renamed "Metis_Tactics" to "Metis_Tactic", now that there is only one Metis tactic ("metisFT" is legacy)
(* Author: Lukas Bulwahn, TU Muenchen *)
header {* A simple counterexample generator performing exhaustive testing *}
theory Quickcheck_Exhaustive
imports Quickcheck
uses ("Tools/Quickcheck/exhaustive_generators.ML")
begin
subsection {* basic operations for exhaustive generators *}
definition orelse :: "'a option => 'a option => 'a option" (infixr "orelse" 55)
where
[code_unfold]: "x orelse y = (case x of Some x' => Some x' | None => y)"
subsection {* exhaustive generator type classes *}
class exhaustive = term_of +
fixes exhaustive :: "('a \<Rightarrow> term list option) \<Rightarrow> code_numeral \<Rightarrow> term list option"
class full_exhaustive = term_of +
fixes full_exhaustive :: "('a * (unit => term) \<Rightarrow> term list option) \<Rightarrow> code_numeral \<Rightarrow> term list option"
instantiation code_numeral :: full_exhaustive
begin
function full_exhaustive_code_numeral' :: "(code_numeral * (unit => term) => term list option) => code_numeral => code_numeral => term list option"
where "full_exhaustive_code_numeral' f d i =
(if d < i then None
else (f (i, %_. Code_Evaluation.term_of i)) orelse (full_exhaustive_code_numeral' f d (i + 1)))"
by pat_completeness auto
termination
by (relation "measure (%(_, d, i). Code_Numeral.nat_of (d + 1 - i))") auto
definition "full_exhaustive f d = full_exhaustive_code_numeral' f d 0"
instance ..
end
instantiation code_numeral :: exhaustive
begin
function exhaustive_code_numeral' :: "(code_numeral => term list option) => code_numeral => code_numeral => term list option"
where "exhaustive_code_numeral' f d i =
(if d < i then None
else (f i orelse exhaustive_code_numeral' f d (i + 1)))"
by pat_completeness auto
termination
by (relation "measure (%(_, d, i). Code_Numeral.nat_of (d + 1 - i))") auto
definition "exhaustive f d = exhaustive_code_numeral' f d 0"
instance ..
end
instantiation nat :: exhaustive
begin
definition "exhaustive f d = exhaustive (%x. f (Code_Numeral.nat_of x)) d"
instance ..
end
instantiation nat :: full_exhaustive
begin
definition "full_exhaustive f d = full_exhaustive (%(x, xt). f (Code_Numeral.nat_of x, %_. Code_Evaluation.term_of (Code_Numeral.nat_of x))) d"
instance ..
end
instantiation int :: exhaustive
begin
function exhaustive' :: "(int => term list option) => int => int => term list option"
where "exhaustive' f d i = (if d < i then None else (f i orelse exhaustive' f d (i + 1)))"
by pat_completeness auto
termination
by (relation "measure (%(_, d, i). nat (d + 1 - i))") auto
definition "exhaustive f d = exhaustive' f (Code_Numeral.int_of d) (- (Code_Numeral.int_of d))"
instance ..
end
instantiation int :: full_exhaustive
begin
function full_exhaustive' :: "(int * (unit => term) => term list option) => int => int => term list option"
where "full_exhaustive' f d i = (if d < i then None else (case f (i, %_. Code_Evaluation.term_of i) of Some t => Some t | None => full_exhaustive' f d (i + 1)))"
by pat_completeness auto
termination
by (relation "measure (%(_, d, i). nat (d + 1 - i))") auto
definition "full_exhaustive f d = full_exhaustive' f (Code_Numeral.int_of d) (- (Code_Numeral.int_of d))"
instance ..
end
instantiation prod :: (exhaustive, exhaustive) exhaustive
begin
definition
"exhaustive f d = exhaustive (%x. exhaustive (%y. f ((x, y))) d) d"
instance ..
