--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Lattice_Constructions.thy Tue Aug 19 15:19:16 2014 +0200
@@ -0,0 +1,417 @@
+(* Title: HOL/Library/Lattice_Constructions.thy
+ Author: Lukas Bulwahn
+ Copyright 2010 TU Muenchen
+*)
+
+theory Lattice_Constructions
+imports Main
+begin
+
+subsection {* Values extended by a bottom element *}
+
+datatype 'a bot = Value 'a | Bot
+
+instantiation bot :: (preorder) preorder
+begin
+
+definition less_eq_bot where
+ "x \<le> y \<longleftrightarrow> (case x of Bot \<Rightarrow> True | Value x \<Rightarrow> (case y of Bot \<Rightarrow> False | Value y \<Rightarrow> x \<le> y))"
+
+definition less_bot where
+ "x < y \<longleftrightarrow> (case y of Bot \<Rightarrow> False | Value y \<Rightarrow> (case x of Bot \<Rightarrow> True | Value x \<Rightarrow> x < y))"
+
+lemma less_eq_bot_Bot [simp]: "Bot \<le> x"
+ by (simp add: less_eq_bot_def)
+
+lemma less_eq_bot_Bot_code [code]: "Bot \<le> x \<longleftrightarrow> True"
+ by simp
+
+lemma less_eq_bot_Bot_is_Bot: "x \<le> Bot \<Longrightarrow> x = Bot"
+ by (cases x) (simp_all add: less_eq_bot_def)
+
+lemma less_eq_bot_Value_Bot [simp, code]: "Value x \<le> Bot \<longleftrightarrow> False"
+ by (simp add: less_eq_bot_def)
+
+lemma less_eq_bot_Value [simp, code]: "Value x \<le> Value y \<longleftrightarrow> x \<le> y"
+ by (simp add: less_eq_bot_def)
+
+lemma less_bot_Bot [simp, code]: "x < Bot \<longleftrightarrow> False"
+ by (simp add: less_bot_def)
+
+lemma less_bot_Bot_is_Value: "Bot < x \<Longrightarrow> \<exists>z. x = Value z"
+ by (cases x) (simp_all add: less_bot_def)
+
+lemma less_bot_Bot_Value [simp]: "Bot < Value x"
+ by (simp add: less_bot_def)
+
+lemma less_bot_Bot_Value_code [code]: "Bot < Value x \<longleftrightarrow> True"
+ by simp
+
+lemma less_bot_Value [simp, code]: "Value x < Value y \<longleftrightarrow> x < y"
+ by (simp add: less_bot_def)
+
+instance proof
+qed (auto simp add: less_eq_bot_def less_bot_def less_le_not_le elim: order_trans split: bot.splits)
+
+end
+
+instance bot :: (order) order proof
+qed (auto simp add: less_eq_bot_def less_bot_def split: bot.splits)
+
+instance bot :: (linorder) linorder proof
+qed (auto simp add: less_eq_bot_def less_bot_def split: bot.splits)
+
+instantiation bot :: (order) bot
+begin
+
+definition "bot = Bot"
+
+instance ..
+
+end
+
+instantiation bot :: (top) top
+begin
+
+definition "top = Value top"
+
+instance ..
