src/HOL/ex/Sqrt.thy
 author nipkow Fri Mar 06 17:38:47 2009 +0100 (2009-03-06) changeset 30313 b2441b0c8d38 parent 28952 15a4b2cf8c34 child 30411 9c9b6511ad1b permissions -rw-r--r--
```     1 (*  Title:      HOL/ex/Sqrt.thy
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```     2     Author:     Markus Wenzel, TU Muenchen
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```     3
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```     4 *)
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```     5
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```     6 header {*  Square roots of primes are irrational *}
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```     7
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```     8 theory Sqrt
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```     9 imports Complex_Main Primes
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```    10 begin
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```    11
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```    12 text {* The definition and the key representation theorem for the set of
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```    13 rational numbers has been moved to a core theory.  *}
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```    14
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```    15 declare Rats_abs_nat_div_natE[elim?]
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```    16
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```    17 subsection {* Main theorem *}
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```    18
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```    19 text {*
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```    20   The square root of any prime number (including @{text 2}) is
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```    21   irrational.
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```    22 *}
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```    23
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```    24 theorem sqrt_prime_irrational:
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```    25   assumes "prime p"
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```    26   shows "sqrt (real p) \<notin> \<rat>"
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```    27 proof
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```    28   from `prime p` have p: "1 < p" by (simp add: prime_def)
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```    29   assume "sqrt (real p) \<in> \<rat>"
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```    30   then obtain m n where
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```    31       n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt (real p)\<bar> = real m / real n"
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```    32     and gcd: "gcd m n = 1" ..
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```    33   have eq: "m\<twosuperior> = p * n\<twosuperior>"
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```    34   proof -
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```    35     from n and sqrt_rat have "real m = \<bar>sqrt (real p)\<bar> * real n" by simp
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```    36     then have "real (m\<twosuperior>) = (sqrt (real p))\<twosuperior> * real (n\<twosuperior>)"
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```    37       by (auto simp add: power2_eq_square)
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```    38     also have "(sqrt (real p))\<twosuperior> = real p" by simp
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```    39     also have "\<dots> * real (n\<twosuperior>) = real (p * n\<twosuperior>)" by simp
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```    40     finally show ?thesis ..
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```    41   qed
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```    42   have "p dvd m \<and> p dvd n"
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```    43   proof
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```    44     from eq have "p dvd m\<twosuperior>" ..
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```    45     with `prime p` show "p dvd m" by (rule prime_dvd_power_two)
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```    46     then obtain k where "m = p * k" ..
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```    47     with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: power2_eq_square mult_ac)
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```    48     with p have "n\<twosuperior> = p * k\<twosuperior>" by (simp add: power2_eq_square)
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```    49     then have "p dvd n\<twosuperior>" ..
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```    50     with `prime p` show "p dvd n" by (rule prime_dvd_power_two)
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```    51   qed
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```    52   then have "p dvd gcd m n" ..
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```    53   with gcd have "p dvd 1" by simp
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```    54   then have "p \<le> 1" by (simp add: dvd_imp_le)
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```    55   with p show False by simp
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```    56 qed
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```    57
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```    58 corollary "sqrt (real (2::nat)) \<notin> \<rat>"
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```    59   by (rule sqrt_prime_irrational) (rule two_is_prime)
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```    60
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```    61
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```    62 subsection {* Variations *}
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```    63
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```    64 text {*
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```    65   Here is an alternative version of the main proof, using mostly
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```    66   linear forward-reasoning.  While this results in less top-down
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```    67   structure, it is probably closer to proofs seen in mathematics.
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```    68 *}
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```    69
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```    70 theorem
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```    71   assumes "prime p"
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```    72   shows "sqrt (real p) \<notin> \<rat>"
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```    73 proof
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```    74   from `prime p` have p: "1 < p" by (simp add: prime_def)
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```    75   assume "sqrt (real p) \<in> \<rat>"
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```    76   then obtain m n where
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```    77       n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt (real p)\<bar> = real m / real n"
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```    78     and gcd: "gcd m n = 1" ..
```
```    79   from n and sqrt_rat have "real m = \<bar>sqrt (real p)\<bar> * real n" by simp
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```    80   then have "real (m\<twosuperior>) = (sqrt (real p))\<twosuperior> * real (n\<twosuperior>)"
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```    81     by (auto simp add: power2_eq_square)
```
```    82   also have "(sqrt (real p))\<twosuperior> = real p" by simp
```
```    83   also have "\<dots> * real (n\<twosuperior>) = real (p * n\<twosuperior>)" by simp
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```    84   finally have eq: "m\<twosuperior> = p * n\<twosuperior>" ..
```
```    85   then have "p dvd m\<twosuperior>" ..
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```    86   with `prime p` have dvd_m: "p dvd m" by (rule prime_dvd_power_two)
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```    87   then obtain k where "m = p * k" ..
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```    88   with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: power2_eq_square mult_ac)
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```    89   with p have "n\<twosuperior> = p * k\<twosuperior>" by (simp add: power2_eq_square)
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```    90   then have "p dvd n\<twosuperior>" ..
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```    91   with `prime p` have "p dvd n" by (rule prime_dvd_power_two)
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```    92   with dvd_m have "p dvd gcd m n" by (rule gcd_greatest)
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```    93   with gcd have "p dvd 1" by simp
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```    94   then have "p \<le> 1" by (simp add: dvd_imp_le)
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```    95   with p show False by simp
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```    96 qed
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```    97
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```    98 end
```