end
instantiation prod :: (full_exhaustive, full_exhaustive) full_exhaustive
begin
definition
"full_exhaustive f d = full_exhaustive (%(x, t1). full_exhaustive (%(y, t2). f ((x, y),
%u. let T1 = (Typerep.typerep (TYPE('a)));
T2 = (Typerep.typerep (TYPE('b)))
in Code_Evaluation.App (Code_Evaluation.App (
Code_Evaluation.Const (STR ''Product_Type.Pair'')
(Typerep.Typerep (STR ''fun'') [T1, Typerep.Typerep (STR ''fun'') [T2, Typerep.Typerep (STR ''Product_Type.prod'') [T1, T2]]]))
(t1 ())) (t2 ()))) d) d"
instance ..
end
instantiation "fun" :: ("{equal, exhaustive}", exhaustive) exhaustive
begin
fun exhaustive_fun' :: "(('a => 'b) => term list option) => code_numeral => code_numeral => term list option"
where
"exhaustive_fun' f i d = (exhaustive (%b. f (%_. b)) d)
orelse (if i > 1 then
exhaustive_fun' (%g. exhaustive (%a. exhaustive (%b.
f (g(a := b))) d) d) (i - 1) d else None)"
definition exhaustive_fun :: "(('a => 'b) => term list option) => code_numeral => term list option"
where
"exhaustive_fun f d = exhaustive_fun' f d d"
instance ..
end
instantiation "fun" :: ("{equal, full_exhaustive}", full_exhaustive) full_exhaustive
begin
fun full_exhaustive_fun' :: "(('a => 'b) * (unit => term) => term list option) => code_numeral => code_numeral => term list option"
where
"full_exhaustive_fun' f i d = (full_exhaustive (%(b, t). f (%_. b, %_. Code_Evaluation.Abs (STR ''x'') (Typerep.typerep TYPE('a)) (t ()))) d)
orelse (if i > 1 then
full_exhaustive_fun' (%(g, gt). full_exhaustive (%(a, at). full_exhaustive (%(b, bt).
f (g(a := b),
(%_. let A = (Typerep.typerep (TYPE('a)));
B = (Typerep.typerep (TYPE('b)));
fun = (%T U. Typerep.Typerep (STR ''fun'') [T, U])
in
Code_Evaluation.App (Code_Evaluation.App (Code_Evaluation.App
(Code_Evaluation.Const (STR ''Fun.fun_upd'') (fun (fun A B) (fun A (fun B (fun A B)))))
(gt ())) (at ())) (bt ())))) d) d) (i - 1) d else None)"
definition full_exhaustive_fun :: "(('a => 'b) * (unit => term) => term list option) => code_numeral => term list option"
where
"full_exhaustive_fun f d = full_exhaustive_fun' f d d"
instance ..
end
subsubsection {* A smarter enumeration scheme for functions over finite datatypes *}
class check_all = enum + term_of +
fixes check_all :: "('a * (unit \<Rightarrow> term) \<Rightarrow> term list option) \<Rightarrow> term list option"
fixes enum_term_of :: "'a itself \<Rightarrow> unit \<Rightarrow> term list"
fun check_all_n_lists :: "(('a :: check_all) list * (unit \<Rightarrow> term list) \<Rightarrow> term list option) \<Rightarrow> code_numeral \<Rightarrow> term list option"
where
"check_all_n_lists f n =
(if n = 0 then f ([], (%_. [])) else check_all (%(x, xt). check_all_n_lists (%(xs, xst). f ((x # xs), (%_. (xt () # xst ())))) (n - 1)))"
definition mk_map_term :: " (unit \<Rightarrow> typerep) \<Rightarrow> (unit \<Rightarrow> typerep) \<Rightarrow> (unit \<Rightarrow> term list) \<Rightarrow> (unit \<Rightarrow> term list) \<Rightarrow> unit \<Rightarrow> term"
where
"mk_map_term T1 T2 domm rng =
(%_. let T1 = T1 ();
T2 = T2 ();
update_term = (%g (a, b).
