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+(*  Title:      HOL/ex/Sqrt.thy
+    Author:     Markus Wenzel, TU Muenchen
+*)
+
+header {*  Square roots of primes are irrational *}
+
+theory Sqrt
+imports Complex_Main "~~/src/HOL/Number_Theory/Primes"
+begin
+
+text {*
+  The square root of any prime number (including @{text 2}) is
+  irrational.
+*}
+
+theorem sqrt_prime_irrational:
+  assumes "prime (p::nat)"
+  shows "sqrt (real p) \<notin> \<rat>"
+proof
+  from `prime p` have p: "1 < p" by (simp add: prime_nat_def)
+  assume "sqrt (real p) \<in> \<rat>"
+  then obtain m n :: nat where
+      n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt (real p)\<bar> = real m / real n"
+    and gcd: "gcd m n = 1" by (rule Rats_abs_nat_div_natE)
+  have eq: "m\<twosuperior> = p * n\<twosuperior>"
+  proof -
+    from n and sqrt_rat have "real m = \<bar>sqrt (real p)\<bar> * real n" by simp
+    then have "real (m\<twosuperior>) = (sqrt (real p))\<twosuperior> * real (n\<twosuperior>)"
+      by (auto simp add: power2_eq_square)
+    also have "(sqrt (real p))\<twosuperior> = real p" by simp
+    also have "\<dots> * real (n\<twosuperior>) = real (p * n\<twosuperior>)" by simp
+    finally show ?thesis ..
+  qed
+  have "p dvd m \<and> p dvd n"
+  proof
+    from eq have "p dvd m\<twosuperior>" ..
+    with `prime p` pos2 show "p dvd m" by (rule prime_dvd_power_nat)
+    then obtain k where "m = p * k" ..
+    with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: power2_eq_square mult_ac)
+    with p have "n\<twosuperior> = p * k\<twosuperior>" by (simp add: power2_eq_square)
+    then have "p dvd n\<twosuperior>" ..
+    with `prime p` pos2 show "p dvd n" by (rule prime_dvd_power_nat)
+  qed
+  then have "p dvd gcd m n" ..
+  with gcd have "p dvd 1" by simp
+  then have "p \<le> 1" by (simp add: dvd_imp_le)
+  with p show False by simp
+qed
+
+corollary "sqrt (real (2::nat)) \<notin> \<rat>"
+  by (rule sqrt_prime_irrational) (rule two_is_prime_nat)
+
+
+subsection {* Variations *}
+
+text {*
+  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.
+*}
+
+theorem
+  assumes "prime (p::nat)"
+  shows "sqrt (real p) \<notin> \<rat>"
+proof
+  from `prime p` have p: "1 < p" by (simp add: prime_nat_def)
+  assume "sqrt (real p) \<in> \<rat>"
+  then obtain m n :: nat where
+      n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt (real p)\<bar> = real m / real n"
+    and gcd: "gcd m n = 1" by (rule Rats_abs_nat_div_natE)
+  from n and sqrt_rat have "real m = \<bar>sqrt (real p)\<bar> * real n" by simp
+  then have "real (m\<twosuperior>) = (sqrt (real p))\<twosuperior> * real (n\<twosuperior>)"
+    by (auto simp add: power2_eq_square)
+  also have "(sqrt (real p))\<twosuperior> = real p" by simp
+  also have "\<dots> * real (n\<twosuperior>) = real (p * n\<twosuperior>)" by simp
+  finally have eq: "m\<twosuperior> = p * n\<twosuperior>" ..
+  then have "p dvd m\<twosuperior>" ..
+  with `prime p` pos2 have dvd_m: "p dvd m" by (rule prime_dvd_power_nat)
+  then obtain k where "m = p * k" ..
+  with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: power2_eq_square mult_ac)
+  with p have "n\<twosuperior> = p * k\<twosuperior>" by (simp add: power2_eq_square)
+  then have "p dvd n\<twosuperior>" ..
+  with `prime p` pos2 have "p dvd n" by (rule prime_dvd_power_nat)
+  with dvd_m have "p dvd gcd m n" by (rule gcd_greatest_nat)
+  with gcd have "p dvd 1" by simp
+  then have "p \<le> 1" by (simp add: dvd_imp_le)
+  with p show False by simp
+qed
+
+end