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--
1 (*  Title:      HOL/ex/Sqrt.thy
2     Author:     Markus Wenzel, Tobias Nipkow, TU Muenchen
3 *)
5 header {*  Square roots of primes are irrational *}
7 theory Sqrt
8 imports Complex_Main "~~/src/HOL/Number_Theory/Primes"
9 begin
11 text {*
12   The square root of any prime number (including @{text 2}) is
13   irrational.
14 *}
16 theorem sqrt_prime_irrational:
17   assumes "prime (p::nat)"
18   shows "sqrt (real p) \<notin> \<rat>"
19 proof
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
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 -
27     from n and sqrt_rat have "real m = \<bar>sqrt (real p)\<bar> * real n" by simp
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"
35   proof
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
46   then have "p \<le> 1" by (simp add: dvd_imp_le)
47   with p show False by simp
48 qed
50 corollary sqrt_real_2_not_rat: "sqrt (real (2::nat)) \<notin> \<rat>"
51   by (rule sqrt_prime_irrational) (rule two_is_prime_nat)
54 subsection {* Variations *}
56 text {*
57   Here is an alternative version of the main proof, using mostly
58   linear forward-reasoning.  While this results in less top-down
59   structure, it is probably closer to proofs seen in mathematics.
60 *}
62 theorem
63   assumes "prime (p::nat)"
64   shows "sqrt (real p) \<notin> \<rat>"
65 proof
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
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
91 text{* Another old chestnut, which is a consequence of the irrationality of 2. *}
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")
94 proof cases
95   assume "sqrt 2 powr sqrt 2 \<in> \<rat>"
96   hence "?P (sqrt 2) (sqrt 2)" by(metis sqrt_real_2_not_rat[simplified])
97   thus ?thesis by blast
98 next
99   assume 1: "sqrt 2 powr sqrt 2 \<notin> \<rat>"
100   have "(sqrt 2 powr sqrt 2) powr sqrt 2 = 2"
101     using powr_realpow[of _ 2]
102     by (simp add: powr_powr power2_eq_square[symmetric])
103   hence "?P (sqrt 2 powr sqrt 2) (sqrt 2)"
104     by (metis 1 Rats_number_of sqrt_real_2_not_rat[simplified])
105   thus ?thesis by blast
106 qed
108 end