m + t * n)"
apply (simp add: xzgcd_correct)
apply safe
apply (rule exI)+
@@ -804,8 +804,8 @@
done
lemma zcong_lineq_ex:
- "Numeral0 < n ==> zgcd (a, n) = Numeral1 ==> \<exists>x. [a * x = Numeral1] (mod n)"
- apply (cut_tac m = a and n = n and k = "Numeral1" in zgcd_ex_linear)
+ "0 < n ==> zgcd (a, n) = 1 ==> \<exists>x. [a * x = 1] (mod n)"
+ apply (cut_tac m = a and n = n and k = "1" in zgcd_ex_linear)
apply safe
apply (rule_tac x = s in exI)
apply (rule_tac b = "s * a + t * n" in zcong_trans)
@@ -816,8 +816,8 @@
done
lemma zcong_lineq_unique:
- "Numeral0 < n ==>
- zgcd (a, n) = Numeral1 ==> \<exists>!x. Numeral0 \<le> x \<and> x < n \<and> [a * x = b] (mod n)"
+ "0 < n ==>
+ zgcd (a, n) = 1 ==> \<exists>!x. 0 \<le> x \<and> x < n \<and> [a * x = b] (mod n)"
apply auto
apply (rule_tac [2] zcong_zless_imp_eq)
apply (tactic {* stac (thm "zcong_cancel2" RS sym) 6 *})
@@ -833,7 +833,7 @@
apply (subst zcong_zmod)
apply (subst zmod_zmult1_eq [symmetric])
apply (subst zcong_zmod [symmetric])
- apply (subgoal_tac "[a * x * b = Numeral1 * b] (mod n)")
+ apply (subgoal_tac "[a * x * b = 1 * b] (mod n)")
apply (rule_tac [2] zcong_zmult)
apply (simp_all add: zmult_assoc)
done