--- /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
+ License: GPL (GNU GENERAL PUBLIC LICENSE)
+*)
+
+header {* Square roots of primes are irrational *}
+
+theory Sqrt_Irrational = Real + Primes:
+
+syntax (xsymbols) (* FIXME move to main HOL (!?) *)
+ "_square" :: "'a \<Rightarrow> 'a" ("(_\<twosuperior>)" [1000] 999)
+syntax (HTML output)
+ "_square" :: "'a \<Rightarrow> 'a" ("(_\<twosuperior>)" [1000] 999)
+syntax (output)
+ "_square" :: "'a \<Rightarrow> 'a" ("(_^2)" [1000] 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)"
+ by (simp add: real_mult_div_cancel1)
+ 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