--- a/src/HOL/Presburger.thy Tue Jan 15 16:19:21 2008 +0100
+++ b/src/HOL/Presburger.thy Tue Jan 15 16:19:23 2008 +0100
@@ -496,53 +496,53 @@
*}
lemma eq_Pls_Pls:
- "Numeral.Pls = Numeral.Pls \<longleftrightarrow> True" by presburger
+ "Int.Pls = Int.Pls \<longleftrightarrow> True" by presburger
lemma eq_Pls_Min:
- "Numeral.Pls = Numeral.Min \<longleftrightarrow> False"
- unfolding Pls_def Numeral.Min_def by presburger
+ "Int.Pls = Int.Min \<longleftrightarrow> False"
+ unfolding Pls_def Int.Min_def by presburger
lemma eq_Pls_Bit0:
- "Numeral.Pls = Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls = k"
+ "Int.Pls = Int.Bit k bit.B0 \<longleftrightarrow> Int.Pls = k"
unfolding Pls_def Bit_def bit.cases by presburger
lemma eq_Pls_Bit1:
- "Numeral.Pls = Numeral.Bit k bit.B1 \<longleftrightarrow> False"
+ "Int.Pls = Int.Bit k bit.B1 \<longleftrightarrow> False"
unfolding Pls_def Bit_def bit.cases by presburger
lemma eq_Min_Pls:
- "Numeral.Min = Numeral.Pls \<longleftrightarrow> False"
- unfolding Pls_def Numeral.Min_def by presburger
+ "Int.Min = Int.Pls \<longleftrightarrow> False"
+ unfolding Pls_def Int.Min_def by presburger
lemma eq_Min_Min:
- "Numeral.Min = Numeral.Min \<longleftrightarrow> True" by presburger
+ "Int.Min = Int.Min \<longleftrightarrow> True" by presburger
lemma eq_Min_Bit0:
- "Numeral.Min = Numeral.Bit k bit.B0 \<longleftrightarrow> False"
- unfolding Numeral.Min_def Bit_def bit.cases by presburger
+ "Int.Min = Int.Bit k bit.B0 \<longleftrightarrow> False"
+ unfolding Int.Min_def Bit_def bit.cases by presburger
lemma eq_Min_Bit1:
- "Numeral.Min = Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min = k"
- unfolding Numeral.Min_def Bit_def bit.cases by presburger
+ "Int.Min = Int.Bit k bit.B1 \<longleftrightarrow> Int.Min = k"
+ unfolding Int.Min_def Bit_def bit.cases by presburger
lemma eq_Bit0_Pls:
- "Numeral.Bit k bit.B0 = Numeral.Pls \<longleftrightarrow> Numeral.Pls = k"
+ "Int.Bit k bit.B0 = Int.Pls \<longleftrightarrow> Int.Pls = k"
unfolding Pls_def Bit_def bit.cases by presburger
lemma eq_Bit1_Pls:
- "Numeral.Bit k bit.B1 = Numeral.Pls \<longleftrightarrow> False"
+ "Int.Bit k bit.B1 = Int.Pls \<longleftrightarrow> False"
unfolding Pls_def Bit_def bit.cases by presburger
lemma eq_Bit0_Min:
- "Numeral.Bit k bit.B0 = Numeral.Min \<longleftrightarrow> False"
- unfolding Numeral.Min_def Bit_def bit.cases by presburger
+ "Int.Bit k bit.B0 = Int.Min \<longleftrightarrow> False"
+ unfolding Int.Min_def Bit_def bit.cases by presburger
lemma eq_Bit1_Min:
- "(Numeral.Bit k bit.B1) = Numeral.Min \<longleftrightarrow> Numeral.Min = k"
- unfolding Numeral.Min_def Bit_def bit.cases by presburger
+ "(Int.Bit k bit.B1) = Int.Min \<longleftrightarrow> Int.Min = k"
+ unfolding Int.Min_def Bit_def bit.cases by presburger
lemma eq_Bit_Bit:
- "Numeral.Bit k1 v1 = Numeral.Bit k2 v2 \<longleftrightarrow>
+ "Int.Bit k1 v1 = Int.Bit k2 v2 \<longleftrightarrow>
v1 = v2 \<and> k1 = k2"