+
+end
+
+instantiation bot :: (semilattice_inf) semilattice_inf
+begin
+
+definition inf_bot
+where
+ "inf x y = (case x of Bot => Bot | Value v => (case y of Bot => Bot | Value v' => Value (inf v v')))"
+
+instance proof
+qed (auto simp add: inf_bot_def less_eq_bot_def split: bot.splits)
+
+end
+
+instantiation bot :: (semilattice_sup) semilattice_sup
+begin
+
+definition sup_bot
+where
+ "sup x y = (case x of Bot => y | Value v => (case y of Bot => x | Value v' => Value (sup v v')))"
+
+instance proof
+qed (auto simp add: sup_bot_def less_eq_bot_def split: bot.splits)
+
+end
+
+instance bot :: (lattice) bounded_lattice_bot
+by(intro_classes)(simp add: bot_bot_def)
+
+section {* Values extended by a top element *}
+
+datatype 'a top = Value 'a | Top
+
+instantiation top :: (preorder) preorder
+begin
+
+definition less_eq_top where
+ "x \<le> y \<longleftrightarrow> (case y of Top \<Rightarrow> True | Value y \<Rightarrow> (case x of Top \<Rightarrow> False | Value x \<Rightarrow> x \<le> y))"
+
+definition less_top where
+ "x < y \<longleftrightarrow> (case x of Top \<Rightarrow> False | Value x \<Rightarrow> (case y of Top \<Rightarrow> True | Value y \<Rightarrow> x < y))"
+
+lemma less_eq_top_Top [simp]: "x <= Top"
+ by (simp add: less_eq_top_def)
+
+lemma less_eq_top_Top_code [code]: "x \<le> Top \<longleftrightarrow> True"
+ by simp
+
+lemma less_eq_top_is_Top: "Top \<le> x \<Longrightarrow> x = Top"
+ by (cases x) (simp_all add: less_eq_top_def)
+
+lemma less_eq_top_Top_Value [simp, code]: "Top \<le> Value x \<longleftrightarrow> False"
+ by (simp add: less_eq_top_def)
+
+lemma less_eq_top_Value_Value [simp, code]: "Value x \<le> Value y \<longleftrightarrow> x \<le> y"
+ by (simp add: less_eq_top_def)
+
+lemma less_top_Top [simp, code]: "Top < x \<longleftrightarrow> False"
+ by (simp add: less_top_def)
+
+lemma less_top_Top_is_Value: "x < Top \<Longrightarrow> \<exists>z. x = Value z"
+ by (cases x) (simp_all add: less_top_def)
+
+lemma less_top_Value_Top [simp]: "Value x < Top"
+ by (simp add: less_top_def)
+
+lemma less_top_Value_Top_code [code]: "Value x < Top \<longleftrightarrow> True"
+ by simp
+
+lemma less_top_Value [simp, code]: "Value x < Value y \<longleftrightarrow> x < y"
+ by (simp add: less_top_def)
+
+instance proof
+qed (auto simp add: less_eq_top_def less_top_def less_le_not_le elim: order_trans split: top.splits)
+
+end
+
+instance top :: (order) order proof
+qed (auto simp add: less_eq_top_def less_top_def split: top.splits)
+
+instance top :: (linorder) linorder proof
+qed (auto simp add: less_eq_top_def less_top_def split: top.splits)
+
+instantiation top :: (order) top
+begin
+
+definition "top = Top"
+
+instance ..
+
+end
+
+instantiation top :: (bot) bot
+begin
+
+definition "bot = Value bot"
+
+instance ..