Code_Evaluation.App (Code_Evaluation.App (Code_Evaluation.App
(Code_Evaluation.Const (STR ''Fun.fun_upd'')
(Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''fun'') [T1, T2],
Typerep.Typerep (STR ''fun'') [T1,
Typerep.Typerep (STR ''fun'') [T2, Typerep.Typerep (STR ''fun'') [T1, T2]]]]))
g) a) b)
in
List.foldl update_term (Code_Evaluation.Abs (STR ''x'') T1 (Code_Evaluation.Const (STR ''HOL.undefined'') T2)) (zip (domm ()) (rng ())))"
instantiation "fun" :: ("{equal, check_all}", check_all) check_all
begin
definition
"check_all f =
(let
mk_term = mk_map_term (%_. Typerep.typerep (TYPE('a))) (%_. Typerep.typerep (TYPE('b))) (enum_term_of (TYPE('a)));
enum = (Enum.enum :: 'a list)
in check_all_n_lists (\<lambda>(ys, yst). f (the o map_of (zip enum ys), mk_term yst)) (Code_Numeral.of_nat (length enum)))"
definition enum_term_of_fun :: "('a => 'b) itself => unit => term list"
where
"enum_term_of_fun = (%_ _. let
enum_term_of_a = enum_term_of (TYPE('a));
mk_term = mk_map_term (%_. Typerep.typerep (TYPE('a))) (%_. Typerep.typerep (TYPE('b))) enum_term_of_a
in map (%ys. mk_term (%_. ys) ()) (Enum.n_lists (length (enum_term_of_a ())) (enum_term_of (TYPE('b)) ())))"
instance ..
end
instantiation unit :: check_all
begin
definition
"check_all f = f (Code_Evaluation.valtermify ())"
definition enum_term_of_unit :: "unit itself => unit => term list"
where
"enum_term_of_unit = (%_ _. [Code_Evaluation.term_of ()])"
instance ..
end
instantiation bool :: check_all
begin
definition
"check_all f = (case f (Code_Evaluation.valtermify False) of Some x' \<Rightarrow> Some x' | None \<Rightarrow> f (Code_Evaluation.valtermify True))"
definition enum_term_of_bool :: "bool itself => unit => term list"
where
"enum_term_of_bool = (%_ _. map Code_Evaluation.term_of (Enum.enum :: bool list))"
instance ..
end
instantiation prod :: (check_all, check_all) check_all
begin
definition
"check_all f = check_all (%(x, t1). check_all (%(y, t2). f ((x, y),
%u. let T1 = (Typerep.typerep (TYPE('a)));
T2 = (Typerep.typerep (TYPE('b)))
in Code_Evaluation.App (Code_Evaluation.App (
Code_Evaluation.Const (STR ''Product_Type.Pair'')
(Typerep.Typerep (STR ''fun'') [T1, Typerep.Typerep (STR ''fun'') [T2, Typerep.Typerep (STR ''Product_Type.prod'') [T1, T2]]]))
(t1 ())) (t2 ()))))"
definition enum_term_of_prod :: "('a * 'b) itself => unit => term list"
where
"enum_term_of_prod = (%_ _. map (%(x, y).
let T1 = (Typerep.typerep (TYPE('a)));
T2 = (Typerep.typerep (TYPE('b)))
in Code_Evaluation.App (Code_Evaluation.App (
Code_Evaluation.Const (STR ''Product_Type.Pair'')
(Typerep.Typerep (STR ''fun'') [T1, Typerep.Typerep (STR ''fun'') [T2, Typerep.Typerep (STR ''Product_Type.prod'') [T1, T2]]])) x) y)
(Enum.product (enum_term_of (TYPE('a)) ()) (enum_term_of (TYPE('b)) ()))) "
instance ..
