src/HOL/Complex/ex/Sqrt.thy
 author paulson Thu Jul 01 12:29:53 2004 +0200 (2004-07-01) changeset 15013 34264f5e4691 parent 14981 e73f8140af78 child 15149 c5c4884634b7 permissions -rw-r--r--
new treatment of binary numerals
```     1 (*  Title:      HOL/Hyperreal/ex/Sqrt.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Markus Wenzel, TU Muenchen
```
```     4
```
```     5 *)
```
```     6
```
```     7 header {*  Square roots of primes are irrational *}
```
```     8
```
```     9 theory Sqrt = Primes + Complex_Main:
```
```    10
```
```    11 subsection {* Preliminaries *}
```
```    12
```
```    13 text {*
```
```    14   The set of rational numbers, including the key representation
```
```    15   theorem.
```
```    16 *}
```
```    17
```
```    18 constdefs
```
```    19   rationals :: "real set"    ("\<rat>")
```
```    20   "\<rat> \<equiv> {x. \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real (m::nat) / real (n::nat)}"
```
```    21
```
```    22 theorem rationals_rep: "x \<in> \<rat> \<Longrightarrow>
```
```    23   \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real m / real n \<and> gcd (m, n) = 1"
```
```    24 proof -
```
```    25   assume "x \<in> \<rat>"
```
```    26   then obtain m n :: nat where
```
```    27       n: "n \<noteq> 0" and x_rat: "\<bar>x\<bar> = real m / real n"
```
```    28     by (unfold rationals_def) blast
```
```    29   let ?gcd = "gcd (m, n)"
```
```    30   from n have gcd: "?gcd \<noteq> 0" by (simp add: gcd_zero)
```
```    31   let ?k = "m div ?gcd"
```
```    32   let ?l = "n div ?gcd"
```
```    33   let ?gcd' = "gcd (?k, ?l)"
```
```    34   have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m"
```
```    35     by (rule dvd_mult_div_cancel)
```
```    36   have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n"
```
```    37     by (rule dvd_mult_div_cancel)
```
```    38
```
```    39   from n and gcd_l have "?l \<noteq> 0"
```
```    40     by (auto iff del: neq0_conv)
```
```    41   moreover
```
```    42   have "\<bar>x\<bar> = real ?k / real ?l"
```
```    43   proof -
```
```    44     from gcd have "real ?k / real ?l =
```
```    45         real (?gcd * ?k) / real (?gcd * ?l)"
```
```    46       by (simp add: mult_divide_cancel_left)
```
```    47     also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp
```
```    48     also from x_rat have "\<dots> = \<bar>x\<bar>" ..
```
```    49     finally show ?thesis ..
```
```    50   qed
```
```    51   moreover
```
```    52   have "?gcd' = 1"
```
```    53   proof -
```
```    54     have "?gcd * ?gcd' = gcd (?gcd * ?k, ?gcd * ?l)"
```
```    55       by (rule gcd_mult_distrib2)
```
```    56     with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp
```
```    57     with gcd show ?thesis by simp
```
```    58   qed
```
```    59   ultimately show ?thesis by blast
```
```    60 qed
```
```    61
```
```    62 lemma [elim?]: "r \<in> \<rat> \<Longrightarrow>
```
```    63   (\<And>m n. n \<noteq> 0 \<Longrightarrow> \<bar>r\<bar> = real m / real n \<Longrightarrow> gcd (m, n) = 1 \<Longrightarrow> C)
```
```    64     \<Longrightarrow> C"
```
```    65   using rationals_rep by blast
```
```    66
```
```    67
```
```    68 subsection {* Main theorem *}
```
```    69
```
```    70 text {*
```
```    71   The square root of any prime number (including @{text 2}) is
```
```    72   irrational.
```
```    73 *}
```
```    74
```
```    75 theorem sqrt_prime_irrational: "p \<in> prime \<Longrightarrow> sqrt (real p) \<notin> \<rat>"
```
```    76 proof
```
```    77   assume p_prime: "p \<in> prime"
```
```    78   then have p: "1 < p" by (simp add: prime_def)
```
```    79   assume "sqrt (real p) \<in> \<rat>"
```
```    80   then obtain m n where
```
```    81       n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt (real p)\<bar> = real m / real n"
```
```    82     and gcd: "gcd (m, n) = 1" ..
