src/HOL/ex/Sqrt.thy
 author paulson Fri Nov 13 12:27:13 2015 +0000 (2015-11-13) changeset 61649 268d88ec9087 parent 60690 a9e45c9588c3 child 61762 d50b993b4fb9 permissions -rw-r--r--
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
```     1 (*  Title:      HOL/ex/Sqrt.thy
```
```     2     Author:     Markus Wenzel, Tobias Nipkow, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 section \<open>Square roots of primes are irrational\<close>
```
```     6
```
```     7 theory Sqrt
```
```     8 imports Complex_Main "~~/src/HOL/Number_Theory/Primes"
```
```     9 begin
```
```    10
```
```    11 text \<open>The square root of any prime number (including 2) is irrational.\<close>
```
```    12
```
```    13 theorem sqrt_prime_irrational:
```
```    14   assumes "prime (p::nat)"
```
```    15   shows "sqrt p \<notin> \<rat>"
```
```    16 proof
```
```    17   from \<open>prime p\<close> have p: "1 < p" by (simp add: prime_nat_def)
```
```    18   assume "sqrt p \<in> \<rat>"
```
```    19   then obtain m n :: nat where
```
```    20       n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt p\<bar> = m / n"
```
```    21     and "coprime m n" by (rule Rats_abs_nat_div_natE)
```
```    22   have eq: "m\<^sup>2 = p * n\<^sup>2"
```
```    23   proof -
```
```    24     from n and sqrt_rat have "m = \<bar>sqrt p\<bar> * n" by simp
```
```    25     then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2"
```
```    26       by (auto simp add: power2_eq_square)
```
```    27     also have "(sqrt p)\<^sup>2 = p" by simp
```
```    28     also have "\<dots> * n\<^sup>2 = p * n\<^sup>2" by simp
```
```    29     finally show ?thesis using of_nat_eq_iff by blast
```
```    30   qed
```
```    31   have "p dvd m \<and> p dvd n"
```
```    32   proof
```
```    33     from eq have "p dvd m\<^sup>2" ..
```
```    34     with \<open>prime p\<close> show "p dvd m" by (rule prime_dvd_power_nat)
```
```    35     then obtain k where "m = p * k" ..
```
```    36     with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by (auto simp add: power2_eq_square ac_simps)
```
```    37     with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square)
```
```    38     then have "p dvd n\<^sup>2" ..
```
```    39     with \<open>prime p\<close> show "p dvd n" by (rule prime_dvd_power_nat)
```
```    40   qed
```
```    41   then have "p dvd gcd m n" by simp
```
```    42   with \<open>coprime m n\<close> have "p = 1" by simp
```
```    43   with p show False by simp
```
```    44 qed
```
```    45
```
```    46 corollary sqrt_2_not_rat: "sqrt 2 \<notin> \<rat>"
```
```    47   using sqrt_prime_irrational[of 2] by simp
```
```    48
```
```    49
```
```    50 subsection \<open>Variations\<close>
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```    51
```
```    52 text \<open>
```
```    53   Here is an alternative version of the main proof, using mostly
```
```    54   linear forward-reasoning.  While this results in less top-down
```
```    55   structure, it is probably closer to proofs seen in mathematics.
```
```    56 \<close>
```
```    57
```
```    58 theorem
```
```    59   assumes "prime (p::nat)"
```
```    60   shows "sqrt p \<notin> \<rat>"
```
```    61 proof
```
```    62   from \<open>prime p\<close> have p: "1 < p" by (simp add: prime_nat_def)
```
```    63   assume "sqrt p \<in> \<rat>"
```
```    64   then obtain m n :: nat where
```
```    65       n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt p\<bar> = m / n"
```
```    66     and "coprime m n" by (rule Rats_abs_nat_div_natE)
```
```    67   from n and sqrt_rat have "m = \<bar>sqrt p\<bar> * n" by simp
```
```    68   then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2"
```
```    69     by (auto simp add: power2_eq_square)
```
```    70   also have "(sqrt p)\<^sup>2 = p" by simp
```
```    71   also have "\<dots> * n\<^sup>2 = p * n\<^sup>2" by simp
```
```    72   finally have eq: "m\<^sup>2 = p * n\<^sup>2" using of_nat_eq_iff by blast
```
```    73   then have "p dvd m\<^sup>2" ..
```
```    74   with \<open>prime p\<close> have dvd_m: "p dvd m" by (rule prime_dvd_power_nat)
```
```    75   then obtain k where "m = p * k" ..
```
```    76   with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by (auto simp add: power2_eq_square ac_simps)
```
```    77   with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square)
```
```    78   then have "p dvd n\<^sup>2" ..
```
```    79   with \<open>prime p\<close> have "p dvd n" by (rule prime_dvd_power_nat)
```
```    80   with dvd_m have "p dvd gcd m n" by (rule gcd_greatest_nat)
```
```    81   with \<open>coprime m n\<close> have "p = 1" by simp
```
```    82   with p show False by simp
```
```    83 qed
```
```    84
```
```    85
```
```    86 text \<open>Another old chestnut, which is a consequence of the irrationality of 2.\<close>
```
```    87
```
```    88 lemma "\<exists>a b::real. a \<notin> \<rat> \<and> b \<notin> \<rat> \<and> a powr b \<in> \<rat>" (is "\<exists>a b. ?P a b")
```
```    89 proof cases
```
```    90   assume "sqrt 2 powr sqrt 2 \<in> \<rat>"
```
```    91   then have "?P (sqrt 2) (sqrt 2)"
```
```    92     by (metis sqrt_2_not_rat)
```
```    93   then show ?thesis by blast
```
```    94 next
```
```    95   assume 1: "sqrt 2 powr sqrt 2 \<notin> \<rat>"
```
```    96   have "(sqrt 2 powr sqrt 2) powr sqrt 2 = 2"
```
```    97     using powr_realpow [of _ 2]
```
```    98     by (simp add: powr_powr power2_eq_square [symmetric])
```
```    99   then have "?P (sqrt 2 powr sqrt 2) (sqrt 2)"
```
```   100     by (metis 1 Rats_number_of sqrt_2_not_rat)
```
```   101   then show ?thesis by blast
```
```   102 qed
```
```   103
```
```   104 end
```