+
+end
+
+instantiation top :: (semilattice_inf) semilattice_inf
+begin
+
+definition inf_top
+where
+ "inf x y = (case x of Top => y | Value v => (case y of Top => x | Value v' => Value (inf v v')))"
+
+instance proof
+qed (auto simp add: inf_top_def less_eq_top_def split: top.splits)
+
+end
+
+instantiation top :: (semilattice_sup) semilattice_sup
+begin
+
+definition sup_top
+where
+ "sup x y = (case x of Top => Top | Value v => (case y of Top => Top | Value v' => Value (sup v v')))"
+
+instance proof
+qed (auto simp add: sup_top_def less_eq_top_def split: top.splits)
+
+end
+
+instance top :: (lattice) bounded_lattice_top
+by(intro_classes)(simp add: top_top_def)
+
+subsection {* Values extended by a top and a bottom element *}
+
+datatype 'a flat_complete_lattice = Value 'a | Bot | Top
+
+instantiation flat_complete_lattice :: (type) order
+begin
+
+definition less_eq_flat_complete_lattice where
+ "x \<le> y == (case x of Bot => True | Value v1 => (case y of Bot => False | Value v2 => (v1 = v2) | Top => True) | Top => (y = Top))"
+
+definition less_flat_complete_lattice where
+ "x < y = (case x of Bot => \<not> (y = Bot) | Value v1 => (y = Top) | Top => False)"
+
+lemma [simp]: "Bot <= y"
+unfolding less_eq_flat_complete_lattice_def by auto
+
+lemma [simp]: "y <= Top"
+unfolding less_eq_flat_complete_lattice_def by (auto split: flat_complete_lattice.splits)
+
+lemma greater_than_two_values:
+ assumes "a ~= aa" "Value a <= z" "Value aa <= z"
+ shows "z = Top"
+using assms
+by (cases z) (auto simp add: less_eq_flat_complete_lattice_def)
+
+lemma lesser_than_two_values:
+ assumes "a ~= aa" "z <= Value a" "z <= Value aa"
+ shows "z = Bot"
+using assms
+by (cases z) (auto simp add: less_eq_flat_complete_lattice_def)
+
+instance proof
+qed (auto simp add: less_eq_flat_complete_lattice_def less_flat_complete_lattice_def split: flat_complete_lattice.splits)
+
+end
+
+instantiation flat_complete_lattice :: (type) bot
+begin
+
+definition "bot = Bot"
+
+instance ..
+
+end
+
+instantiation flat_complete_lattice :: (type) top
+begin
+
+definition "top = Top"
+
+instance ..
+
+end
+
+instantiation flat_complete_lattice :: (type) lattice
+begin
+
+definition inf_flat_complete_lattice
+where
+ "inf x y = (case x of Bot => Bot | Value v1 => (case y of Bot => Bot | Value v2 => if (v1 = v2) then x else Bot | Top => x) | Top => y)"
+
+definition sup_flat_complete_lattice
+where
+ "sup x y = (case x of Bot => y | Value v1 => (case y of Bot => x | Value v2 => if v1 = v2 then x else Top | Top => Top) | Top => Top)"
+
+instance proof
+qed (auto simp add: inf_flat_complete_lattice_def sup_flat_complete_lattice_def less_eq_flat_complete_lattice_def split: flat_complete_lattice.splits)
+
+end
+
+instantiation flat_complete_lattice :: (type) complete_lattice
+begin
+
+definition Sup_flat_complete_lattice
+where
+ "Sup A = (if (A = {} \<or> A = {Bot}) then Bot else (if (\<exists> v. A - {Bot} = {Value v}) then Value (THE v. A - {Bot} = {Value v}) else Top))"
+
+definition Inf_flat_complete_lattice
+where
+ "Inf A = (if (A = {} \<or> A = {Top}) then Top else (if (\<exists> v. A - {Top} = {Value v}) then Value (THE v. A - {Top} = {Value v}) else Bot))"
+
+instance
+proof
+ fix x A
+ assume "(x :: 'a flat_complete_lattice) : A"
+ {
+ fix v
+ assume "A - {Top} = {Value v}"
+ from this have "(THE v. A - {Top} = {Value v}) = v"
+ by (auto intro!: the1_equality)
+ moreover
+ from `x : A` `A - {Top} = {Value v}` have "x = Top \<or> x = Value v"
+ by auto
+ ultimately have "Value (THE v. A - {Top} = {Value v}) <= x"
