(* 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 \<open>0 \<le> r\<close> 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 \<open>2 < r\<close> have "2 * r < r * r" by (rule mult_strict_right_mono)
with \<open>r * r \<le> n\<close> and \<open>n \<le> 2147483647\<close> show ?thesis
by simp
qed auto
then show "2 * r \<le> 2147483647" by simp
qed
spark_end
end