src/HOL/ex/Sqrt.thy
 author wenzelm Tue Aug 13 16:25:47 2013 +0200 (2013-08-13) changeset 53015 a1119cf551e8 parent 51708 5188a18c33b1 child 53598 2bd8d459ebae permissions -rw-r--r--
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
```     1 (*  Title:      HOL/ex/Sqrt.thy
```
```     2     Author:     Markus Wenzel, Tobias Nipkow, TU Muenchen
```
```     3 *)
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```     4
```
```     5 header {*  Square roots of primes are irrational *}
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```     6
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```     7 theory Sqrt
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```     8 imports Complex_Main "~~/src/HOL/Number_Theory/Primes"
```
```     9 begin
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```    10
```
```    11 text {* The square root of any prime number (including 2) is irrational. *}
```
```    12
```
```    13 theorem sqrt_prime_irrational:
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```    14   assumes "prime (p::nat)"
```
```    15   shows "sqrt p \<notin> \<rat>"
```
```    16 proof
```
```    17   from `prime p` 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 gcd: "gcd m n = 1" 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 ..
```
```    30   qed
```
```    31   have "p dvd m \<and> p dvd n"
```
```    32   proof
```
```    33     from eq have "p dvd m\<^sup>2" ..
```
```    34     with `prime p` pos2 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 mult_ac)
```
```    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 `prime p` pos2 show "p dvd n" by (rule prime_dvd_power_nat)
```
```    40   qed
```
```    41   then have "p dvd gcd m n" ..
```
```    42   with gcd have "p dvd 1" by simp
```
```    43   then have "p \<le> 1" by (simp add: dvd_imp_le)
```
```    44   with p show False by simp
```
```    45 qed
```
```    46
```
```    47 corollary sqrt_2_not_rat: "sqrt 2 \<notin> \<rat>"
```
```    48   using sqrt_prime_irrational[of 2] by simp
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```    49
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```    50 subsection {* Variations *}
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```    51
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```    52 text {*
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```    53   Here is an alternative version of the main proof, using mostly
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```    54   linear forward-reasoning.  While this results in less top-down
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```    55   structure, it is probably closer to proofs seen in mathematics.
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```    56 *}
```
```    57
```
```    58 theorem
```
```    59   assumes "prime (p::nat)"
```
```    60   shows "sqrt p \<notin> \<rat>"
```
```    61 proof
```
```    62   from `prime p` 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 gcd: "gcd m n = 1" 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" ..
```
```    73   then have "p dvd m\<^sup>2" ..
```
```    74   with `prime p` pos2 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 mult_ac)
```
```    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 `prime p` pos2 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 gcd have "p dvd 1" by simp
```
```    82   then have "p \<le> 1" by (simp add: dvd_imp_le)
```
```    83   with p show False by simp
```
```    84 qed
```
```    85
```
```    86
```
```    87 text {* Another old chestnut, which is a consequence of the irrationality of 2. *}
```
```    88
```
```    89 lemma "\<exists>a b::real. a \<notin> \<rat> \<and> b \<notin> \<rat> \<and> a powr b \<in> \<rat>" (is "EX a b. ?P a b")
```
```    90 proof cases
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```    91   assume "sqrt 2 powr sqrt 2 \<in> \<rat>"
```
```    92   then have "?P (sqrt 2) (sqrt 2)"
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```    93     by (metis sqrt_2_not_rat)
```
```    94   then show ?thesis by blast
```
```    95 next
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```    96   assume 1: "sqrt 2 powr sqrt 2 \<notin> \<rat>"
```
```    97   have "(sqrt 2 powr sqrt 2) powr sqrt 2 = 2"
```
```    98     using powr_realpow [of _ 2]
```
```    99     by (simp add: powr_powr power2_eq_square [symmetric])
```
```   100   then have "?P (sqrt 2 powr sqrt 2) (sqrt 2)"
```
```   101     by (metis 1 Rats_number_of sqrt_2_not_rat)
```
```   102   then show ?thesis by blast
```
```   103 qed
```
```   104
```
```   105 end
```
```   106
```