(* 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 {* A non-distributive lattice *}
datatype non_distrib_lattice = Zero | A | B | C | One
instantiation non_distrib_lattice :: order
begin
definition less_eq_non_distrib_lattice
where
"a <= b = (case a of Zero => True | A => (b = A) \<or> (b = One) | B => (b = B) \<or> (b = One) | C => (b = C) \<or> (b = One) | One => (b = One))"
definition less_non_distrib_lattice
where
"a < b = (case a of Zero => (b \<noteq> Zero) | A => (b = One) | B => (b = One) | C => (b = One) | One => False)"
instance proof
qed (auto simp add: less_eq_non_distrib_lattice_def less_non_distrib_lattice_def split: non_distrib_lattice.split non_distrib_lattice.split_asm)
end
instantiation non_distrib_lattice :: lattice
begin
definition sup_non_distrib_lattice
where
"sup a b = (if a = b then a else (if a = Zero then b else (if b = Zero then a else One)))"
definition inf_non_distrib_lattice
where
"inf a b = (if a = b then a else (if a = One then b else (if b = One then a else Zero)))"
instance proof
qed (auto simp add: inf_non_distrib_lattice_def sup_non_distrib_lattice_def less_eq_non_distrib_lattice_def split: split_if non_distrib_lattice.split non_distrib_lattice.split_asm)
end
hide_const Zero A B C One
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 :: (preorder) bot
begin
definition "bot = Bot"
instance proof
qed (simp add: bot_bot_def)
end
instantiation bot :: (top) top
begin
definition "top = Value top"
instance proof
qed (simp add: top_bot_def less_eq_bot_def split: bot.split)
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 ..
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 :: (preorder) top
begin
definition "top = Top"
instance proof
qed (simp add: top_top_def)
end
instantiation top :: (bot) bot
begin
definition "bot = Value bot"
instance proof
qed (simp add: bot_top_def less_eq_top_def split: top.split)
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 ..
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 proof
qed (simp add: bot_flat_complete_lattice_def)
end
instantiation flat_complete_lattice :: (type) top
begin
definition "top = Top"
instance proof
qed (auto simp add: less_eq_flat_complete_lattice_def top_flat_complete_lattice_def split: flat_complete_lattice.splits)
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 fastsimp
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 fastsimp
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
qed
end
section {* Quickcheck configuration *}
quickcheck_params[finite_types = false, default_type = ["int", "non_distrib_lattice", "int bot", "int top", "int flat_complete_lattice"]]
hide_type non_distrib_lattice flat_complete_lattice bot top
end