(* Title: HOL/SPARK/Examples/Gcd/Greatest_Common_Divisor.thy
Author: Stefan Berghofer
Copyright: secunet Security Networks AG
*)
theory Greatest_Common_Divisor
imports "../../SPARK" GCD
begin
spark_proof_functions
gcd = "gcd :: int \<Rightarrow> int \<Rightarrow> int"
spark_open "greatest_common_divisor/g_c_d"
spark_vc procedure_g_c_d_4
proof -
from `0 < d` have "0 \<le> c mod d" by (rule pos_mod_sign)
with `0 \<le> c` `0 < d` `c - c sdiv d * d \<noteq> 0` show ?C1
by (simp add: sdiv_pos_pos zmod_zdiv_equality')
next
from `0 \<le> c` `0 < d` `gcd c d = gcd m n` show ?C2
by (simp add: sdiv_pos_pos zmod_zdiv_equality' gcd_non_0_int)
qed
spark_vc procedure_g_c_d_11
proof -
from `0 \<le> c` `0 < d` `c - c sdiv d * d = 0`
have "d dvd c"
by (auto simp add: sdiv_pos_pos dvd_def ac_simps)
with `0 < d` `gcd c d = gcd m n` show ?C1
by simp
qed
spark_end
end