unfolding Bit_def
apply (cases v1)
@@ -562,53 +562,53 @@
lemma less_eq_Pls_Pls:
- "Numeral.Pls \<le> Numeral.Pls \<longleftrightarrow> True" by rule+
+ "Int.Pls \<le> Int.Pls \<longleftrightarrow> True" by rule+
lemma less_eq_Pls_Min:
- "Numeral.Pls \<le> Numeral.Min \<longleftrightarrow> False"
- unfolding Pls_def Numeral.Min_def by presburger
+ "Int.Pls \<le> Int.Min \<longleftrightarrow> False"
+ unfolding Pls_def Int.Min_def by presburger
lemma less_eq_Pls_Bit:
- "Numeral.Pls \<le> Numeral.Bit k v \<longleftrightarrow> Numeral.Pls \<le> k"
+ "Int.Pls \<le> Int.Bit k v \<longleftrightarrow> Int.Pls \<le> k"
unfolding Pls_def Bit_def by (cases v) auto
lemma less_eq_Min_Pls:
- "Numeral.Min \<le> Numeral.Pls \<longleftrightarrow> True"
- unfolding Pls_def Numeral.Min_def by presburger
+ "Int.Min \<le> Int.Pls \<longleftrightarrow> True"
+ unfolding Pls_def Int.Min_def by presburger
lemma less_eq_Min_Min:
- "Numeral.Min \<le> Numeral.Min \<longleftrightarrow> True" by rule+
+ "Int.Min \<le> Int.Min \<longleftrightarrow> True" by rule+
lemma less_eq_Min_Bit0:
- "Numeral.Min \<le> Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Min < k"
- unfolding Numeral.Min_def Bit_def by auto
+ "Int.Min \<le> Int.Bit k bit.B0 \<longleftrightarrow> Int.Min < k"
+ unfolding Int.Min_def Bit_def by auto
lemma less_eq_Min_Bit1:
- "Numeral.Min \<le> Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min \<le> k"
- unfolding Numeral.Min_def Bit_def by auto
+ "Int.Min \<le> Int.Bit k bit.B1 \<longleftrightarrow> Int.Min \<le> k"
+ unfolding Int.Min_def Bit_def by auto
lemma less_eq_Bit0_Pls:
- "Numeral.Bit k bit.B0 \<le> Numeral.Pls \<longleftrightarrow> k \<le> Numeral.Pls"
+ "Int.Bit k bit.B0 \<le> Int.Pls \<longleftrightarrow> k \<le> Int.Pls"
unfolding Pls_def Bit_def by simp
lemma less_eq_Bit1_Pls:
- "Numeral.Bit k bit.B1 \<le> Numeral.Pls \<longleftrightarrow> k < Numeral.Pls"
+ "Int.Bit k bit.B1 \<le> Int.Pls \<longleftrightarrow> k < Int.Pls"
unfolding Pls_def Bit_def by auto
lemma less_eq_Bit_Min:
- "Numeral.Bit k v \<le> Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min"
- unfolding Numeral.Min_def Bit_def by (cases v) auto
+ "Int.Bit k v \<le> Int.Min \<longleftrightarrow> k \<le> Int.Min"
+ unfolding Int.Min_def Bit_def by (cases v) auto
lemma less_eq_Bit0_Bit:
- "Numeral.Bit k1 bit.B0 \<le> Numeral.Bit k2 v \<longleftrightarrow> k1 \<le> k2"
+ "Int.Bit k1 bit.B0 \<le> Int.Bit k2 v \<longleftrightarrow> k1 \<le> k2"
unfolding Bit_def bit.cases by (cases v) auto
lemma less_eq_Bit_Bit1:
- "Numeral.Bit k1 v \<le> Numeral.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2"
+ "Int.Bit k1 v \<le> Int.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2"
unfolding Bit_def bit.cases by (cases v) auto
lemma less_eq_Bit1_Bit0:
- "Numeral.Bit k1 bit.B1 \<le> Numeral.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2"
+ "Int.Bit k1 bit.B1 \<le> Int.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2"
