Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
to their abstract counterparts, while other binary numerals work correctly.
(* 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 => 'a" ("(_\<twosuperior>)" [1000] 999)
syntax (HTML output)
"_square" :: "'a => 'a" ("(_\<twosuperior>)" [1000] 999)
syntax (output)
"_square" :: "'a => 'a" ("(_^2)" [1000] 999)
translations
"x\<twosuperior>" == "x^Suc (Suc 0)"
subsection {* The set of rational numbers *}
constdefs
rationals :: "real set" ("\<rat>")
"\<rat> == {x. \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real (m::nat) / real (n::nat)}"
theorem rationals_rep: "x \<in> \<rat> ==>
\<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 "... = real m / real n" by simp
also from x_rat have "... = \<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> ==>
(!!m n. n \<noteq> 0 ==> \<bar>r\<bar> = real m / real n ==> gcd (m, n) = 1 ==> C) ==> 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 ==> p \<in> prime ==> 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 "... = 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::nat) ==> x \<notin> \<rat>"
proof (rule sqrt_prime_irrational)
{
fix m :: nat assume dvd: "m dvd 2"
hence "m \<le> 2" by (simp add: dvd_imp_le)
moreover from dvd have "m \<noteq> 0" by (auto 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 ==> p \<in> prime ==> 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 "... = 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