4 *) |
4 *) |
5 |
5 |
6 header {* Square roots of primes are irrational (script version) *} |
6 header {* Square roots of primes are irrational (script version) *} |
7 |
7 |
8 theory Sqrt_Script |
8 theory Sqrt_Script |
9 imports Complex_Main Primes |
9 imports Complex_Main "~~/src/HOL/Number_Theory/Primes" |
10 begin |
10 begin |
11 |
11 |
12 text {* |
12 text {* |
13 \medskip Contrast this linear Isabelle/Isar script with Markus |
13 \medskip Contrast this linear Isabelle/Isar script with Markus |
14 Wenzel's more mathematical version. |
14 Wenzel's more mathematical version. |
15 *} |
15 *} |
16 |
16 |
17 subsection {* Preliminaries *} |
17 subsection {* Preliminaries *} |
18 |
18 |
19 lemma prime_nonzero: "prime p \<Longrightarrow> p \<noteq> 0" |
19 lemma prime_nonzero: "prime (p::nat) \<Longrightarrow> p \<noteq> 0" |
20 by (force simp add: prime_def) |
20 by (force simp add: prime_nat_def) |
21 |
21 |
22 lemma prime_dvd_other_side: |
22 lemma prime_dvd_other_side: |
23 "n * n = p * (k * k) \<Longrightarrow> prime p \<Longrightarrow> p dvd n" |
23 "(n::nat) * n = p * (k * k) \<Longrightarrow> prime p \<Longrightarrow> p dvd n" |
24 apply (subgoal_tac "p dvd n * n", blast dest: prime_dvd_mult) |
24 apply (subgoal_tac "p dvd n * n", blast dest: prime_dvd_mult_nat) |
25 apply auto |
25 apply auto |
26 done |
26 done |
27 |
27 |
28 lemma reduction: "prime p \<Longrightarrow> |
28 lemma reduction: "prime (p::nat) \<Longrightarrow> |
29 0 < k \<Longrightarrow> k * k = p * (j * j) \<Longrightarrow> k < p * j \<and> 0 < j" |
29 0 < k \<Longrightarrow> k * k = p * (j * j) \<Longrightarrow> k < p * j \<and> 0 < j" |
30 apply (rule ccontr) |
30 apply (rule ccontr) |
31 apply (simp add: linorder_not_less) |
31 apply (simp add: linorder_not_less) |
32 apply (erule disjE) |
32 apply (erule disjE) |
33 apply (frule mult_le_mono, assumption) |
33 apply (frule mult_le_mono, assumption) |
34 apply auto |
34 apply auto |
35 apply (force simp add: prime_def) |
35 apply (force simp add: prime_nat_def) |
36 done |
36 done |
37 |
37 |
38 lemma rearrange: "(j::nat) * (p * j) = k * k \<Longrightarrow> k * k = p * (j * j)" |
38 lemma rearrange: "(j::nat) * (p * j) = k * k \<Longrightarrow> k * k = p * (j * j)" |
39 by (simp add: mult_ac) |
39 by (simp add: mult_ac) |
40 |
40 |
41 lemma prime_not_square: |
41 lemma prime_not_square: |
42 "prime p \<Longrightarrow> (\<And>k. 0 < k \<Longrightarrow> m * m \<noteq> p * (k * k))" |
42 "prime (p::nat) \<Longrightarrow> (\<And>k. 0 < k \<Longrightarrow> m * m \<noteq> p * (k * k))" |
43 apply (induct m rule: nat_less_induct) |
43 apply (induct m rule: nat_less_induct) |
44 apply clarify |
44 apply clarify |
45 apply (frule prime_dvd_other_side, assumption) |
45 apply (frule prime_dvd_other_side, assumption) |
46 apply (erule dvdE) |
46 apply (erule dvdE) |
47 apply (simp add: nat_mult_eq_cancel_disj prime_nonzero) |
47 apply (simp add: nat_mult_eq_cancel_disj prime_nonzero) |
55 The square root of any prime number (including @{text 2}) is |
55 The square root of any prime number (including @{text 2}) is |
56 irrational. |
56 irrational. |
57 *} |
57 *} |
58 |
58 |
59 theorem prime_sqrt_irrational: |
59 theorem prime_sqrt_irrational: |
60 "prime p \<Longrightarrow> x * x = real p \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<notin> \<rat>" |
60 "prime (p::nat) \<Longrightarrow> x * x = real p \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<notin> \<rat>" |
61 apply (rule notI) |
61 apply (rule notI) |
62 apply (erule Rats_abs_nat_div_natE) |
62 apply (erule Rats_abs_nat_div_natE) |
63 apply (simp del: real_of_nat_mult |
63 apply (simp del: real_of_nat_mult |
64 add: real_abs_def divide_eq_eq prime_not_square real_of_nat_mult [symmetric]) |
64 add: real_abs_def divide_eq_eq prime_not_square real_of_nat_mult [symmetric]) |
65 done |
65 done |
66 |
66 |
67 lemmas two_sqrt_irrational = |
67 lemmas two_sqrt_irrational = |
68 prime_sqrt_irrational [OF two_is_prime] |
68 prime_sqrt_irrational [OF two_is_prime_nat] |
69 |
69 |
70 end |
70 end |