src/HOL/ex/Sqrt.thy
 author nipkow Mon Dec 19 17:10:45 2011 +0100 (2011-12-19) changeset 45917 1ce1bc9ff64a parent 32479 521cc9bf2958 child 46495 8e8a339e176f permissions -rw-r--r--
added old chestnut
```     1 (*  Title:      HOL/ex/Sqrt.thy
```
```     2     Author:     Markus Wenzel, Tobias Nipkow, TU Muenchen
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```     3 *)
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```     4
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```     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"
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```     9 begin
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```    10
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```    11 text {*
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```    12   The square root of any prime number (including @{text 2}) is
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```    13   irrational.
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```    14 *}
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```    15
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```    16 theorem sqrt_prime_irrational:
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```    17   assumes "prime (p::nat)"
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```    18   shows "sqrt (real p) \<notin> \<rat>"
```
```    19 proof
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```    20   from `prime p` have p: "1 < p" by (simp add: prime_nat_def)
```
```    21   assume "sqrt (real p) \<in> \<rat>"
```
```    22   then obtain m n :: nat where
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```    23       n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt (real p)\<bar> = real m / real n"
```
```    24     and gcd: "gcd m n = 1" by (rule Rats_abs_nat_div_natE)
```
```    25   have eq: "m\<twosuperior> = p * n\<twosuperior>"
```
```    26   proof -
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```    27     from n and sqrt_rat have "real m = \<bar>sqrt (real p)\<bar> * real n" by simp
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```    28     then have "real (m\<twosuperior>) = (sqrt (real p))\<twosuperior> * real (n\<twosuperior>)"
```
```    29       by (auto simp add: power2_eq_square)
```
```    30     also have "(sqrt (real p))\<twosuperior> = real p" by simp
```
```    31     also have "\<dots> * real (n\<twosuperior>) = real (p * n\<twosuperior>)" by simp
```
```    32     finally show ?thesis ..
```
```    33   qed
```
```    34   have "p dvd m \<and> p dvd n"
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```    35   proof
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```    36     from eq have "p dvd m\<twosuperior>" ..
```
```    37     with `prime p` pos2 show "p dvd m" by (rule prime_dvd_power_nat)
```
```    38     then obtain k where "m = p * k" ..
```
```    39     with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: power2_eq_square mult_ac)
```
```    40     with p have "n\<twosuperior> = p * k\<twosuperior>" by (simp add: power2_eq_square)
```
```    41     then have "p dvd n\<twosuperior>" ..
```
```    42     with `prime p` pos2 show "p dvd n" by (rule prime_dvd_power_nat)
```
```    43   qed
```
```    44   then have "p dvd gcd m n" ..
```
```    45   with gcd have "p dvd 1" by simp
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```    46   then have "p \<le> 1" by (simp add: dvd_imp_le)
```
```    47   with p show False by simp
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```    48 qed
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```    49
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```    50 corollary sqrt_real_2_not_rat: "sqrt (real (2::nat)) \<notin> \<rat>"
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```    51   by (rule sqrt_prime_irrational) (rule two_is_prime_nat)
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```    52
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```    53
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```    54 subsection {* Variations *}
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```    55
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```    56 text {*
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```    57   Here is an alternative version of the main proof, using mostly
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```    58   linear forward-reasoning.  While this results in less top-down
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```    59   structure, it is probably closer to proofs seen in mathematics.
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```    60 *}
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```    61
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```    62 theorem
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```    63   assumes "prime (p::nat)"
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```    64   shows "sqrt (real p) \<notin> \<rat>"
```
```    65 proof
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```    66   from `prime p` have p: "1 < p" by (simp add: prime_nat_def)
```
```    67   assume "sqrt (real p) \<in> \<rat>"
```
```    68   then obtain m n :: nat where
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```    69       n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt (real p)\<bar> = real m / real n"
```
```    70     and gcd: "gcd m n = 1" by (rule Rats_abs_nat_div_natE)
```
```    71   from n and sqrt_rat have "real m = \<bar>sqrt (real p)\<bar> * real n" by simp
```
```    72   then have "real (m\<twosuperior>) = (sqrt (real p))\<twosuperior> * real (n\<twosuperior>)"
```
```    73     by (auto simp add: power2_eq_square)
```
```    74   also have "(sqrt (real p))\<twosuperior> = real p" by simp
```
```    75   also have "\<dots> * real (n\<twosuperior>) = real (p * n\<twosuperior>)" by simp
```
```    76   finally have eq: "m\<twosuperior> = p * n\<twosuperior>" ..
```
```    77   then have "p dvd m\<twosuperior>" ..
```
```    78   with `prime p` pos2 have dvd_m: "p dvd m" by (rule prime_dvd_power_nat)
```
```    79   then obtain k where "m = p * k" ..
```
```    80   with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: power2_eq_square mult_ac)
```
```    81   with p have "n\<twosuperior> = p * k\<twosuperior>" by (simp add: power2_eq_square)
```
```    82   then have "p dvd n\<twosuperior>" ..
```
```    83   with `prime p` pos2 have "p dvd n" by (rule prime_dvd_power_nat)
```
```    84   with dvd_m have "p dvd gcd m n" by (rule gcd_greatest_nat)
```
```    85   with gcd have "p dvd 1" by simp
```
```    86   then have "p \<le> 1" by (simp add: dvd_imp_le)
```
```    87   with p show False by simp
```
```    88 qed
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```    89
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```    90
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```    91 text{* Another old chestnut, which is a consequence of the irrationality of 2. *}
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```    92
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```    93 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")
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```    94 proof cases
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```    95   assume "sqrt 2 powr sqrt 2 \<in> \<rat>"
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```    96   hence "?P (sqrt 2) (sqrt 2)" by(metis sqrt_real_2_not_rat[simplified])
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```    97   thus ?thesis by blast
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```    98 next
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```    99   assume 1: "sqrt 2 powr sqrt 2 \<notin> \<rat>"
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```   100   have "(sqrt 2 powr sqrt 2) powr sqrt 2 = 2"
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```   101     using powr_realpow[of _ 2]
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```   102     by (simp add: powr_powr power2_eq_square[symmetric])
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```   103   hence "?P (sqrt 2 powr sqrt 2) (sqrt 2)"
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```   104     by (metis 1 Rats_number_of sqrt_real_2_not_rat[simplified])
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```   105   thus ?thesis by blast
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```   106 qed
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```   107
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```   108 end
```