end
instantiation sum :: (check_all, check_all) check_all
begin
definition
"check_all f = (case check_all (%(a, t). f (Inl a, %_.
let T1 = (Typerep.typerep (TYPE('a)));
T2 = (Typerep.typerep (TYPE('b)))
in Code_Evaluation.App (Code_Evaluation.Const (STR ''Sum_Type.Inl'')
(Typerep.Typerep (STR ''fun'') [T1, Typerep.Typerep (STR ''Sum_Type.sum'') [T1, T2]])) (t ()))) of Some x' => Some x'
| None => check_all (%(b, t). f (Inr b, %_. let
T1 = (Typerep.typerep (TYPE('a)));
T2 = (Typerep.typerep (TYPE('b)))
in Code_Evaluation.App (Code_Evaluation.Const (STR ''Sum_Type.Inr'')
(Typerep.Typerep (STR ''fun'') [T2, Typerep.Typerep (STR ''Sum_Type.sum'') [T1, T2]])) (t ()))))"
definition enum_term_of_sum :: "('a + 'b) itself => unit => term list"
where
"enum_term_of_sum = (%_ _.
let
T1 = (Typerep.typerep (TYPE('a)));
T2 = (Typerep.typerep (TYPE('b)))
in
map (Code_Evaluation.App (Code_Evaluation.Const (STR ''Sum_Type.Inl'')
(Typerep.Typerep (STR ''fun'') [T1, Typerep.Typerep (STR ''Sum_Type.sum'') [T1, T2]])))
(enum_term_of (TYPE('a)) ()) @
map (Code_Evaluation.App (Code_Evaluation.Const (STR ''Sum_Type.Inr'')
(Typerep.Typerep (STR ''fun'') [T2, Typerep.Typerep (STR ''Sum_Type.sum'') [T1, T2]])))
(enum_term_of (TYPE('b)) ()))"
instance ..
end
instantiation nibble :: check_all
begin
definition
"check_all f =
f (Code_Evaluation.valtermify Nibble0) orelse
f (Code_Evaluation.valtermify Nibble1) orelse
f (Code_Evaluation.valtermify Nibble2) orelse
f (Code_Evaluation.valtermify Nibble3) orelse
f (Code_Evaluation.valtermify Nibble4) orelse
f (Code_Evaluation.valtermify Nibble5) orelse
f (Code_Evaluation.valtermify Nibble6) orelse
f (Code_Evaluation.valtermify Nibble7) orelse
f (Code_Evaluation.valtermify Nibble8) orelse
f (Code_Evaluation.valtermify Nibble9) orelse
f (Code_Evaluation.valtermify NibbleA) orelse
f (Code_Evaluation.valtermify NibbleB) orelse
f (Code_Evaluation.valtermify NibbleC) orelse
f (Code_Evaluation.valtermify NibbleD) orelse
f (Code_Evaluation.valtermify NibbleE) orelse
f (Code_Evaluation.valtermify NibbleF)"
definition enum_term_of_nibble :: "nibble itself => unit => term list"
where
"enum_term_of_nibble = (%_ _. map Code_Evaluation.term_of (Enum.enum :: nibble list))"
instance ..
end
instantiation char :: check_all
begin
definition
"check_all f = check_all (%(x, t1). check_all (%(y, t2). f (Char x y, %_. Code_Evaluation.App (Code_Evaluation.App (Code_Evaluation.term_of Char) (t1 ())) (t2 ()))))"
definition enum_term_of_char :: "char itself => unit => term list"
where
"enum_term_of_char = (%_ _. map Code_Evaluation.term_of (Enum.enum :: char list))"
instance ..
