--- a/src/HOL/Integ/Presburger.thy Fri Mar 02 15:43:24 2007 +0100
+++ b/src/HOL/Integ/Presburger.thy Fri Mar 02 15:43:25 2007 +0100
@@ -1069,11 +1069,6 @@
lemma eq_number_of [code func]:
"((number_of k)\<Colon>int) = number_of l \<longleftrightarrow> k = l"
unfolding number_of_is_id ..
-
-lemma less_eq_number_of [code func]:
- "((number_of k)\<Colon>int) <= number_of l \<longleftrightarrow> k <= l"
- unfolding number_of_is_id ..
-
lemma eq_Pls_Pls:
"Numeral.Pls = Numeral.Pls" ..
@@ -1140,6 +1135,10 @@
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_Bit_Bit
+lemma less_eq_number_of [code func]:
+ "(number_of k \<Colon> int) \<le> number_of l \<longleftrightarrow> k \<le> l"
+ unfolding number_of_is_id ..
+
lemma less_eq_Pls_Pls:
"Numeral.Pls \<le> Numeral.Pls" ..
@@ -1180,14 +1179,76 @@
lemma less_eq_Bit0_Bit:
"Numeral.Bit k1 bit.B0 \<le> Numeral.Bit k2 v \<longleftrightarrow> k1 \<le> k2"
- unfolding Min_def Bit_def bit.cases by (cases v) auto
+ 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"
- unfolding Min_def Bit_def bit.cases by (cases v) auto
+ 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"
+ unfolding Bit_def by (auto split: bit.split)
lemmas less_eq_numeral_code [code func] = less_eq_Pls_Pls less_eq_Pls_Min less_eq_Pls_Bit
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_Bit_Min less_eq_Bit0_Bit less_eq_Bit_Bit1
+ less_eq_Bit0_Pls less_eq_Bit1_Pls less_eq_Bit_Min less_eq_Bit0_Bit less_eq_Bit_Bit1 less_eq_Bit1_Bit0
+
+lemma less_eq_number_of [code func]:
+ "(number_of k \<Colon> int) < number_of l \<longleftrightarrow> k < l"
+ unfolding number_of_is_id ..
+
+lemma less_Pls_Pls:
+ "\<not> (Numeral.Pls < Numeral.Pls)" by auto
+
+lemma less_Pls_Min:
+ "\<not> (Numeral.Pls < Numeral.Min)"
+ unfolding Pls_def Min_def by auto
+
+lemma less_Pls_Bit0:
+ "Numeral.Pls < Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.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"
+ unfolding Pls_def Bit_def by auto
+
+lemma less_Min_Pls:
+ "Numeral.Min < Numeral.Pls"
+ unfolding Pls_def Min_def by auto
+
+lemma less_Min_Min:
+ "\<not> (Numeral.Min < Numeral.Min)" by auto
+
+lemma less_Min_Bit:
+ "Numeral.Min < Numeral.Bit k v \<longleftrightarrow> Numeral.Min < k"
+ unfolding Min_def Bit_def by (auto split: bit.split)
+
+lemma less_Bit_Pls:
+ "Numeral.Bit k v < Numeral.Pls \<longleftrightarrow> k < Numeral.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 Min_def Bit_def by auto
+
+lemma less_Bit1_Min:
+ "Numeral.Bit k bit.B1 < Numeral.Min \<longleftrightarrow> k < Numeral.Min"
+ unfolding Min_def Bit_def by auto
+
+lemma less_Bit_Bit0:
+ "Numeral.Bit k1 v < Numeral.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"
+ 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"
+ unfolding Bit_def bit.cases by auto
+
+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_Bit less_Bit_Pls
+ less_Bit0_Min less_Bit1_Min less_Bit_Bit0 less_Bit1_Bit less_Bit0_Bit1
end
--- a/src/HOL/Presburger.thy Fri Mar 02 15:43:24 2007 +0100
+++ b/src/HOL/Presburger.thy Fri Mar 02 15:43:25 2007 +0100
@@ -1069,11 +1069,6 @@
lemma eq_number_of [code func]:
"((number_of k)\<Colon>int) = number_of l \<longleftrightarrow> k = l"
unfolding number_of_is_id ..
