src/HOL/Presburger.thy
changeset 22026 cc60e54aa7cb
parent 21454 a1937c51ed88
child 22394 54ea68b5a92f
     1.1 --- a/src/HOL/Presburger.thy	Fri Jan 05 17:38:05 2007 +0100
     1.2 +++ b/src/HOL/Presburger.thy	Sat Jan 06 20:47:09 2007 +0100
     1.3 @@ -926,18 +926,6 @@
     1.4  theorem all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))"
     1.5    by (simp split add: split_nat)
     1.6  
     1.7 -theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
     1.8 -  apply (simp split add: split_nat)
     1.9 -  apply (rule iffI)
    1.10 -  apply (erule exE)
    1.11 -  apply (rule_tac x = "int x" in exI)
    1.12 -  apply simp
    1.13 -  apply (erule exE)
    1.14 -  apply (rule_tac x = "nat x" in exI)
    1.15 -  apply (erule conjE)
    1.16 -  apply (erule_tac x = "nat x" in allE)
    1.17 -  apply simp
    1.18 -  done
    1.19  
    1.20  theorem zdiff_int_split: "P (int (x - y)) =
    1.21    ((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
    1.22 @@ -945,22 +933,6 @@
    1.23    apply (simp_all add: zdiff_int)
    1.24    done
    1.25  
    1.26 -theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
    1.27 -  apply (simp only: dvd_def ex_nat int_int_eq [symmetric] zmult_int [symmetric]
    1.28 -    nat_0_le cong add: conj_cong)
    1.29 -  apply (rule iffI)
    1.30 -  apply iprover
    1.31 -  apply (erule exE)
    1.32 -  apply (case_tac "x=0")
    1.33 -  apply (rule_tac x=0 in exI)
    1.34 -  apply simp
    1.35 -  apply (case_tac "0 \<le> k")
    1.36 -  apply iprover
    1.37 -  apply (simp add: linorder_not_le)
    1.38 -  apply (drule mult_strict_left_mono_neg [OF iffD2 [OF zero_less_int_conv]])
    1.39 -  apply assumption
    1.40 -  apply (simp add: mult_ac)
    1.41 -  done
    1.42  
    1.43  theorem number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)"
    1.44    by simp