theory gcd_GCD2_lemmas = "/home/hoenicke/jahob/lib/Jahob":

lemma GCD_minus [simp]:
  "((\<exists>q. (c::int) * q = a - b) \<and> (\<exists>q. c * q = b)) = 
  ((\<exists>q. (c::int) * q = a) \<and> (\<exists>q. c * q = b))"
proof
  assume "(\<exists>q. c * q = a) \<and> (\<exists>q. c * q = b)"
  then obtain q1 q2 where
    "((c * q1 = a) \<and> (c * q2 = b))" by auto
  hence "c * (q1 - q2) = a - b \<and> c * q2 = b" 
    by (auto simp add:zdiff_zmult_distrib2)
  thus "(\<exists>q. c * q = a - b) \<and> (\<exists>q. c * q = b)" by auto
next
  assume "(\<exists>q. c * q = a - b) \<and> (\<exists>q. c * q = b)" 
  then obtain q1 q2 where
    "((c * q1 = a - b) \<and> (c * q2 = b))" by auto
  hence "((c * (q1 + q2) = a) \<and> (c * q2 = b))"
    by (auto simp add:zadd_zmult_distrib2)
  thus "(\<exists>q. c * q = a) \<and> (\<exists>q. c * q = b)" by auto
qed

lemma GCD_minus2 [simp]:
  "((\<exists>q. (c::int) * q = a) \<and> (\<exists>q. c * q = b - a)) = 
  ((\<exists>q. (c::int) * q = a) \<and> (\<exists>q. c * q = b))"
proof -
  have "((\<exists>q. (c::int) * q = a) \<and> (\<exists>q. c * q = b - a)) = 
    ((\<exists>q. c * q = b - a) \<and> (\<exists>q. (c::int) * q = a))" by blast
  also have "\<dots> = ((\<exists>q. c * q = b) \<and> (\<exists>q. (c::int) * q = a))" by simp
  also have "\<dots> = ((\<exists>q. (c::int) * q = a) \<and> (\<exists>q. c * q = b))" by blast
  finally show "?thesis" .
qed


lemma GCD2_gcd_InitialPartOfProcedure3: "([|(ALL xObj. (xObj : Object));
((GCD2 Int Array) = {null});
(null : Object_alloc);
(0 <= olda);
(0 <= oldb);
(0 <= a_18);
(0 <= b_16);
(ALL c. (((EX q. ((inttimes c q) = olda)) & (EX q. ((inttimes c q) = oldb))) <-> ((EX q. ((inttimes c q) = a_18)) & (EX q. ((inttimes c q) = b_16)))));
(a_18 ~= 0);
(b_16 ~= 0);
(intless b_16 a_18)|] ==>
    comment ''InvPreservation'' (((EX q. ((inttimes c_bv350 q) = olda)) & (EX q. ((inttimes c_bv350 q) = oldb))) <-> ((EX q. ((inttimes c_bv350 q) = (a_18 - b_16))) & (EX q. ((inttimes c_bv350 q) = b_16)))))"
  by auto

lemma GCD2_gcd_InitialPartOfProcedure5: "([|(ALL xObj. (xObj : Object));
((GCD2 Int Array) = {null});
(null : Object_alloc);
(0 <= olda);
(0 <= oldb);
(0 <= a_18);
(0 <= b_16);
(ALL c. (((EX q. ((inttimes c q) = olda)) & (EX q. ((inttimes c q) = oldb))) <-> ((EX q. ((inttimes c q) = a_18)) & (EX q. ((inttimes c q) = b_16)))));
(a_18 ~= 0);
(b_16 ~= 0);
(~(intless b_16 a_18))|] ==>
    comment ''InvPreservation'' (((EX q. ((inttimes c_bv289 q) = olda)) & (EX q. ((inttimes c_bv289 q) = oldb))) <-> ((EX q. ((inttimes c_bv289 q) = a_18)) & (EX q. ((inttimes c_bv289 q) = (b_16 - a_18))))))"
by auto

end