equal
deleted
inserted
replaced
9 |
9 |
10 spark_open "sqrt/isqrt" |
10 spark_open "sqrt/isqrt" |
11 |
11 |
12 spark_vc function_isqrt_4 |
12 spark_vc function_isqrt_4 |
13 proof - |
13 proof - |
14 from `0 \<le> r` have "(r = 0 \<or> r = 1 \<or> r = 2) \<or> 2 < r" by auto |
14 from \<open>0 \<le> r\<close> have "(r = 0 \<or> r = 1 \<or> r = 2) \<or> 2 < r" by auto |
15 then show "2 * r \<le> 2147483646" |
15 then show "2 * r \<le> 2147483646" |
16 proof |
16 proof |
17 assume "2 < r" |
17 assume "2 < r" |
18 then have "0 < r" by simp |
18 then have "0 < r" by simp |
19 with `2 < r` have "2 * r < r * r" by (rule mult_strict_right_mono) |
19 with \<open>2 < r\<close> have "2 * r < r * r" by (rule mult_strict_right_mono) |
20 with `r * r \<le> n` and `n \<le> 2147483647` show ?thesis |
20 with \<open>r * r \<le> n\<close> and \<open>n \<le> 2147483647\<close> show ?thesis |
21 by simp |
21 by simp |
22 qed auto |
22 qed auto |
23 then show "2 * r \<le> 2147483647" by simp |
23 then show "2 * r \<le> 2147483647" by simp |
24 qed |
24 qed |
25 |
25 |