```
```    83   have eq: "m\<twosuperior> = p * n\<twosuperior>"
```
```    84   proof -
```
```    85     from n and sqrt_rat have "real m = \<bar>sqrt (real p)\<bar> * real n" by simp
```
```    86     then have "real (m\<twosuperior>) = (sqrt (real p))\<twosuperior> * real (n\<twosuperior>)"
```
```    87       by (auto simp add: power2_eq_square)
```
```    88     also have "(sqrt (real p))\<twosuperior> = real p" by simp
```
```    89     also have "\<dots> * real (n\<twosuperior>) = real (p * n\<twosuperior>)" by simp
```
```    90     finally show ?thesis ..
```
```    91   qed
```
```    92   have "p dvd m \<and> p dvd n"
```
```    93   proof
```
```    94     from eq have "p dvd m\<twosuperior>" ..
```
```    95     with p_prime show "p dvd m" by (rule prime_dvd_power_two)
```
```    96     then obtain k where "m = p * k" ..
```
```    97     with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: power2_eq_square mult_ac)
```
```    98     with p have "n\<twosuperior> = p * k\<twosuperior>" by (simp add: power2_eq_square)
```
```    99     then have "p dvd n\<twosuperior>" ..
```
```   100     with p_prime show "p dvd n" by (rule prime_dvd_power_two)
```
```   101   qed
```
```   102   then have "p dvd gcd (m, n)" ..
```
```   103   with gcd have "p dvd 1" by simp
```
```   104   then have "p \<le> 1" by (simp add: dvd_imp_le)
```
```   105   with p show False by simp
```
```   106 qed
```
```   107
```
```   108 corollary "sqrt (real (2::nat)) \<notin> \<rat>"
```
```   109   by (rule sqrt_prime_irrational) (rule two_is_prime)
```
```   110
```
```   111
```
```   112 subsection {* Variations *}
```
```   113
```
```   114 text {*
```
```   115   Here is an alternative version of the main proof, using mostly
```
```   116   linear forward-reasoning.  While this results in less top-down
```
```   117   structure, it is probably closer to proofs seen in mathematics.
```
```   118 *}
```
```   119
```
```   120 theorem "p \<in> prime \<Longrightarrow> sqrt (real p) \<notin> \<rat>"
```
```   121 proof
```
```   122   assume p_prime: "p \<in> prime"
```
```   123   then have p: "1 < p" by (simp add: prime_def)
```
```   124   assume "sqrt (real p) \<in> \<rat>"
```
```   125   then obtain m n where
```
```   126       n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt (real p)\<bar> = real m / real n"
```
```   127     and gcd: "gcd (m, n) = 1" ..
```
```   128   from n and sqrt_rat have "real m = \<bar>sqrt (real p)\<bar> * real n" by simp
```
```   129   then have "real (m\<twosuperior>) = (sqrt (real p))\<twosuperior> * real (n\<twosuperior>)"
```
```   130     by (auto simp add: power2_eq_square)
```
```   131   also have "(sqrt (real p))\<twosuperior> = real p" by simp
```
```   132   also have "\<dots> * real (n\<twosuperior>) = real (p * n\<twosuperior>)" by simp
```
```   133   finally have eq: "m\<twosuperior> = p * n\<twosuperior>" ..
```
```   134   then have "p dvd m\<twosuperior>" ..
```
```   135   with p_prime have dvd_m: "p dvd m" by (rule prime_dvd_power_two)
```
```   136   then obtain k where "m = p * k" ..
```
```   137   with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: power2_eq_square mult_ac)
```
```   138   with p have "n\<twosuperior> = p * k\<twosuperior>" by (simp add: power2_eq_square)
```
```   139   then have "p dvd n\<twosuperior>" ..
```
```   140   with p_prime have "p dvd n" by (rule prime_dvd_power_two)
```
```   141   with dvd_m have "p dvd gcd (m, n)" by (rule gcd_greatest)
```
```   142   with gcd have "p dvd 1" by simp
```
```   143   then have "p \<le> 1" by (simp add: dvd_imp_le)
```
```   144   with p show False by simp
```
```   145 qed
```
```   146
```
```   147 end
```