(* Title: HOL/Quickcheck_Examples/Completeness.thy
Author: Lukas Bulwahn
Copyright 2012 TU Muenchen
*)
header {* Proving completeness of exhaustive generators *}
theory Completeness
imports Main
begin
subsection {* Preliminaries *}
primrec is_some
where
"is_some (Some t) = True"
| "is_some None = False"
setup {* Exhaustive_Generators.setup_exhaustive_datatype_interpretation *}
subsection {* Defining the size of values *}
hide_const size
class size =
fixes size :: "'a => nat"
instantiation int :: size
begin
definition size_int :: "int => nat"
where
"size_int n = nat (abs n)"
instance ..
end
instantiation code_numeral :: size
begin
definition size_code_numeral :: "code_numeral => nat"
where
"size_code_numeral = Code_Numeral.nat_of"
instance ..
end
instantiation nat :: size
begin
definition size_nat :: "nat => nat"
where
"size_nat n = n"
instance ..
end
instantiation list :: (size) size
begin
primrec size_list :: "'a list => nat"
where
"size [] = 1"
| "size (x # xs) = max (size x) (size xs) + 1"
instance ..
end
subsection {* Completeness *}
class complete = exhaustive + size +
assumes
complete: "(\<exists> v. size v \<le> n \<and> is_some (f v)) \<longleftrightarrow> is_some (exhaustive_class.exhaustive f (Code_Numeral.of_nat n))"
lemma complete_imp1:
"size (v :: 'a :: complete) \<le> n \<Longrightarrow> is_some (f v) \<Longrightarrow> is_some (exhaustive_class.exhaustive f (Code_Numeral.of_nat n))"
by (metis complete)
lemma complete_imp2:
assumes "is_some (exhaustive_class.exhaustive f (Code_Numeral.of_nat n))"
obtains v where "size (v :: 'a :: complete) \<le> n" "is_some (f v)"
using assms by (metis complete)
subsubsection {* Instance Proofs *}
declare exhaustive'.simps [simp del]
lemma complete_exhaustive':
"(\<exists> i. j <= i & i <= k & is_some (f i)) \<longleftrightarrow> is_some (Quickcheck_Exhaustive.exhaustive' f k j)"
proof (induct rule: Quickcheck_Exhaustive.exhaustive'.induct[of _ f k j])
case (1 f d i)
show ?case
proof (cases "f i")
case None
from this 1 show ?thesis
unfolding exhaustive'.simps[of _ _ i] Quickcheck_Exhaustive.orelse_def
apply auto
apply (metis is_some.simps(2) order_le_neq_trans zless_imp_add1_zle)
apply (metis add1_zle_eq less_int_def)
done
next
case Some
from this show ?thesis
unfolding exhaustive'.simps[of _ _ i] Quickcheck_Exhaustive.orelse_def by auto
qed
qed
lemma int_of_nat:
"Code_Numeral.int_of (Code_Numeral.of_nat n) = int n"
unfolding int_of_def by simp
instance int :: complete
proof
fix n f
show "(\<exists>v. size (v :: int) \<le> n \<and> is_some (f v)) = is_some (exhaustive_class.exhaustive f (Code_Numeral.of_nat n))"
unfolding exhaustive_int_def complete_exhaustive'[symmetric]
apply auto apply (rule_tac x="v" in exI)
unfolding size_int_def by (metis int_of_nat abs_le_iff minus_le_iff nat_le_iff)+
qed
declare exhaustive_code_numeral'.simps[simp del]
lemma complete_code_numeral':
"(\<exists>n. j \<le> n \<and> n \<le> k \<and> is_some (f n)) =
is_some (Quickcheck_Exhaustive.exhaustive_code_numeral' f k j)"
proof (induct rule: exhaustive_code_numeral'.induct[of _ f k j])
case (1 f k j)
show "(\<exists>n\<ge>j. n \<le> k \<and> is_some (f n)) = is_some (Quickcheck_Exhaustive.exhaustive_code_numeral' f k j)"
unfolding exhaustive_code_numeral'.simps[of f k j] Quickcheck_Exhaustive.orelse_def
