--- a/src/HOL/Presburger.thy Wed Apr 02 15:58:26 2008 +0200
+++ b/src/HOL/Presburger.thy Wed Apr 02 15:58:27 2008 +0200
@@ -489,286 +489,4 @@
then show ?thesis by simp
qed
-
-subsection {* Code generator setup *}
-
-text {*
- Presburger arithmetic is convenient to prove some
- of the following code lemmas on integer numerals:
-*}
-
-lemma eq_Pls_Pls:
- "Int.Pls = Int.Pls \<longleftrightarrow> True" by presburger
-
-lemma eq_Pls_Min:
- "Int.Pls = Int.Min \<longleftrightarrow> False"
- unfolding Pls_def Int.Min_def by presburger
-
-lemma eq_Pls_Bit0:
- "Int.Pls = Int.Bit0 k \<longleftrightarrow> Int.Pls = k"
- unfolding Pls_def Bit0_def by presburger
-
-lemma eq_Pls_Bit1:
- "Int.Pls = Int.Bit1 k \<longleftrightarrow> False"
- unfolding Pls_def Bit1_def by presburger
-
-lemma eq_Min_Pls:
- "Int.Min = Int.Pls \<longleftrightarrow> False"
- unfolding Pls_def Int.Min_def by presburger
-
-lemma eq_Min_Min:
- "Int.Min = Int.Min \<longleftrightarrow> True" by presburger
-
-lemma eq_Min_Bit0:
- "Int.Min = Int.Bit0 k \<longleftrightarrow> False"
- unfolding Int.Min_def Bit0_def by presburger
-
-lemma eq_Min_Bit1:
- "Int.Min = Int.Bit1 k \<longleftrightarrow> Int.Min = k"
- unfolding Int.Min_def Bit1_def by presburger
-
-lemma eq_Bit0_Pls:
- "Int.Bit0 k = Int.Pls \<longleftrightarrow> Int.Pls = k"
- unfolding Pls_def Bit0_def by presburger
-
-lemma eq_Bit1_Pls:
- "Int.Bit1 k = Int.Pls \<longleftrightarrow> False"
- unfolding Pls_def Bit1_def by presburger
-
-lemma eq_Bit0_Min:
- "Int.Bit0 k = Int.Min \<longleftrightarrow> False"
- unfolding Int.Min_def Bit0_def by presburger
-
-lemma eq_Bit1_Min:
- "Int.Bit1 k = Int.Min \<longleftrightarrow> Int.Min = k"
- unfolding Int.Min_def Bit1_def by presburger
-
-lemma eq_Bit0_Bit0:
- "Int.Bit0 k1 = Int.Bit0 k2 \<longleftrightarrow> k1 = k2"
- unfolding Bit0_def by presburger
-
-lemma eq_Bit0_Bit1:
- "Int.Bit0 k1 = Int.Bit1 k2 \<longleftrightarrow> False"
- unfolding Bit0_def Bit1_def by presburger
-
-lemma eq_Bit1_Bit0:
- "Int.Bit1 k1 = Int.Bit0 k2 \<longleftrightarrow> False"
- unfolding Bit0_def Bit1_def by presburger
-
-lemma eq_Bit1_Bit1:
- "Int.Bit1 k1 = Int.Bit1 k2 \<longleftrightarrow> k1 = k2"
- unfolding Bit1_def by presburger
-
-lemma eq_number_of:
- "(number_of k \<Colon> int) = number_of l \<longleftrightarrow> k = l"
- unfolding number_of_is_id ..