+ by auto
+ }
+ from `x : A` this show "Inf A <= x"
+ unfolding Inf_flat_complete_lattice_def
+ by fastforce
+next
+ fix z A
+ assume z: "\<And>x. x : A ==> z <= (x :: 'a flat_complete_lattice)"
+ {
+ fix v
+ assume "A - {Top} = {Value v}"
+ moreover
+ from this have "(THE v. A - {Top} = {Value v}) = v"
+ by (auto intro!: the1_equality)
+ moreover
+ note z
+ moreover
+ ultimately have "z <= Value (THE v::'a. A - {Top} = {Value v})"
+ by auto
+ } moreover
+ {
+ assume not_one_value: "A ~= {}" "A ~= {Top}" "~ (EX v::'a. A - {Top} = {Value v})"
+ have "z <= Bot"
+ proof (cases "A - {Top} = {Bot}")
+ case True
+ from this z show ?thesis
+ by auto
+ next
+ case False
+ from not_one_value
+ obtain a1 where a1: "a1 : A - {Top}" by auto
+ from not_one_value False a1
+ obtain a2 where "a2 : A - {Top} \<and> a1 \<noteq> a2"
+ by (cases a1) auto
+ from this a1 z[of "a1"] z[of "a2"] show ?thesis
+ apply (cases a1)
+ apply auto
+ apply (cases a2)
+ apply auto
+ apply (auto dest!: lesser_than_two_values)
+ done
+ qed
+ } moreover
+ note z moreover
+ ultimately show "z <= Inf A"
+ unfolding Inf_flat_complete_lattice_def
+ by auto
+next
+ fix x A
+ assume "(x :: 'a flat_complete_lattice) : A"
+ {
+ fix v
+ assume "A - {Bot} = {Value v}"
+ from this have "(THE v. A - {Bot} = {Value v}) = v"
+ by (auto intro!: the1_equality)
+ moreover
+ from `x : A` `A - {Bot} = {Value v}` have "x = Bot \<or> x = Value v"
+ by auto
+ ultimately have "x <= Value (THE v. A - {Bot} = {Value v})"
+ by auto
+ }
+ from `x : A` this show "x <= Sup A"
+ unfolding Sup_flat_complete_lattice_def
+ by fastforce
+next
+ fix z A
+ assume z: "\<And>x. x : A ==> x <= (z :: 'a flat_complete_lattice)"
+ {
+ fix v
+ assume "A - {Bot} = {Value v}"
+ moreover
+ from this have "(THE v. A - {Bot} = {Value v}) = v"
+ by (auto intro!: the1_equality)
+ moreover
+ note z
+ moreover
+ ultimately have "Value (THE v::'a. A - {Bot} = {Value v}) <= z"
+ by auto
+ } moreover
+ {
+ assume not_one_value: "A ~= {}" "A ~= {Bot}" "~ (EX v::'a. A - {Bot} = {Value v})"
+ have "Top <= z"
+ proof (cases "A - {Bot} = {Top}")
+ case True
+ from this z show ?thesis
+ by auto
+ next
+ case False
+ from not_one_value
+ obtain a1 where a1: "a1 : A - {Bot}" by auto
+ from not_one_value False a1
+ obtain a2 where "a2 : A - {Bot} \<and> a1 \<noteq> a2"
+ by (cases a1) auto
+ from this a1 z[of "a1"] z[of "a2"] show ?thesis
+ apply (cases a1)
+ apply auto
+ apply (cases a2)
+ apply (auto dest!: greater_than_two_values)
+ done
+ qed
+ } moreover
+ note z moreover
+ ultimately show "Sup A <= z"
+ unfolding Sup_flat_complete_lattice_def
+ by auto
+next
+ show "Inf {} = (top :: 'a flat_complete_lattice)"
+ by(simp add: Inf_flat_complete_lattice_def top_flat_complete_lattice_def)
+next
+ show "Sup {} = (bot :: 'a flat_complete_lattice)"
+ by(simp add: Sup_flat_complete_lattice_def bot_flat_complete_lattice_def)
+qed
+
+end
+
+end
\ No newline at end of file
--- a/src/HOL/Library/Library.thy Tue Aug 19 14:58:38 2014 +0200
+++ b/src/HOL/Library/Library.thy Tue Aug 19 15:19:16 2014 +0200
@@ -32,6 +32,7 @@
IArray
Lattice_Algebras
Lattice_Syntax
+ Lattice_Constructions
ListVector
Lubs_Glbs
Mapping
--- a/src/HOL/Library/Quickcheck_Types.thy Tue Aug 19 14:58:38 2014 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,430 +0,0 @@
-(* Title: HOL/Library/Quickcheck_Types.thy
- Author: Lukas Bulwahn
- Copyright 2010 TU Muenchen
-*)
-
-theory Quickcheck_Types
-imports Main
-begin
-
-text {*
-This theory provides some default types for the quickcheck execution.
