author wenzelm Thu, 27 Sep 2001 18:43:40 +0200 changeset 11594 3ccea743e5e7 parent 11593 ab08d61966b1 child 11595 6ef2535fff93
Square roots of primes are irrational;
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Real/ex/Sqrt_Irrational.thy	Thu Sep 27 18:43:40 2001 +0200
@@ -0,0 +1,159 @@
+(*  Title:      HOL/Real/ex/Sqrt_Irrational.thy
+    ID:         \$Id\$
+    Author:     Markus Wenzel, TU Muenchen
+*)
+
+header {*  Square roots of primes are irrational *}
+
+theory Sqrt_Irrational = Real + Primes:
+
+syntax (xsymbols)                        (* FIXME move to main HOL (!?) *)
+  "_square" :: "'a \<Rightarrow> 'a"  ("(_\<twosuperior>)"  999)
+syntax (HTML output)
+  "_square" :: "'a \<Rightarrow> 'a"  ("(_\<twosuperior>)"  999)
+syntax (output)
+  "_square" :: "'a \<Rightarrow> 'a"  ("(_^2)"  999)
+translations
+  "x\<twosuperior>" \<rightleftharpoons> "x^2"
+
+
+subsection {* The set of rational numbers *}
+
+constdefs
+  rationals :: "real set"    ("\<rat>")
+  "\<rat> \<equiv> {x. \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real (m::nat) / real (n::nat)}"
+
+theorem rationals_rep: "x \<in> \<rat> \<Longrightarrow>
+  \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real m / real n \<and> gcd (m, n) = 1"
+proof -
+  assume "x \<in> \<rat>"
+  then obtain m n :: nat where n: "n \<noteq> 0" and x_rat: "\<bar>x\<bar> = real m / real n"
+    by (unfold rationals_def) blast
+  let ?gcd = "gcd (m, n)"
+  from n have gcd: "?gcd \<noteq> 0" by (simp add: gcd_zero)
+  let ?k = "m div ?gcd"
+  let ?l = "n div ?gcd"
+  let ?gcd' = "gcd (?k, ?l)"
+  have "?gcd dvd m" .. hence gcd_k: "?gcd * ?k = m"
+    by (rule dvd_mult_div_cancel)
+  have "?gcd dvd n" .. hence gcd_l: "?gcd * ?l = n"
+    by (rule dvd_mult_div_cancel)
+
+  from n gcd_l have "?l \<noteq> 0"
+    by (auto iff del: neq0_conv)
+  moreover
+  have "\<bar>x\<bar> = real ?k / real ?l"
+  proof -
+    from gcd have "real ?k / real ?l = real (?gcd * ?k) / real (?gcd * ?l)"
+    also from gcd_k gcd_l have "\<dots> = real m / real n" by simp
+    also from x_rat have "\<dots> = \<bar>x\<bar>" ..
+    finally show ?thesis ..
+  qed
+  moreover
+  have "?gcd' = 1"
+  proof -
+    have "?gcd * ?gcd' = gcd (?gcd * ?k, ?gcd * ?l)"
+      by (rule gcd_mult_distrib2)
+    with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp
+    with gcd show ?thesis by simp
+  qed
+  ultimately show ?thesis by blast
+qed
+
+lemma [elim?]: "r \<in> \<rat> \<Longrightarrow>
+    (\<And>m n. n \<noteq> 0 \<Longrightarrow> \<bar>r\<bar> = real m / real n \<Longrightarrow> gcd (m, n) = 1 \<Longrightarrow> C) \<Longrightarrow> C"
+  by (insert rationals_rep) blast
+
+
+subsection {* Main theorem *}
+
+text {*
+  The square root of any prime number (including @{text 2}) is
+  irrational.
+*}
+
+theorem sqrt_prime_irrational: "x\<twosuperior> = real p \<Longrightarrow> p \<in> prime \<Longrightarrow> x \<notin> \<rat>"
+proof
+  assume x_sqrt: "x\<twosuperior> = real p"
+  assume p_prime: "p \<in> prime"
+  hence p: "1 < p" by (simp add: prime_def)
+  assume "x \<in> \<rat>"
+  then obtain m n where
+    n: "n \<noteq> 0" and x_rat: "\<bar>x\<bar> = real m / real n" and gcd: "gcd (m, n) = 1" ..
+  have eq: "m\<twosuperior> = p * n\<twosuperior>"
+  proof -
+    from n x_rat have "real m = \<bar>x\<bar> * real n" by simp
+    hence "real (m\<twosuperior>) = x\<twosuperior> * real (n\<twosuperior>)" by (simp split: abs_split)
+    also from x_sqrt have "\<dots> = 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 p_prime show "p dvd m" by (rule prime_dvd_square)
+    then obtain k where "m = p * k" ..
+    with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: mult_ac)
+    with p have "n\<twosuperior> = p * k\<twosuperior>" by simp
+    hence "p dvd n\<twosuperior>" ..
+    with p_prime show "p dvd n" by (rule prime_dvd_square)
+  qed
+  hence "p dvd gcd (m, n)" ..
+  with gcd have "p dvd 1" by simp
+  hence "p \<le> 1" by (simp add: dvd_imp_le)
+  with p show False by simp
+qed
+
+
+subsection {* Variations *}
+
+text {*
+  Just for the record: we instantiate the main theorem for the
+  specific prime number @{text 2} (real mathematicians would never do
+  this formally :-).
+*}
+
+theorem "x\<twosuperior> = real 2 \<Longrightarrow> x \<notin> \<rat>"
+proof (rule sqrt_prime_irrational)
+  {
+    fix m assume dvd: "m dvd 2"
+    hence "m \<le> 2" by (simp add: dvd_imp_le)
+    moreover from dvd have "m \<noteq> 0" by (auto dest: dvd_0_left iff del: neq0_conv)
+    ultimately have "m = 1 \<or> m = 2" by arith
+  }
+  thus "2 \<in> prime" by (simp add: prime_def)
+qed
+
+text {*
+  \medskip 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 "x\<twosuperior> = real p \<Longrightarrow> p \<in> prime \<Longrightarrow> x \<notin> \<rat>"
+proof
+  assume x_sqrt: "x\<twosuperior> = real p"
+  assume p_prime: "p \<in> prime"
+  hence p: "1 < p" by (simp add: prime_def)
+  assume "x \<in> \<rat>"
+  then obtain m n where
+    n: "n \<noteq> 0" and x_rat: "\<bar>x\<bar> = real m / real n" and gcd: "gcd (m, n) = 1" ..
+  from n x_rat have "real m = \<bar>x\<bar> * real n" by simp
+  hence "real (m\<twosuperior>) = x\<twosuperior> * real (n\<twosuperior>)" by (simp split: abs_split)
+  also from x_sqrt have "\<dots> = real (p * n\<twosuperior>)" by simp
+  finally have eq: "m\<twosuperior> = p * n\<twosuperior>" ..
+  hence "p dvd m\<twosuperior>" ..
+  with p_prime have dvd_m: "p dvd m" by (rule prime_dvd_square)
+  then obtain k where "m = p * k" ..
+  with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: mult_ac)
+  with p have "n\<twosuperior> = p * k\<twosuperior>" by simp
+  hence "p dvd n\<twosuperior>" ..
+  with p_prime have "p dvd n" by (rule prime_dvd_square)
+  with dvd_m have "p dvd gcd (m, n)" by (rule gcd_greatest)
+  with gcd have "p dvd 1" by simp
+  hence "p \<le> 1" by (simp add: dvd_imp_le)
+  with p show False by simp
+qed
+
+end```