-
-
-lemma less_eq_Pls_Pls:
- "Int.Pls \<le> Int.Pls \<longleftrightarrow> True" by rule+
-
-lemma less_eq_Pls_Min:
- "Int.Pls \<le> Int.Min \<longleftrightarrow> False"
- unfolding Pls_def Int.Min_def by presburger
-
-lemma less_eq_Pls_Bit0:
- "Int.Pls \<le> Int.Bit0 k \<longleftrightarrow> Int.Pls \<le> k"
- unfolding Pls_def Bit0_def by auto
-
-lemma less_eq_Pls_Bit1:
- "Int.Pls \<le> Int.Bit1 k \<longleftrightarrow> Int.Pls \<le> k"
- unfolding Pls_def Bit1_def by auto
-
-lemma less_eq_Min_Pls:
- "Int.Min \<le> Int.Pls \<longleftrightarrow> True"
- unfolding Pls_def Int.Min_def by presburger
-
-lemma less_eq_Min_Min:
- "Int.Min \<le> Int.Min \<longleftrightarrow> True" by rule+
-
-lemma less_eq_Min_Bit0:
- "Int.Min \<le> Int.Bit0 k \<longleftrightarrow> Int.Min < k"
- unfolding Int.Min_def Bit0_def by auto
-
-lemma less_eq_Min_Bit1:
- "Int.Min \<le> Int.Bit1 k \<longleftrightarrow> Int.Min \<le> k"
- unfolding Int.Min_def Bit1_def by auto
-
-lemma less_eq_Bit0_Pls:
- "Int.Bit0 k \<le> Int.Pls \<longleftrightarrow> k \<le> Int.Pls"
- unfolding Pls_def Bit0_def by simp
-
-lemma less_eq_Bit1_Pls:
- "Int.Bit1 k \<le> Int.Pls \<longleftrightarrow> k < Int.Pls"
- unfolding Pls_def Bit1_def by auto
-
-lemma less_eq_Bit0_Min:
- "Int.Bit0 k \<le> Int.Min \<longleftrightarrow> k \<le> Int.Min"
- unfolding Int.Min_def Bit0_def by auto
-
-lemma less_eq_Bit1_Min:
- "Int.Bit1 k \<le> Int.Min \<longleftrightarrow> k \<le> Int.Min"
- unfolding Int.Min_def Bit1_def by auto
-
-lemma less_eq_Bit0_Bit0:
- "Int.Bit0 k1 \<le> Int.Bit0 k2 \<longleftrightarrow> k1 \<le> k2"
- unfolding Bit0_def by auto
-
-lemma less_eq_Bit0_Bit1:
- "Int.Bit0 k1 \<le> Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"
- unfolding Bit0_def Bit1_def by auto
-
-lemma less_eq_Bit1_Bit0:
- "Int.Bit1 k1 \<le> Int.Bit0 k2 \<longleftrightarrow> k1 < k2"
- unfolding Bit0_def Bit1_def by auto
-
-lemma less_eq_Bit1_Bit1:
- "Int.Bit1 k1 \<le> Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"
- unfolding Bit1_def by auto
-
-lemma less_eq_number_of:
- "(number_of k \<Colon> int) \<le> number_of l \<longleftrightarrow> k \<le> l"
- unfolding number_of_is_id ..
-
-
-lemma less_Pls_Pls:
- "Int.Pls < Int.Pls \<longleftrightarrow> False" by simp
-
-lemma less_Pls_Min:
- "Int.Pls < Int.Min \<longleftrightarrow> False"
- unfolding Pls_def Int.Min_def by presburger
-
-lemma less_Pls_Bit0:
- "Int.Pls < Int.Bit0 k \<longleftrightarrow> Int.Pls < k"
- unfolding Pls_def Bit0_def by auto
-
-lemma less_Pls_Bit1:
- "Int.Pls < Int.Bit1 k \<longleftrightarrow> Int.Pls \<le> k"
- unfolding Pls_def Bit1_def by auto
-
-lemma less_Min_Pls:
- "Int.Min < Int.Pls \<longleftrightarrow> True"
- unfolding Pls_def Int.Min_def by presburger
-
-lemma less_Min_Min:
- "Int.Min < Int.Min \<longleftrightarrow> False" by simp
-
-lemma less_Min_Bit0:
- "Int.Min < Int.Bit0 k \<longleftrightarrow> Int.Min < k"
- unfolding Int.Min_def Bit0_def by auto
-
-lemma less_Min_Bit1:
- "Int.Min < Int.Bit1 k \<longleftrightarrow> Int.Min < k"
- unfolding Int.Min_def Bit1_def by auto
-
-lemma less_Bit0_Pls:
- "Int.Bit0 k < Int.Pls \<longleftrightarrow> k < Int.Pls"
- unfolding Pls_def Bit0_def by auto
-
-lemma less_Bit1_Pls:
- "Int.Bit1 k < Int.Pls \<longleftrightarrow> k < Int.Pls"
- unfolding Pls_def Bit1_def by auto
-
-lemma less_Bit0_Min:
- "Int.Bit0 k < Int.Min \<longleftrightarrow> k \<le> Int.Min"
- unfolding Int.Min_def Bit0_def by auto
-
-lemma less_Bit1_Min:
- "Int.Bit1 k < Int.Min \<longleftrightarrow> k < Int.Min"
- unfolding Int.Min_def Bit1_def by auto
-
-lemma less_Bit0_Bit0:
- "Int.Bit0 k1 < Int.Bit0 k2 \<longleftrightarrow> k1 < k2"
- unfolding Bit0_def by auto
-
-lemma less_Bit0_Bit1:
- "Int.Bit0 k1 < Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"
- unfolding Bit0_def Bit1_def by auto
-
-lemma less_Bit1_Bit0:
- "Int.Bit1 k1 < Int.Bit0 k2 \<longleftrightarrow> k1 < k2"
- unfolding Bit0_def Bit1_def by auto
-
-lemma less_Bit1_Bit1:
- "Int.Bit1 k1 < Int.Bit1 k2 \<longleftrightarrow> k1 < k2"
- unfolding Bit1_def by auto
-
-lemma less_number_of:
- "(number_of k \<Colon> int) < number_of l \<longleftrightarrow> k < l"
- unfolding number_of_is_id ..