apply auto
apply (auto split: option.split)
apply (insert 1[symmetric])
apply simp
apply (metis is_some.simps(2) less_eq_code_numeral_def not_less_eq_eq order_antisym)
apply simp
apply (split option.split_asm)
defer apply fastforce
apply simp by (metis Suc_leD)
qed
instance code_numeral :: complete
proof
fix n f
show "(\<exists>v. size (v :: code_numeral) \<le> n \<and> is_some (f v)) \<longleftrightarrow> is_some (exhaustive_class.exhaustive f (Code_Numeral.of_nat n))"
unfolding exhaustive_code_numeral_def complete_code_numeral'[symmetric]
by (auto simp add: size_code_numeral_def)
qed
instance nat :: complete
proof
fix n f
show "(\<exists>v. size (v :: nat) \<le> n \<and> is_some (f v)) \<longleftrightarrow> is_some (exhaustive_class.exhaustive f (Code_Numeral.of_nat n))"
unfolding exhaustive_nat_def complete[of n "%x. f (Code_Numeral.nat_of x)", symmetric]
apply auto
apply (rule_tac x="Code_Numeral.of_nat v" in exI)
apply (auto simp add: size_code_numeral_def size_nat_def) done
qed
instance list :: (complete) complete
proof
fix n f
show "(\<exists> v. size (v :: 'a list) \<le> n \<and> is_some (f (v :: 'a list))) \<longleftrightarrow> is_some (exhaustive_class.exhaustive f (Code_Numeral.of_nat n))"
proof (induct n arbitrary: f)
case 0
{ fix v have "size (v :: 'a list) > 0" by (induct v) auto }
from this show ?case by (simp add: list.exhaustive_list.simps)
next
case (Suc n)
show ?case
proof
assume "\<exists>v. Completeness.size_class.size v \<le> Suc n \<and> is_some (f v)"
from this guess v .. note v = this
show "is_some (exhaustive_class.exhaustive f (Code_Numeral.of_nat (Suc n)))"
proof (cases "v")
case Nil
from this v show ?thesis
unfolding list.exhaustive_list.simps[of _ "Code_Numeral.of_nat (Suc n)"] Quickcheck_Exhaustive.orelse_def
by (auto split: option.split)
next
case (Cons v' vs')
from Cons v have size_v': "Completeness.size_class.size v' \<le> n"
and "Completeness.size_class.size vs' \<le> n" by auto
from Suc v Cons this have "is_some (exhaustive_class.exhaustive (\<lambda>xs. f (v' # xs)) (Code_Numeral.of_nat n))"
by metis
from complete_imp1[OF size_v', of "%x. (exhaustive_class.exhaustive (\<lambda>xs. f (x # xs)) (Code_Numeral.of_nat n))", OF this]
show ?thesis
unfolding list.exhaustive_list.simps[of _ "Code_Numeral.of_nat (Suc n)"] Quickcheck_Exhaustive.orelse_def
by (auto split: option.split)
qed
next
assume is_some: "is_some (exhaustive_class.exhaustive f (Code_Numeral.of_nat (Suc n)))"
show "\<exists>v. size v \<le> Suc n \<and> is_some (f v)"
proof (cases "f []")
case Some
from this show ?thesis
by (metis Suc_eq_plus1 is_some.simps(1) le_add2 size_list.simps(1))
next
case None
from this is_some have
"is_some (exhaustive_class.exhaustive (\<lambda>x. exhaustive_class.exhaustive (\<lambda>xs. f (x # xs)) (Code_Numeral.of_nat n)) (Code_Numeral.of_nat n))"
unfolding list.exhaustive_list.simps[of _ "Code_Numeral.of_nat (Suc n)"] Quickcheck_Exhaustive.orelse_def
by simp
from complete_imp2[OF this] guess v' . note v = this
from Suc[of "%xs. f (v' # xs)"] this(2) obtain vs' where "size vs' \<le> n" "is_some (f (v' # vs'))"
by metis
note vs' = this
from this v have "size (v' # vs') \<le> Suc n" by auto
from this vs' v show ?thesis by metis
qed
qed
qed
qed
end