-
-lemma less_eq_number_of [code func]:
- "((number_of k)\<Colon>int) <= number_of l \<longleftrightarrow> k <= l"
- unfolding number_of_is_id ..
-
lemma eq_Pls_Pls:
"Numeral.Pls = Numeral.Pls" ..
@@ -1140,6 +1135,10 @@
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_Bit_Bit
+lemma less_eq_number_of [code func]:
+ "(number_of k \<Colon> int) \<le> number_of l \<longleftrightarrow> k \<le> l"
+ unfolding number_of_is_id ..
+
lemma less_eq_Pls_Pls:
"Numeral.Pls \<le> Numeral.Pls" ..
@@ -1180,14 +1179,76 @@
lemma less_eq_Bit0_Bit:
"Numeral.Bit k1 bit.B0 \<le> Numeral.Bit k2 v \<longleftrightarrow> k1 \<le> k2"
- unfolding Min_def Bit_def bit.cases by (cases v) auto
+ 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"
- unfolding Min_def Bit_def bit.cases by (cases v) auto
+ 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"
+ unfolding Bit_def by (auto split: bit.split)
lemmas less_eq_numeral_code [code func] = less_eq_Pls_Pls less_eq_Pls_Min less_eq_Pls_Bit
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_Bit_Min less_eq_Bit0_Bit less_eq_Bit_Bit1
+ less_eq_Bit0_Pls less_eq_Bit1_Pls less_eq_Bit_Min less_eq_Bit0_Bit less_eq_Bit_Bit1 less_eq_Bit1_Bit0
+
+lemma less_eq_number_of [code func]:
+ "(number_of k \<Colon> int) < number_of l \<longleftrightarrow> k < l"
+ unfolding number_of_is_id ..
+
+lemma less_Pls_Pls:
+ "\<not> (Numeral.Pls < Numeral.Pls)" by auto
+
+lemma less_Pls_Min:
+ "\<not> (Numeral.Pls < Numeral.Min)"
+ unfolding Pls_def Min_def by auto
+
+lemma less_Pls_Bit0:
+ "Numeral.Pls < Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.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"
+ unfolding Pls_def Bit_def by auto
+
+lemma less_Min_Pls:
+ "Numeral.Min < Numeral.Pls"
+ unfolding Pls_def Min_def by auto
+
+lemma less_Min_Min:
+ "\<not> (Numeral.Min < Numeral.Min)" by auto
+
+lemma less_Min_Bit:
+ "Numeral.Min < Numeral.Bit k v \<longleftrightarrow> Numeral.Min < k"
+ unfolding Min_def Bit_def by (auto split: bit.split)
+
+lemma less_Bit_Pls:
+ "Numeral.Bit k v < Numeral.Pls \<longleftrightarrow> k < Numeral.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 Min_def Bit_def by auto
+
+lemma less_Bit1_Min:
+ "Numeral.Bit k bit.B1 < Numeral.Min \<longleftrightarrow> k < Numeral.Min"
+ unfolding Min_def Bit_def by auto
+
+lemma less_Bit_Bit0:
+ "Numeral.Bit k1 v < Numeral.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"
+ 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"
+ unfolding Bit_def bit.cases by auto
+
+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_Bit less_Bit_Pls
+ less_Bit0_Min less_Bit1_Min less_Bit_Bit0 less_Bit1_Bit less_Bit0_Bit1
end
--- a/src/HOL/ex/NormalForm.thy Fri Mar 02 15:43:24 2007 +0100
+++ b/src/HOL/ex/NormalForm.thy Fri Mar 02 15:43:25 2007 +0100
@@ -104,11 +104,9 @@
lemma "(2::int) <= 3" by normalization
lemma "abs ((-4::int) + 2 * 1) = 2" by normalization
lemma "4 - 42 * abs (3 + (-7\<Colon>int)) = -164" by normalization
-normal_form "min 0 x"
-normal_form "min 0 (x::nat)"
lemma "(if (0\<Colon>nat) \<le> (x\<Colon>nat) then 0\<Colon>nat else x) = 0" by normalization
-
-normal_form "nat 4 = Suc (Suc (Suc (Suc 0)))"
+lemma "4 = Suc (Suc (Suc (Suc 0)))" by normalization
+lemma "nat 4 = Suc (Suc (Suc (Suc 0)))" by normalization
normal_form "Suc 0 \<in> set ms"