src/HOL/ex/Sqrt.thy

 (*  Title:      HOL/ex/Sqrt.thy
-    Author:     Markus Wenzel, Tobias Nipkow, TU Muenchen
+    Author:     Makarius
+    Author:     Tobias Nipkow, TU Muenchen
 *)
 
 section \<open>Square roots of primes are irrational\<close>
 
 theory Sqrt
-imports Complex_Main "HOL-Computational_Algebra.Primes"
+  imports Complex_Main "HOL-Computational_Algebra.Primes"
 begin
 
-text \<open>The square root of any prime number (including 2) is irrational.\<close>
+text \<open>
+  The square root of any prime number (including 2) is irrational.
+\<close>
 
 theorem sqrt_prime_irrational:
-  assumes "prime (p::nat)"
+  fixes p :: nat
+  assumes "prime p"
   shows "sqrt p \<notin> \<rat>"
 proof
-  from \<open>prime p\<close> have p: "1 < p" by (simp add: prime_nat_iff)
+  from \<open>prime p\<close> have p: "p > 1" by (rule prime_gt_1_nat)
   assume "sqrt p \<in> \<rat>"
-  then obtain m n :: nat where
-      n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt p\<bar> = m / n"
-    and "coprime m n" by (rule Rats_abs_nat_div_natE)
+  then obtain m n :: nat
+    where n: "n \<noteq> 0"
+      and sqrt_rat: "\<bar>sqrt p\<bar> = m / n"
+      and "coprime m n" by (rule Rats_abs_nat_div_natE)
   have eq: "m\<^sup>2 = p * n\<^sup>2"
   proof -
     from n and sqrt_rat have "m = \<bar>sqrt p\<bar> * n" by simp
-    then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2"
-      by (auto simp add: power2_eq_square)
+    then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2" by (simp add: power_mult_distrib)
     also have "(sqrt p)\<^sup>2 = p" by simp
     also have "\<dots> * n\<^sup>2 = p * n\<^sup>2" by simp
-    finally show ?thesis using of_nat_eq_iff by blast
+    finally show ?thesis by linarith
   qed
   have "p dvd m \<and> p dvd n"
   proof
     from eq have "p dvd m\<^sup>2" ..
-    with \<open>prime p\<close> show "p dvd m" by (rule prime_dvd_power_nat)
+    with \<open>prime p\<close> show "p dvd m" by (rule prime_dvd_power)
     then obtain k where "m = p * k" ..
-    with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by (auto simp add: power2_eq_square ac_simps)
+    with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by algebra
     with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square)
     then have "p dvd n\<^sup>2" ..
-    with \<open>prime p\<close> show "p dvd n" by (rule prime_dvd_power_nat)
+    with \<open>prime p\<close> show "p dvd n" by (rule prime_dvd_power)
   qed
   then have "p dvd gcd m n" by simp
   with \<open>coprime m n\<close> have "p = 1" by simp
   with p show False by simp
 qed
 
 corollary sqrt_2_not_rat: "sqrt 2 \<notin> \<rat>"
-  using sqrt_prime_irrational[of 2] by simp
-
-
-subsection \<open>Variations\<close>
+  using sqrt_prime_irrational [of 2] by simp
 
 text \<open>
-  Here is an alternative version of the main proof, using mostly
-  linear forward-reasoning.  While this results in less top-down
-  structure, it is probably closer to proofs seen in mathematics.
+  Here is an alternative version of the main proof, using mostly linear
+  forward-reasoning. While this results in less top-down structure, it is
+  probably closer to proofs seen in mathematics.
 \<close>
 
 theorem
-  assumes "prime (p::nat)"
+  fixes p :: nat
+  assumes "prime p"
   shows "sqrt p \<notin> \<rat>"
 proof
-  from \<open>prime p\<close> have p: "1 < p" by (simp add: prime_nat_iff)
+  from \<open>prime p\<close> have p: "p > 1" by (rule prime_gt_1_nat)
   assume "sqrt p \<in> \<rat>"
-  then obtain m n :: nat where
-      n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt p\<bar> = m / n"
-    and "coprime m n" by (rule Rats_abs_nat_div_natE)
+  then obtain m n :: nat
+    where n: "n \<noteq> 0"
+      and sqrt_rat: "\<bar>sqrt p\<bar> = m / n"
+      and "coprime m n" by (rule Rats_abs_nat_div_natE)
   from n and sqrt_rat have "m = \<bar>sqrt p\<bar> * n" by simp
-  then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2"
-    by (auto simp add: power2_eq_square)
+  then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2" by (auto simp add: power2_eq_square)
   also have "(sqrt p)\<^sup>2 = p" by simp
   also have "\<dots> * n\<^sup>2 = p * n\<^sup>2" by simp
-  finally have eq: "m\<^sup>2 = p * n\<^sup>2" using of_nat_eq_iff by blast
+  finally have eq: "m\<^sup>2 = p * n\<^sup>2" by linarith
   then have "p dvd m\<^sup>2" ..
-  with \<open>prime p\<close> have dvd_m: "p dvd m" by (rule prime_dvd_power_nat)
+  with \<open>prime p\<close> have dvd_m: "p dvd m" by (rule prime_dvd_power)
   then obtain k where "m = p * k" ..
-  with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by (auto simp add: power2_eq_square ac_simps)
+  with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by algebra
   with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square)
   then have "p dvd n\<^sup>2" ..
-  with \<open>prime p\<close> have "p dvd n" by (rule prime_dvd_power_nat)
+  with \<open>prime p\<close> have "p dvd n" by (rule prime_dvd_power)
   with dvd_m have "p dvd gcd m n" by (rule gcd_greatest)
   with \<open>coprime m n\<close> have "p = 1" by simp
   with p show False by simp
 qed
 
 
-text \<open>Another old chestnut, which is a consequence of the irrationality of \<^term>\<open>sqrt 2\<close>.\<close>
+text \<open>
+  Another old chestnut, which is a consequence of the irrationality of
+  \<^term>\<open>sqrt 2\<close>.
+\<close>
 
 lemma "\<exists>a b::real. a \<notin> \<rat> \<and> b \<notin> \<rat> \<and> a powr b \<in> \<rat>" (is "\<exists>a b. ?P a b")
 proof (cases "sqrt 2 powr sqrt 2 \<in> \<rat>")
   case True
   with sqrt_2_not_rat have "?P (sqrt 2) (sqrt 2)" by simp