unfolding Bit_def by (auto split: bit.split)
lemma less_eq_number_of:
@@ -617,53 +617,53 @@
lemma less_Pls_Pls:
- "Numeral.Pls < Numeral.Pls \<longleftrightarrow> False" by simp
+ "Int.Pls < Int.Pls \<longleftrightarrow> False" by simp
lemma less_Pls_Min:
- "Numeral.Pls < Numeral.Min \<longleftrightarrow> False"
- unfolding Pls_def Numeral.Min_def by presburger
+ "Int.Pls < Int.Min \<longleftrightarrow> False"
+ unfolding Pls_def Int.Min_def by presburger
lemma less_Pls_Bit0:
- "Numeral.Pls < Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls < k"
+ "Int.Pls < Int.Bit k bit.B0 \<longleftrightarrow> Int.Pls < k"
unfolding Pls_def Bit_def by auto
lemma less_Pls_Bit1:
- "Numeral.Pls < Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Pls \<le> k"
+ "Int.Pls < Int.Bit k bit.B1 \<longleftrightarrow> Int.Pls \<le> k"
unfolding Pls_def Bit_def by auto
lemma less_Min_Pls:
- "Numeral.Min < Numeral.Pls \<longleftrightarrow> True"
- unfolding Pls_def Numeral.Min_def by presburger
+ "Int.Min < Int.Pls \<longleftrightarrow> True"
+ unfolding Pls_def Int.Min_def by presburger
lemma less_Min_Min:
- "Numeral.Min < Numeral.Min \<longleftrightarrow> False" by simp
+ "Int.Min < Int.Min \<longleftrightarrow> False" by simp
lemma less_Min_Bit:
- "Numeral.Min < Numeral.Bit k v \<longleftrightarrow> Numeral.Min < k"
- unfolding Numeral.Min_def Bit_def by (auto split: bit.split)
+ "Int.Min < Int.Bit k v \<longleftrightarrow> Int.Min < k"
+ unfolding Int.Min_def Bit_def by (auto split: bit.split)
lemma less_Bit_Pls:
- "Numeral.Bit k v < Numeral.Pls \<longleftrightarrow> k < Numeral.Pls"
+ "Int.Bit k v < Int.Pls \<longleftrightarrow> k < Int.Pls"
unfolding Pls_def Bit_def by (auto split: bit.split)
lemma less_Bit0_Min:
- "Numeral.Bit k bit.B0 < Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min"
- unfolding Numeral.Min_def Bit_def by auto
+ "Int.Bit k bit.B0 < Int.Min \<longleftrightarrow> k \<le> Int.Min"
+ unfolding Int.Min_def Bit_def by auto
lemma less_Bit1_Min:
- "Numeral.Bit k bit.B1 < Numeral.Min \<longleftrightarrow> k < Numeral.Min"
- unfolding Numeral.Min_def Bit_def by auto
+ "Int.Bit k bit.B1 < Int.Min \<longleftrightarrow> k < Int.Min"
+ unfolding Int.Min_def Bit_def by auto
lemma less_Bit_Bit0:
- "Numeral.Bit k1 v < Numeral.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2"
+ "Int.Bit k1 v < Int.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2"
unfolding Bit_def by (auto split: bit.split)
lemma less_Bit1_Bit:
- "Numeral.Bit k1 bit.B1 < Numeral.Bit k2 v \<longleftrightarrow> k1 < k2"
+ "Int.Bit k1 bit.B1 < Int.Bit k2 v \<longleftrightarrow> k1 < k2"
unfolding Bit_def by (auto split: bit.split)
lemma less_Bit0_Bit1:
- "Numeral.Bit k1 bit.B0 < Numeral.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2"
+ "Int.Bit k1 bit.B0 < Int.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2"
unfolding Bit_def bit.cases by arith
lemma less_number_of: