author  haftmann 
Fri, 18 Jul 2008 18:25:53 +0200  
changeset 27651  16a26996c30e 
parent 21404  eb85850d3eb7 
child 28001  4642317e0deb 
permissions  rwrr 
13957  1 
(* Title: HOL/Hyperreal/ex/Sqrt_Script.thy 
2 
ID: $Id$ 

3 
Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

4 
Copyright 2001 University of Cambridge 

5 
*) 

6 

7 
header {* Square roots of primes are irrational (script version) *} 

8 

15149  9 
theory Sqrt_Script 
10 
imports Primes Complex_Main 

11 
begin 

13957  12 

13 
text {* 

14 
\medskip Contrast this linear Isabelle/Isar script with Markus 

15 
Wenzel's more mathematical version. 

16 
*} 

17 

18 
subsection {* Preliminaries *} 

19 

16663  20 
lemma prime_nonzero: "prime p \<Longrightarrow> p \<noteq> 0" 
13957  21 
by (force simp add: prime_def) 
22 

23 
lemma prime_dvd_other_side: 

16663  24 
"n * n = p * (k * k) \<Longrightarrow> prime p \<Longrightarrow> p dvd n" 
13957  25 
apply (subgoal_tac "p dvd n * n", blast dest: prime_dvd_mult) 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
21404
diff
changeset

26 
apply auto 
13957  27 
done 
28 

16663  29 
lemma reduction: "prime p \<Longrightarrow> 
13957  30 
0 < k \<Longrightarrow> k * k = p * (j * j) \<Longrightarrow> k < p * j \<and> 0 < j" 
31 
apply (rule ccontr) 

32 
apply (simp add: linorder_not_less) 

33 
apply (erule disjE) 

34 
apply (frule mult_le_mono, assumption) 

35 
apply auto 

36 
apply (force simp add: prime_def) 

37 
done 

38 

39 
lemma rearrange: "(j::nat) * (p * j) = k * k \<Longrightarrow> k * k = p * (j * j)" 

40 
by (simp add: mult_ac) 

41 

42 
lemma prime_not_square: 

16663  43 
"prime p \<Longrightarrow> (\<And>k. 0 < k \<Longrightarrow> m * m \<noteq> p * (k * k))" 
13957  44 
apply (induct m rule: nat_less_induct) 
45 
apply clarify 

46 
apply (frule prime_dvd_other_side, assumption) 

47 
apply (erule dvdE) 

48 
apply (simp add: nat_mult_eq_cancel_disj prime_nonzero) 

49 
apply (blast dest: rearrange reduction) 

50 
done 

51 

52 

53 
subsection {* The set of rational numbers *} 

54 

19736  55 
definition 
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
19736
diff
changeset

56 
rationals :: "real set" ("\<rat>") where 
19736  57 
"\<rat> = {x. \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real (m::nat) / real (n::nat)}" 
13957  58 

59 

60 
subsection {* Main theorem *} 

61 

62 
text {* 

63 
The square root of any prime number (including @{text 2}) is 

64 
irrational. 

65 
*} 

66 

67 
theorem prime_sqrt_irrational: 

16663  68 
"prime p \<Longrightarrow> x * x = real p \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<notin> \<rat>" 
13957  69 
apply (simp add: rationals_def real_abs_def) 
70 
apply clarify 

71 
apply (erule_tac P = "real m / real n * ?x = ?y" in rev_mp) 

72 
apply (simp del: real_of_nat_mult 

14288  73 
add: divide_eq_eq prime_not_square real_of_nat_mult [symmetric]) 
13957  74 
done 
75 

76 
lemmas two_sqrt_irrational = 

77 
prime_sqrt_irrational [OF two_is_prime] 

78 

79 
end 