# Theory Completeness

theory Completeness
imports Main
```(*  Title:      HOL/Quickcheck_Examples/Completeness.thy
Author:     Lukas Bulwahn
Copyright   2012 TU Muenchen
*)

section ‹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"

lemma is_some_is_not_None:
"is_some x ⟷ x ≠ None"
by (cases x) simp_all

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 natural :: size
begin

definition size_natural :: "natural => nat"
where
"size_natural = nat_of_natural"

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: "(∃ v. size v ≤ n ∧ is_some (f v)) ⟷ is_some (exhaustive_class.exhaustive f (natural_of_nat n))"

lemma complete_imp1:
"size (v :: 'a :: complete) ≤ n ⟹ is_some (f v) ⟹ is_some (exhaustive_class.exhaustive f (natural_of_nat n))"
by (metis complete)

lemma complete_imp2:
assumes "is_some (exhaustive_class.exhaustive f (natural_of_nat n))"
obtains v where "size (v :: 'a :: complete) ≤ n" "is_some (f v)"
using assms by (metis complete)

subsubsection ‹Instance Proofs›

declare exhaustive_int'.simps [simp del]

lemma complete_exhaustive':
"(∃ i. j <= i & i <= k & is_some (f i)) ⟷ is_some (Quickcheck_Exhaustive.exhaustive_int' f k j)"
proof (induct rule: Quickcheck_Exhaustive.exhaustive_int'.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_int'.simps[of _ _ i] Quickcheck_Exhaustive.orelse_def
apply (auto simp add: add1_zle_eq dest: less_imp_le)
apply auto
done
next
case Some
from this show ?thesis
unfolding exhaustive_int'.simps[of _ _ i] Quickcheck_Exhaustive.orelse_def by auto
qed
qed

instance int :: complete
proof
fix n f
show "(∃v. size (v :: int) ≤ n ∧ is_some (f v)) = is_some (exhaustive_class.exhaustive f (natural_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 abs_le_iff minus_le_iff nat_le_iff)+
qed

declare exhaustive_natural'.simps[simp del]

lemma complete_natural':
"(∃n. j ≤ n ∧ n ≤ k ∧ is_some (f n)) =
is_some (Quickcheck_Exhaustive.exhaustive_natural' f k j)"
proof (induct rule: exhaustive_natural'.induct[of _ f k j])
case (1 f k j)
show "(∃n≥j. n ≤ k ∧ is_some (f n)) = is_some (Quickcheck_Exhaustive.exhaustive_natural' f k j)"
unfolding exhaustive_natural'.simps [of f k j] Quickcheck_Exhaustive.orelse_def
apply (auto split: option.split)
apply (auto simp add: less_eq_natural_def less_natural_def 1 [symmetric] dest: Suc_leD)
apply (metis is_some.simps(2) natural_eqI not_less_eq_eq order_antisym)
done
qed

instance natural :: complete
proof
fix n f
show "(∃v. size (v :: natural) ≤ n ∧ is_some (f v)) ⟷ is_some (exhaustive_class.exhaustive f (natural_of_nat n))"
unfolding exhaustive_natural_def complete_natural' [symmetric]
by (auto simp add: size_natural_def less_eq_natural_def)
qed

instance nat :: complete
proof
fix n f
show "(∃v. size (v :: nat) ≤ n ∧ is_some (f v)) ⟷ is_some (exhaustive_class.exhaustive f (natural_of_nat n))"
unfolding exhaustive_nat_def complete[of n "%x. f (nat_of_natural x)", symmetric]
apply auto
apply (rule_tac x="natural_of_nat v" in exI)
apply (auto simp add: size_natural_def size_nat_def) done
qed

instance list :: (complete) complete
proof
fix n f
show "(∃ v. size (v :: 'a list) ≤ n ∧ is_some (f (v :: 'a list))) ⟷ is_some (exhaustive_class.exhaustive f (natural_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 "∃v. Completeness.size_class.size v ≤ Suc n ∧ is_some (f v)"
then obtain v where v: "size v ≤ Suc n" "is_some (f v)" by blast
show "is_some (exhaustive_class.exhaustive f (natural_of_nat (Suc n)))"
proof (cases "v")
case Nil
from this v show ?thesis
unfolding list.exhaustive_list.simps[of _ "natural_of_nat (Suc n)"] Quickcheck_Exhaustive.orelse_def
by (auto split: option.split simp add: less_natural_def)
next
case (Cons v' vs')
from Cons v have size_v': "Completeness.size_class.size v' ≤ n"
and "Completeness.size_class.size vs' ≤ n" by auto
from Suc v Cons this have "is_some (exhaustive_class.exhaustive (λxs. f (v' # xs)) (natural_of_nat n))"
by metis
from complete_imp1[OF size_v', of "%x. (exhaustive_class.exhaustive (λxs. f (x # xs)) (natural_of_nat n))", OF this]
show ?thesis
unfolding list.exhaustive_list.simps[of _ "natural_of_nat (Suc n)"] Quickcheck_Exhaustive.orelse_def
by (auto split: option.split simp add: less_natural_def)
qed
next
assume is_some: "is_some (exhaustive_class.exhaustive f (natural_of_nat (Suc n)))"
show "∃v. size v ≤ Suc n ∧ is_some (f v)"
proof (cases "f []")
case Some
then show ?thesis
by (metis Suc_eq_plus1 is_some.simps(1) le_add2 size_list.simps(1))
next
case None
with is_some have
"is_some (exhaustive_class.exhaustive (λx. exhaustive_class.exhaustive (λxs. f (x # xs)) (natural_of_nat n)) (natural_of_nat n))"
unfolding list.exhaustive_list.simps[of _ "natural_of_nat (Suc n)"] Quickcheck_Exhaustive.orelse_def
by (simp add: less_natural_def)
then obtain v' where
v: "size v' ≤ n"
"is_some (exhaustive_class.exhaustive (λxs. f (v' # xs)) (natural_of_nat n))"
by (rule complete_imp2)
with Suc[of "%xs. f (v' # xs)"]
obtain vs' where vs': "size vs' ≤ n" "is_some (f (v' # vs'))"
by metis
with v have "size (v' # vs') ≤ Suc n" by auto
with vs' v show ?thesis by metis
qed
qed
qed
qed

end

```