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(* Title: HOL/Real/ex/Sqrt.thy
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ID: $Id$
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Author: Markus Wenzel, TU Muenchen
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License: GPL (GNU GENERAL PUBLIC LICENSE)
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*)
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header {* Square roots of primes are irrational *}
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theory Sqrt = Primes + Real:
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syntax (xsymbols) (* FIXME move to main HOL (!?) *)
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"_square" :: "'a => 'a" ("(_\<twosuperior>)" [1000] 999)
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syntax (HTML output)
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"_square" :: "'a => 'a" ("(_\<twosuperior>)" [1000] 999)
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syntax (output)
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"_square" :: "'a => 'a" ("(_^2)" [1000] 999)
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translations
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"x\<twosuperior>" == "x^Suc (Suc 0)"
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subsection {* The set of rational numbers *}
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constdefs
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rationals :: "real set" ("\<rat>")
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"\<rat> == {x. \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real (m::nat) / real (n::nat)}"
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theorem rationals_rep: "x \<in> \<rat> ==>
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\<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real m / real n \<and> gcd (m, n) = 1"
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proof -
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assume "x \<in> \<rat>"
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then obtain m n :: nat where n: "n \<noteq> 0" and x_rat: "\<bar>x\<bar> = real m / real n"
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by (unfold rationals_def) blast
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let ?gcd = "gcd (m, n)"
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from n have gcd: "?gcd \<noteq> 0" by (simp add: gcd_zero)
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let ?k = "m div ?gcd"
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let ?l = "n div ?gcd"
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let ?gcd' = "gcd (?k, ?l)"
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have "?gcd dvd m" .. hence gcd_k: "?gcd * ?k = m"
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by (rule dvd_mult_div_cancel)
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have "?gcd dvd n" .. hence gcd_l: "?gcd * ?l = n"
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by (rule dvd_mult_div_cancel)
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from n gcd_l have "?l \<noteq> 0"
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by (auto iff del: neq0_conv)
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moreover
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have "\<bar>x\<bar> = real ?k / real ?l"
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proof -
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from gcd have "real ?k / real ?l = real (?gcd * ?k) / real (?gcd * ?l)"
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by (simp add: real_mult_div_cancel1)
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also from gcd_k gcd_l have "... = real m / real n" by simp
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also from x_rat have "... = \<bar>x\<bar>" ..
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finally show ?thesis ..
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qed
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moreover
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have "?gcd' = 1"
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proof -
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have "?gcd * ?gcd' = gcd (?gcd * ?k, ?gcd * ?l)"
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by (rule gcd_mult_distrib2)
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with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp
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with gcd show ?thesis by simp
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qed
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ultimately show ?thesis by blast
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qed
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lemma [elim?]: "r \<in> \<rat> ==>
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(!!m n. n \<noteq> 0 ==> \<bar>r\<bar> = real m / real n ==> gcd (m, n) = 1 ==> C)
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==> C"
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by (insert rationals_rep) blast
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subsection {* Main theorem *}
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text {*
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The square root of any prime number (including @{text 2}) is
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irrational.
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*}
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theorem sqrt_prime_irrational: "x\<twosuperior> = real p ==> p \<in> prime ==> x \<notin> \<rat>"
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proof
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assume x_sqrt: "x\<twosuperior> = real p"
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assume p_prime: "p \<in> prime"
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hence p: "1 < p" by (simp add: prime_def)
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assume "x \<in> \<rat>"
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then obtain m n where
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n: "n \<noteq> 0" and x_rat: "\<bar>x\<bar> = real m / real n" and gcd: "gcd (m, n) = 1" ..
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have eq: "m\<twosuperior> = p * n\<twosuperior>"
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proof -
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from n x_rat have "real m = \<bar>x\<bar> * real n" by simp
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hence "real (m\<twosuperior>) = x\<twosuperior> * real (n\<twosuperior>)" by (simp split: abs_split)
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also from x_sqrt have "... = real (p * n\<twosuperior>)" by simp
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finally show ?thesis ..
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qed
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have "p dvd m \<and> p dvd n"
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proof
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from eq have "p dvd m\<twosuperior>" ..
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with p_prime show "p dvd m" by (rule prime_dvd_square)
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then obtain k where "m = p * k" ..
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with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: mult_ac)
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with p have "n\<twosuperior> = p * k\<twosuperior>" by simp
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hence "p dvd n\<twosuperior>" ..
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with p_prime show "p dvd n" by (rule prime_dvd_square)
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qed
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hence "p dvd gcd (m, n)" ..
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with gcd have "p dvd 1" by simp
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hence "p \<le> 1" by (simp add: dvd_imp_le)
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with p show False by simp
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qed
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subsection {* Variations *}
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text {*
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Just for the record: we instantiate the main theorem for the
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specific prime number @{text 2} (real mathematicians would never do
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this formally :-).
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*}
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theorem "x\<twosuperior> = real (2::nat) ==> x \<notin> \<rat>"
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proof (rule sqrt_prime_irrational)
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{
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fix m :: nat assume dvd: "m dvd 2"
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hence "m \<le> 2" by (simp add: dvd_imp_le)
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moreover from dvd have "m \<noteq> 0" by (auto iff del: neq0_conv)
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ultimately have "m = 1 \<or> m = 2" by arith
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}
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thus "2 \<in> prime" by (simp add: prime_def)
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qed
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text {*
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\medskip An alternative version of the main proof, using mostly
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linear forward-reasoning. While this results in less top-down
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structure, it is probably closer to proofs seen in mathematics.
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*}
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theorem "x\<twosuperior> = real p ==> p \<in> prime ==> x \<notin> \<rat>"
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proof
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assume x_sqrt: "x\<twosuperior> = real p"
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assume p_prime: "p \<in> prime"
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hence p: "1 < p" by (simp add: prime_def)
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assume "x \<in> \<rat>"
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then obtain m n where
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n: "n \<noteq> 0" and x_rat: "\<bar>x\<bar> = real m / real n" and gcd: "gcd (m, n) = 1" ..
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from n x_rat have "real m = \<bar>x\<bar> * real n" by simp
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hence "real (m\<twosuperior>) = x\<twosuperior> * real (n\<twosuperior>)" by (simp split: abs_split)
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also from x_sqrt have "... = real (p * n\<twosuperior>)" by simp
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finally have eq: "m\<twosuperior> = p * n\<twosuperior>" ..
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hence "p dvd m\<twosuperior>" ..
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with p_prime have dvd_m: "p dvd m" by (rule prime_dvd_square)
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then obtain k where "m = p * k" ..
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with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: mult_ac)
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with p have "n\<twosuperior> = p * k\<twosuperior>" by simp
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hence "p dvd n\<twosuperior>" ..
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with p_prime have "p dvd n" by (rule prime_dvd_square)
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with dvd_m have "p dvd gcd (m, n)" by (rule gcd_greatest)
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with gcd have "p dvd 1" by simp
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hence "p \<le> 1" by (simp add: dvd_imp_le)
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with p show False by simp
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qed
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end
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