-In most cases, the default type @{typ "int"} meets the sort constraints
-of the proposition.
-But for the type classes bot and top, the type @{typ "int"} is insufficient.
-Hence, we provide other types than @{typ "int"} as further default types.
-*}
-
-subsection {* Values extended by a bottom element *}
-
-datatype 'a bot = Value 'a | Bot
-
-instantiation bot :: (preorder) preorder
-begin
-
-definition less_eq_bot where
- "x \<le> y \<longleftrightarrow> (case x of Bot \<Rightarrow> True | Value x \<Rightarrow> (case y of Bot \<Rightarrow> False | Value y \<Rightarrow> x \<le> y))"
-
-definition less_bot where
- "x < y \<longleftrightarrow> (case y of Bot \<Rightarrow> False | Value y \<Rightarrow> (case x of Bot \<Rightarrow> True | Value x \<Rightarrow> x < y))"
-
-lemma less_eq_bot_Bot [simp]: "Bot \<le> x"
- by (simp add: less_eq_bot_def)
-
-lemma less_eq_bot_Bot_code [code]: "Bot \<le> x \<longleftrightarrow> True"
- by simp
-
-lemma less_eq_bot_Bot_is_Bot: "x \<le> Bot \<Longrightarrow> x = Bot"
- by (cases x) (simp_all add: less_eq_bot_def)
-
-lemma less_eq_bot_Value_Bot [simp, code]: "Value x \<le> Bot \<longleftrightarrow> False"
- by (simp add: less_eq_bot_def)
-
-lemma less_eq_bot_Value [simp, code]: "Value x \<le> Value y \<longleftrightarrow> x \<le> y"
- by (simp add: less_eq_bot_def)
-
-lemma less_bot_Bot [simp, code]: "x < Bot \<longleftrightarrow> False"
- by (simp add: less_bot_def)
-
-lemma less_bot_Bot_is_Value: "Bot < x \<Longrightarrow> \<exists>z. x = Value z"
- by (cases x) (simp_all add: less_bot_def)
-
-lemma less_bot_Bot_Value [simp]: "Bot < Value x"
- by (simp add: less_bot_def)
-
-lemma less_bot_Bot_Value_code [code]: "Bot < Value x \<longleftrightarrow> True"
- by simp
-
-lemma less_bot_Value [simp, code]: "Value x < Value y \<longleftrightarrow> x < y"
- by (simp add: less_bot_def)
-
-instance proof
-qed (auto simp add: less_eq_bot_def less_bot_def less_le_not_le elim: order_trans split: bot.splits)
-
-end
-
-instance bot :: (order) order proof
-qed (auto simp add: less_eq_bot_def less_bot_def split: bot.splits)
-
-instance bot :: (linorder) linorder proof
-qed (auto simp add: less_eq_bot_def less_bot_def split: bot.splits)
-
-instantiation bot :: (order) bot
-begin
-
-definition "bot = Bot"
-
-instance ..
-
-end
-
-instantiation bot :: (top) top
-begin
-
-definition "top = Value top"
-
-instance ..