end
instantiation option :: (check_all) check_all
begin
definition
"check_all f = f (Code_Evaluation.valtermify (None :: 'a option)) orelse check_all (%(x, t). f (Some x, %_. Code_Evaluation.App
(Code_Evaluation.Const (STR ''Option.option.Some'')
(Typerep.Typerep (STR ''fun'') [Typerep.typerep TYPE('a), Typerep.Typerep (STR ''Option.option'') [Typerep.typerep TYPE('a)]])) (t ())))"
definition enum_term_of_option :: "'a option itself => unit => term list"
where
"enum_term_of_option = (% _ _. (Code_Evaluation.term_of (None :: 'a option)) # (map (Code_Evaluation.App (Code_Evaluation.Const (STR ''Option.option.Some'')
(Typerep.Typerep (STR ''fun'') [Typerep.typerep TYPE('a), Typerep.Typerep (STR ''Option.option'') [Typerep.typerep TYPE('a)]]))) (enum_term_of (TYPE('a)) ())))"
instance ..
end
instantiation Enum.finite_1 :: check_all
begin
definition
"check_all f = f (Code_Evaluation.valtermify Enum.finite_1.a\<^isub>1)"
definition enum_term_of_finite_1 :: "Enum.finite_1 itself => unit => term list"
where
"enum_term_of_finite_1 = (%_ _. [Code_Evaluation.term_of Enum.finite_1.a\<^isub>1])"
instance ..
end
instantiation Enum.finite_2 :: check_all
begin
definition
"check_all f = (case f (Code_Evaluation.valtermify Enum.finite_2.a\<^isub>1) of Some x' \<Rightarrow> Some x' | None \<Rightarrow> f (Code_Evaluation.valtermify Enum.finite_2.a\<^isub>2))"
definition enum_term_of_finite_2 :: "Enum.finite_2 itself => unit => term list"
where
"enum_term_of_finite_2 = (%_ _. map Code_Evaluation.term_of (Enum.enum :: Enum.finite_2 list))"
instance ..
end
instantiation Enum.finite_3 :: check_all
begin
definition
"check_all f = (case f (Code_Evaluation.valtermify Enum.finite_3.a\<^isub>1) of Some x' \<Rightarrow> Some x' | None \<Rightarrow> (case f (Code_Evaluation.valtermify Enum.finite_3.a\<^isub>2) of Some x' \<Rightarrow> Some x' | None \<Rightarrow> f (Code_Evaluation.valtermify Enum.finite_3.a\<^isub>3)))"
definition enum_term_of_finite_3 :: "Enum.finite_3 itself => unit => term list"
where
"enum_term_of_finite_3 = (%_ _. map Code_Evaluation.term_of (Enum.enum :: Enum.finite_3 list))"
instance ..
end
subsection {* Bounded universal quantifiers *}
class bounded_forall =
fixes bounded_forall :: "('a \<Rightarrow> bool) \<Rightarrow> code_numeral \<Rightarrow> bool"
subsection {* Fast exhaustive combinators *}
class fast_exhaustive = term_of +
fixes fast_exhaustive :: "('a \<Rightarrow> unit) \<Rightarrow> code_numeral \<Rightarrow> unit"
consts throw_Counterexample :: "term list => unit"
consts catch_Counterexample :: "unit => term list option"
code_const throw_Counterexample
(Quickcheck "raise (Exhaustive'_Generators.Counterexample _)")
code_const catch_Counterexample
(Quickcheck "(((_); NONE) handle Exhaustive'_Generators.Counterexample ts => SOME ts)")
subsection {* Defining combinators for any first-order data type *}
definition catch_match :: "term list option => term list option => term list option"
where
[code del]: "catch_match t1 t2 = (SOME t. t = t1 \<or> t = t2)"
code_const catch_match
(Quickcheck "(_) handle Match => _")
use "Tools/Quickcheck/exhaustive_generators.ML"
setup {* Exhaustive_Generators.setup *}
declare [[quickcheck_batch_tester = exhaustive]]
hide_fact orelse_def catch_match_def
no_notation orelse (infixr "orelse" 55)
hide_const (open) orelse catch_match mk_map_term check_all_n_lists
end