src/HOL/Number_Theory/Residues.thy
changeset 66304 cde6ceffcbc7
parent 65899 ab7d8c999531
child 66305 7454317f883c
--- a/src/HOL/Number_Theory/Residues.thy	Tue Aug 01 17:30:02 2017 +0200
+++ b/src/HOL/Number_Theory/Residues.thy	Tue Aug 01 17:33:04 2017 +0200
@@ -378,9 +378,9 @@
     by (metis residues_prime.wilson_theorem1 residues_prime.intro le_eq_less_or_eq)
 qed
 
-text {*
+text \<open>
   This result can be transferred to the multiplicative group of
-  $\mathbb{Z}/p\mathbb{Z}$ for $p$ prime. *}
+  $\mathbb{Z}/p\mathbb{Z}$ for $p$ prime.\<close>
 
 lemma mod_nat_int_pow_eq:
   fixes n :: nat and p a :: int
@@ -409,22 +409,22 @@
   have **:"nat ` {1 .. int p - 1} = {1 .. p - 1}" (is "?L = ?R")
   proof
     { fix n assume n: "n \<in> ?L"
-      then have "n \<in> ?R" using `p\<ge>2` by force
+      then have "n \<in> ?R" using \<open>p\<ge>2\<close> by force
     } thus "?L \<subseteq> ?R" by blast
     { fix n assume n: "n \<in> ?R"
-      then have "n \<in> ?L" using `p\<ge>2` Set_Interval.transfer_nat_int_set_functions(2) by fastforce
+      then have "n \<in> ?L" using \<open>p\<ge>2\<close> Set_Interval.transfer_nat_int_set_functions(2) by fastforce
     } thus "?R \<subseteq> ?L" by blast
   qed
   have "nat ` {a^i mod (int p) | i::nat. i \<in> UNIV} = {nat a^i mod p | i . i \<in> UNIV}" (is "?L = ?R")
   proof
     { fix x assume x: "x \<in> ?L"
       then obtain i where i:"x = nat (a^i mod (int p))" by blast
-      hence "x = nat a ^ i mod p" using mod_nat_int_pow_eq[of a "int p" i] a `p\<ge>2` by auto
+      hence "x = nat a ^ i mod p" using mod_nat_int_pow_eq[of a "int p" i] a \<open>p\<ge>2\<close> by auto
       hence "x \<in> ?R" using i by blast
     } thus "?L \<subseteq> ?R" by blast
     { fix x assume x: "x \<in> ?R"
       then obtain i where i:"x = nat a^i mod p" by blast
-      hence "x \<in> ?L" using mod_nat_int_pow_eq[of a "int p" i] a `p\<ge>2` by auto
+      hence "x \<in> ?L" using mod_nat_int_pow_eq[of a "int p" i] a \<open>p\<ge>2\<close> by auto
     } thus "?R \<subseteq> ?L" by blast
   qed
   hence "{1 .. p - 1} = {nat a^i mod p | i. i \<in> UNIV}"