-
-end
-
-instantiation bot :: (semilattice_inf) semilattice_inf
-begin
-
-definition inf_bot
-where
- "inf x y = (case x of Bot => Bot | Value v => (case y of Bot => Bot | Value v' => Value (inf v v')))"
-
-instance proof
-qed (auto simp add: inf_bot_def less_eq_bot_def split: bot.splits)
-
-end
-
-instantiation bot :: (semilattice_sup) semilattice_sup
-begin
-
-definition sup_bot
-where
- "sup x y = (case x of Bot => y | Value v => (case y of Bot => x | Value v' => Value (sup v v')))"
-
-instance proof
-qed (auto simp add: sup_bot_def less_eq_bot_def split: bot.splits)
-
-end
-
-instance bot :: (lattice) bounded_lattice_bot
-by(intro_classes)(simp add: bot_bot_def)
-
-section {* Values extended by a top element *}
-
-datatype 'a top = Value 'a | Top
-
-instantiation top :: (preorder) preorder
-begin
-
-definition less_eq_top where
- "x \<le> y \<longleftrightarrow> (case y of Top \<Rightarrow> True | Value y \<Rightarrow> (case x of Top \<Rightarrow> False | Value x \<Rightarrow> x \<le> y))"
-
-definition less_top where
- "x < y \<longleftrightarrow> (case x of Top \<Rightarrow> False | Value x \<Rightarrow> (case y of Top \<Rightarrow> True | Value y \<Rightarrow> x < y))"
-
-lemma less_eq_top_Top [simp]: "x <= Top"
- by (simp add: less_eq_top_def)
-
-lemma less_eq_top_Top_code [code]: "x \<le> Top \<longleftrightarrow> True"
- by simp
-
-lemma less_eq_top_is_Top: "Top \<le> x \<Longrightarrow> x = Top"
- by (cases x) (simp_all add: less_eq_top_def)
-
-lemma less_eq_top_Top_Value [simp, code]: "Top \<le> Value x \<longleftrightarrow> False"
- by (simp add: less_eq_top_def)
-
-lemma less_eq_top_Value_Value [simp, code]: "Value x \<le> Value y \<longleftrightarrow> x \<le> y"
- by (simp add: less_eq_top_def)
-
-lemma less_top_Top [simp, code]: "Top < x \<longleftrightarrow> False"
- by (simp add: less_top_def)
-
-lemma less_top_Top_is_Value: "x < Top \<Longrightarrow> \<exists>z. x = Value z"
- by (cases x) (simp_all add: less_top_def)
-
-lemma less_top_Value_Top [simp]: "Value x < Top"
- by (simp add: less_top_def)
-
-lemma less_top_Value_Top_code [code]: "Value x < Top \<longleftrightarrow> True"
- by simp
-
-lemma less_top_Value [simp, code]: "Value x < Value y \<longleftrightarrow> x < y"
- by (simp add: less_top_def)
-
-instance proof
-qed (auto simp add: less_eq_top_def less_top_def less_le_not_le elim: order_trans split: top.splits)
-
-end
-
-instance top :: (order) order proof
-qed (auto simp add: less_eq_top_def less_top_def split: top.splits)
-
-instance top :: (linorder) linorder proof
-qed (auto simp add: less_eq_top_def less_top_def split: top.splits)
-
-instantiation top :: (order) top
-begin
-
-definition "top = Top"
-
-instance ..
-
-end
-
-instantiation top :: (bot) bot
-begin
-
-definition "bot = Value bot"
-
-instance ..
