src/HOL/ex/Sqrt.thy
 changeset 28952 15a4b2cf8c34 parent 28001 4642317e0deb child 30411 9c9b6511ad1b
```     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/ex/Sqrt.thy	Wed Dec 03 15:58:44 2008 +0100
1.3 @@ -0,0 +1,98 @@
1.4 +(*  Title:      HOL/ex/Sqrt.thy
1.5 +    Author:     Markus Wenzel, TU Muenchen
1.6 +
1.7 +*)
1.8 +
1.9 +header {*  Square roots of primes are irrational *}
1.10 +
1.11 +theory Sqrt
1.12 +imports Complex_Main Primes
1.13 +begin
1.14 +
1.15 +text {* The definition and the key representation theorem for the set of
1.16 +rational numbers has been moved to a core theory.  *}
1.17 +
1.18 +declare Rats_abs_nat_div_natE[elim?]
1.19 +
1.20 +subsection {* Main theorem *}
1.21 +
1.22 +text {*
1.23 +  The square root of any prime number (including @{text 2}) is
1.24 +  irrational.
1.25 +*}
1.26 +
1.27 +theorem sqrt_prime_irrational:
1.28 +  assumes "prime p"
1.29 +  shows "sqrt (real p) \<notin> \<rat>"
1.30 +proof
1.31 +  from `prime p` have p: "1 < p" by (simp add: prime_def)
1.32 +  assume "sqrt (real p) \<in> \<rat>"
1.33 +  then obtain m n where
1.34 +      n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt (real p)\<bar> = real m / real n"
1.35 +    and gcd: "gcd m n = 1" ..
1.36 +  have eq: "m\<twosuperior> = p * n\<twosuperior>"
1.37 +  proof -
1.38 +    from n and sqrt_rat have "real m = \<bar>sqrt (real p)\<bar> * real n" by simp
1.39 +    then have "real (m\<twosuperior>) = (sqrt (real p))\<twosuperior> * real (n\<twosuperior>)"
1.40 +      by (auto simp add: power2_eq_square)
1.41 +    also have "(sqrt (real p))\<twosuperior> = real p" by simp
1.42 +    also have "\<dots> * real (n\<twosuperior>) = real (p * n\<twosuperior>)" by simp
1.43 +    finally show ?thesis ..
1.44 +  qed
1.45 +  have "p dvd m \<and> p dvd n"
1.46 +  proof
1.47 +    from eq have "p dvd m\<twosuperior>" ..
1.48 +    with `prime p` show "p dvd m" by (rule prime_dvd_power_two)
1.49 +    then obtain k where "m = p * k" ..
1.50 +    with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: power2_eq_square mult_ac)
1.51 +    with p have "n\<twosuperior> = p * k\<twosuperior>" by (simp add: power2_eq_square)
1.52 +    then have "p dvd n\<twosuperior>" ..
1.53 +    with `prime p` show "p dvd n" by (rule prime_dvd_power_two)
1.54 +  qed
1.55 +  then have "p dvd gcd m n" ..
1.56 +  with gcd have "p dvd 1" by simp
1.57 +  then have "p \<le> 1" by (simp add: dvd_imp_le)
1.58 +  with p show False by simp
1.59 +qed
1.60 +
1.61 +corollary "sqrt (real (2::nat)) \<notin> \<rat>"
1.62 +  by (rule sqrt_prime_irrational) (rule two_is_prime)
1.63 +
1.64 +
1.65 +subsection {* Variations *}
1.66 +
1.67 +text {*
1.68 +  Here is an alternative version of the main proof, using mostly
1.69 +  linear forward-reasoning.  While this results in less top-down
1.70 +  structure, it is probably closer to proofs seen in mathematics.
1.71 +*}
1.72 +
1.73 +theorem
1.74 +  assumes "prime p"
1.75 +  shows "sqrt (real p) \<notin> \<rat>"
1.76 +proof
1.77 +  from `prime p` have p: "1 < p" by (simp add: prime_def)
1.78 +  assume "sqrt (real p) \<in> \<rat>"
1.79 +  then obtain m n where
1.80 +      n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt (real p)\<bar> = real m / real n"
1.81 +    and gcd: "gcd m n = 1" ..
1.82 +  from n and sqrt_rat have "real m = \<bar>sqrt (real p)\<bar> * real n" by simp
1.83 +  then have "real (m\<twosuperior>) = (sqrt (real p))\<twosuperior> * real (n\<twosuperior>)"
1.84 +    by (auto simp add: power2_eq_square)
1.85 +  also have "(sqrt (real p))\<twosuperior> = real p" by simp
1.86 +  also have "\<dots> * real (n\<twosuperior>) = real (p * n\<twosuperior>)" by simp
1.87 +  finally have eq: "m\<twosuperior> = p * n\<twosuperior>" ..
1.88 +  then have "p dvd m\<twosuperior>" ..
1.89 +  with `prime p` have dvd_m: "p dvd m" by (rule prime_dvd_power_two)
1.90 +  then obtain k where "m = p * k" ..
1.91 +  with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: power2_eq_square mult_ac)
1.92 +  with p have "n\<twosuperior> = p * k\<twosuperior>" by (simp add: power2_eq_square)
1.93 +  then have "p dvd n\<twosuperior>" ..
1.94 +  with `prime p` have "p dvd n" by (rule prime_dvd_power_two)
1.95 +  with dvd_m have "p dvd gcd m n" by (rule gcd_greatest)
1.96 +  with gcd have "p dvd 1" by simp
1.97 +  then have "p \<le> 1" by (simp add: dvd_imp_le)
1.98 +  with p show False by simp
1.99 +qed
1.100 +
1.101 +end
```