(* Title: HOL/SPARK/Examples/Sqrt/Sqrt.thy
Author: Stefan Berghofer
Copyright: secunet Security Networks AG
*)
theory Sqrt
imports "../../SPARK"
begin
spark_open "sqrt/isqrt"
spark_vc function_isqrt_4
proof -
from `0 \<le> r` have "(r = 0 \<or> r = 1 \<or> r = 2) \<or> 2 < r" by auto
then show "2 * r \<le> 2147483646"
proof
assume "2 < r"
then have "0 < r" by simp
with `2 < r` have "2 * r < r * r" by (rule mult_strict_right_mono)
with `r * r \<le> n` and `n \<le> 2147483647` show ?thesis
by simp
qed auto
then show "2 * r \<le> 2147483647" by simp
qed
spark_end
end