-
-end
-
-instantiation top :: (semilattice_inf) semilattice_inf
-begin
-
-definition inf_top
-where
- "inf x y = (case x of Top => y | Value v => (case y of Top => x | Value v' => Value (inf v v')))"
-
-instance proof
-qed (auto simp add: inf_top_def less_eq_top_def split: top.splits)
-
-end
-
-instantiation top :: (semilattice_sup) semilattice_sup
-begin
-
-definition sup_top
-where
- "sup x y = (case x of Top => Top | Value v => (case y of Top => Top | Value v' => Value (sup v v')))"
-
-instance proof
-qed (auto simp add: sup_top_def less_eq_top_def split: top.splits)
-
-end
-
-instance top :: (lattice) bounded_lattice_top
-by(intro_classes)(simp add: top_top_def)
-
-
-datatype 'a flat_complete_lattice = Value 'a | Bot | Top
-
-instantiation flat_complete_lattice :: (type) order
-begin
-
-definition less_eq_flat_complete_lattice where
- "x \<le> y == (case x of Bot => True | Value v1 => (case y of Bot => False | Value v2 => (v1 = v2) | Top => True) | Top => (y = Top))"
-
-definition less_flat_complete_lattice where
- "x < y = (case x of Bot => \<not> (y = Bot) | Value v1 => (y = Top) | Top => False)"
-
-lemma [simp]: "Bot <= y"
-unfolding less_eq_flat_complete_lattice_def by auto
-
-lemma [simp]: "y <= Top"
-unfolding less_eq_flat_complete_lattice_def by (auto split: flat_complete_lattice.splits)
-
-lemma greater_than_two_values:
- assumes "a ~= aa" "Value a <= z" "Value aa <= z"
- shows "z = Top"
-using assms
-by (cases z) (auto simp add: less_eq_flat_complete_lattice_def)
-
-lemma lesser_than_two_values:
- assumes "a ~= aa" "z <= Value a" "z <= Value aa"
- shows "z = Bot"
-using assms
-by (cases z) (auto simp add: less_eq_flat_complete_lattice_def)
-
-instance proof
-qed (auto simp add: less_eq_flat_complete_lattice_def less_flat_complete_lattice_def split: flat_complete_lattice.splits)
-
-end
-
-instantiation flat_complete_lattice :: (type) bot
-begin
-
-definition "bot = Bot"
-
-instance ..
-
-end
-
-instantiation flat_complete_lattice :: (type) top
-begin
-
-definition "top = Top"
-
-instance ..
-
-end
-
-instantiation flat_complete_lattice :: (type) lattice
-begin
-
-definition inf_flat_complete_lattice
-where
- "inf x y = (case x of Bot => Bot | Value v1 => (case y of Bot => Bot | Value v2 => if (v1 = v2) then x else Bot | Top => x) | Top => y)"
-
-definition sup_flat_complete_lattice
-where
- "sup x y = (case x of Bot => y | Value v1 => (case y of Bot => x | Value v2 => if v1 = v2 then x else Top | Top => Top) | Top => Top)"
-
-instance proof
-qed (auto simp add: inf_flat_complete_lattice_def sup_flat_complete_lattice_def less_eq_flat_complete_lattice_def split: flat_complete_lattice.splits)
-
-end
-
-instantiation flat_complete_lattice :: (type) complete_lattice
-begin
-
-definition Sup_flat_complete_lattice
-where
- "Sup A = (if (A = {} \<or> A = {Bot}) then Bot else (if (\<exists> v. A - {Bot} = {Value v}) then Value (THE v. A - {Bot} = {Value v}) else Top))"
-
-definition Inf_flat_complete_lattice
-where
- "Inf A = (if (A = {} \<or> A = {Top}) then Top else (if (\<exists> v. A - {Top} = {Value v}) then Value (THE v. A - {Top} = {Value v}) else Bot))"
-
-instance
-proof
- fix x A
- assume "(x :: 'a flat_complete_lattice) : A"
- {
- fix v
- assume "A - {Top} = {Value v}"
- from this have "(THE v. A - {Top} = {Value v}) = v"
- by (auto intro!: the1_equality)
- moreover
- from `x : A` `A - {Top} = {Value v}` have "x = Top \<or> x = Value v"
- by auto