-
-lemmas pred_succ_numeral_code [code func] =
- pred_bin_simps succ_bin_simps
-
-lemmas plus_numeral_code [code func] =
- add_bin_simps
- arith_extra_simps(1) [where 'a = int]
-
-lemmas minus_numeral_code [code func] =
- minus_bin_simps
- arith_extra_simps(2) [where 'a = int]
- arith_extra_simps(5) [where 'a = int]
-
-lemmas times_numeral_code [code func] =
- mult_bin_simps
- arith_extra_simps(4) [where 'a = int]
-
-lemmas eq_numeral_code [code func] =
- eq_Pls_Pls eq_Pls_Min eq_Pls_Bit0 eq_Pls_Bit1
- eq_Min_Pls eq_Min_Min eq_Min_Bit0 eq_Min_Bit1
- eq_Bit0_Pls eq_Bit1_Pls eq_Bit0_Min eq_Bit1_Min
- eq_Bit0_Bit0 eq_Bit0_Bit1 eq_Bit1_Bit0 eq_Bit1_Bit1
- eq_number_of
-
-lemmas less_eq_numeral_code [code func] =
- less_eq_Pls_Pls less_eq_Pls_Min less_eq_Pls_Bit0 less_eq_Pls_Bit1
- less_eq_Min_Pls less_eq_Min_Min less_eq_Min_Bit0 less_eq_Min_Bit1
- less_eq_Bit0_Pls less_eq_Bit1_Pls less_eq_Bit0_Min less_eq_Bit1_Min
- less_eq_Bit0_Bit0 less_eq_Bit0_Bit1 less_eq_Bit1_Bit0 less_eq_Bit1_Bit1
- less_eq_number_of
-
-lemmas less_numeral_code [code func] =
- less_Pls_Pls less_Pls_Min less_Pls_Bit0 less_Pls_Bit1
- less_Min_Pls less_Min_Min less_Min_Bit0 less_Min_Bit1
- less_Bit0_Pls less_Bit1_Pls less_Bit0_Min less_Bit1_Min
- less_Bit0_Bit0 less_Bit0_Bit1 less_Bit1_Bit0 less_Bit1_Bit1
- less_number_of
-
-context ring_1
-begin
-
-lemma of_int_num [code func]:
- "of_int k = (if k = 0 then 0 else if k < 0 then
- - of_int (- k) else let
- (l, m) = divAlg (k, 2);
- l' = of_int l
- in if m = 0 then l' + l' else l' + l' + 1)"
-proof -
- have aux1: "k mod (2\<Colon>int) \<noteq> (0\<Colon>int) \<Longrightarrow>
- of_int k = of_int (k div 2 * 2 + 1)"
- proof -
- assume "k mod 2 \<noteq> 0"
- then have "k mod 2 = 1" by arith
- moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp
- ultimately show ?thesis by auto
- qed
- have aux2: "\<And>x. of_int 2 * x = x + x"
- proof -
- fix x
- have int2: "(2::int) = 1 + 1" by arith
- show "of_int 2 * x = x + x"
- unfolding int2 of_int_add left_distrib by simp
- qed
- have aux3: "\<And>x. x * of_int 2 = x + x"
- proof -
- fix x
- have int2: "(2::int) = 1 + 1" by arith
- show "x * of_int 2 = x + x"
- unfolding int2 of_int_add right_distrib by simp
- qed
- from aux1 show ?thesis by (auto simp add: divAlg_mod_div Let_def aux2 aux3)
-qed
-
end
-
-end