src/HOL/Integ/Presburger.thy
 changeset 18202 46af82efd311 parent 17589 58eeffd73be1 child 20051 859e7129961b
```     1.1 --- a/src/HOL/Integ/Presburger.thy	Fri Nov 18 07:10:37 2005 +0100
1.2 +++ b/src/HOL/Integ/Presburger.thy	Fri Nov 18 07:13:58 2005 +0100
1.3 @@ -982,6 +982,99 @@
1.4  theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')"
1.5    by (simp cong: conj_cong)
1.6
1.7 +    (* Theorems used in presburger.ML for the computation simpset*)
1.8 +    (* FIXME: They are present in Float.thy, so may be Float.thy should be lightened.*)
1.9 +
1.10 +lemma lift_bool: "x \<Longrightarrow> x=True"
1.11 +  by simp
1.12 +
1.13 +lemma nlift_bool: "~x \<Longrightarrow> x=False"
1.14 +  by simp
1.15 +
1.16 +lemma not_false_eq_true: "(~ False) = True" by simp
1.17 +
1.18 +lemma not_true_eq_false: "(~ True) = False" by simp
1.19 +
1.20 +
1.21 +lemma int_eq_number_of_eq: "(((number_of v)::int)=(number_of w)) = iszero ((number_of (bin_add v (bin_minus w)))::int)"
1.22 +  by simp
1.23 +lemma int_iszero_number_of_Pls: "iszero (Numeral0::int)"
1.24 +  by (simp only: iszero_number_of_Pls)
1.25 +
1.26 +lemma int_nonzero_number_of_Min: "~(iszero ((-1)::int))"
1.27 +  by simp
1.28 +
1.29 +lemma int_iszero_number_of_0: "iszero ((number_of (w BIT bit.B0))::int) = iszero ((number_of w)::int)"
1.30 +  by simp
1.31 +
1.32 +lemma int_iszero_number_of_1: "\<not> iszero ((number_of (w BIT bit.B1))::int)"
1.33 +  by simp
1.34 +
1.35 +lemma int_less_number_of_eq_neg: "(((number_of x)::int) < number_of y) = neg ((number_of (bin_add x (bin_minus y)))::int)"
1.36 +  by simp
1.37 +
1.38 +lemma int_not_neg_number_of_Pls: "\<not> (neg (Numeral0::int))"
1.39 +  by simp
1.40 +
1.41 +lemma int_neg_number_of_Min: "neg (-1::int)"
1.42 +  by simp
1.43 +
1.44 +lemma int_neg_number_of_BIT: "neg ((number_of (w BIT x))::int) = neg ((number_of w)::int)"
1.45 +  by simp
1.46 +
1.47 +lemma int_le_number_of_eq: "(((number_of x)::int) \<le> number_of y) = (\<not> neg ((number_of (bin_add y (bin_minus x)))::int))"
1.48 +  by simp
1.49 +lemma int_number_of_add_sym: "((number_of v)::int) + number_of w = number_of (bin_add v w)"
1.50 +  by simp
1.51 +
1.52 +lemma int_number_of_diff_sym: "((number_of v)::int) - number_of w = number_of (bin_add v (bin_minus w))"
1.53 +  by simp
1.54 +
1.55 +lemma int_number_of_mult_sym: "((number_of v)::int) * number_of w = number_of (bin_mult v w)"
1.56 +  by simp
1.57 +
1.58 +lemma int_number_of_minus_sym: "- ((number_of v)::int) = number_of (bin_minus v)"
1.59 +  by simp
1.61 +  by simp
1.62 +
1.64 +  by simp
1.65 +
1.66 +lemma mult_left_one: "1 * a = (a::'a::semiring_1)"
1.67 +  by simp
1.68 +
1.69 +lemma mult_right_one: "a * 1 = (a::'a::semiring_1)"
1.70 +  by simp
1.71 +
1.72 +lemma int_pow_0: "(a::int)^(Numeral0) = 1"
1.73 +  by simp
1.74 +
1.75 +lemma int_pow_1: "(a::int)^(Numeral1) = a"
1.76 +  by simp
1.77 +
1.78 +lemma zero_eq_Numeral0_nring: "(0::'a::number_ring) = Numeral0"
1.79 +  by simp
1.80 +
1.81 +lemma one_eq_Numeral1_nring: "(1::'a::number_ring) = Numeral1"
1.82 +  by simp
1.83 +
1.84 +lemma zero_eq_Numeral0_nat: "(0::nat) = Numeral0"
1.85 +  by simp
1.86 +
1.87 +lemma one_eq_Numeral1_nat: "(1::nat) = Numeral1"
1.88 +  by simp
1.89 +
1.90 +lemma zpower_Pls: "(z::int)^Numeral0 = Numeral1"
1.91 +  by simp
1.92 +
1.93 +lemma zpower_Min: "(z::int)^((-1)::nat) = Numeral1"
1.94 +proof -
1.95 +  have 1:"((-1)::nat) = 0"
1.96 +    by simp
1.97 +  show ?thesis by (simp add: 1)
1.98 +qed
1.99 +
1.100  use "cooper_dec.ML"
1.101  use "reflected_presburger.ML"
1.102  use "reflected_cooper.ML"
```