- ultimately have "Value (THE v. A - {Top} = {Value v}) <= x"
- by auto
- }
- from `x : A` this show "Inf A <= x"
- unfolding Inf_flat_complete_lattice_def
- by fastforce
-next
- fix z A
- assume z: "\<And>x. x : A ==> z <= (x :: 'a flat_complete_lattice)"
- {
- fix v
- assume "A - {Top} = {Value v}"
- moreover
- from this have "(THE v. A - {Top} = {Value v}) = v"
- by (auto intro!: the1_equality)
- moreover
- note z
- moreover
- ultimately have "z <= Value (THE v::'a. A - {Top} = {Value v})"
- by auto
- } moreover
- {
- assume not_one_value: "A ~= {}" "A ~= {Top}" "~ (EX v::'a. A - {Top} = {Value v})"
- have "z <= Bot"
- proof (cases "A - {Top} = {Bot}")
- case True
- from this z show ?thesis
- by auto
- next
- case False
- from not_one_value
- obtain a1 where a1: "a1 : A - {Top}" by auto
- from not_one_value False a1
- obtain a2 where "a2 : A - {Top} \<and> a1 \<noteq> a2"
- by (cases a1) auto
- from this a1 z[of "a1"] z[of "a2"] show ?thesis
- apply (cases a1)
- apply auto
- apply (cases a2)
- apply auto
- apply (auto dest!: lesser_than_two_values)
- done
- qed
- } moreover
- note z moreover
- ultimately show "z <= Inf A"
- unfolding Inf_flat_complete_lattice_def
- by auto
-next
- fix x A
- assume "(x :: 'a flat_complete_lattice) : A"
- {
- fix v
- assume "A - {Bot} = {Value v}"
- from this have "(THE v. A - {Bot} = {Value v}) = v"
- by (auto intro!: the1_equality)
- moreover
- from `x : A` `A - {Bot} = {Value v}` have "x = Bot \<or> x = Value v"
- by auto
- ultimately have "x <= Value (THE v. A - {Bot} = {Value v})"
- by auto
- }
- from `x : A` this show "x <= Sup A"
- unfolding Sup_flat_complete_lattice_def
- by fastforce
-next
- fix z A
- assume z: "\<And>x. x : A ==> x <= (z :: 'a flat_complete_lattice)"
- {
- fix v
- assume "A - {Bot} = {Value v}"
- moreover
- from this have "(THE v. A - {Bot} = {Value v}) = v"
- by (auto intro!: the1_equality)
- moreover
- note z
- moreover
- ultimately have "Value (THE v::'a. A - {Bot} = {Value v}) <= z"
- by auto
- } moreover
- {
- assume not_one_value: "A ~= {}" "A ~= {Bot}" "~ (EX v::'a. A - {Bot} = {Value v})"
- have "Top <= z"
- proof (cases "A - {Bot} = {Top}")
- case True
- from this z show ?thesis
- by auto
- next
- case False
- from not_one_value
- obtain a1 where a1: "a1 : A - {Bot}" by auto
- from not_one_value False a1
- obtain a2 where "a2 : A - {Bot} \<and> a1 \<noteq> a2"
- by (cases a1) auto
- from this a1 z[of "a1"] z[of "a2"] show ?thesis
- apply (cases a1)
- apply auto
- apply (cases a2)
- apply (auto dest!: greater_than_two_values)
- done
- qed
- } moreover
- note z moreover
- ultimately show "Sup A <= z"
- unfolding Sup_flat_complete_lattice_def
- by auto
-next
- show "Inf {} = (top :: 'a flat_complete_lattice)"
- by(simp add: Inf_flat_complete_lattice_def top_flat_complete_lattice_def)
-next
- show "Sup {} = (bot :: 'a flat_complete_lattice)"
- by(simp add: Sup_flat_complete_lattice_def bot_flat_complete_lattice_def)
-qed
-
-end
-
-section {* Quickcheck configuration *}
-
-quickcheck_params[finite_types = false, default_type = ["int", "int bot", "int top", "Enum.finite_4 flat_complete_lattice"]]
-
-hide_type flat_complete_lattice bot top
-
-end
\ No newline at end of file
--- a/src/HOL/ROOT Tue Aug 19 14:58:38 2014 +0200
+++ b/src/HOL/ROOT Tue Aug 19 15:19:16 2014 +0200
@@ -38,7 +38,6 @@
Product_Lexorder
Product_Order
Finite_Lattice
- Quickcheck_Types
(*data refinements and dependent applications*)
AList_Mapping
Code_Binary_Nat