(* Author: Lukas Bulwahn, TU Muenchen *)
header {* Counterexample generator preforming narrowing-based testing *}
theory Quickcheck_Narrowing
imports Quickcheck_Exhaustive
uses
("~~/src/HOL/Tools/Quickcheck/PNF_Narrowing_Engine.hs")
("~~/src/HOL/Tools/Quickcheck/Narrowing_Engine.hs")
("~~/src/HOL/Tools/Quickcheck/narrowing_generators.ML")
begin
subsection {* Counterexample generator *}
text {* We create a new target for the necessary code generation setup. *}
setup {* Code_Target.extend_target ("Haskell_Quickcheck", (Code_Haskell.target, K I)) *}
subsubsection {* Code generation setup *}
code_type typerep
(Haskell_Quickcheck "Typerep")
code_const Typerep.Typerep
(Haskell_Quickcheck "Typerep")
code_reserved Haskell_Quickcheck Typerep
code_type char
(Haskell_Quickcheck "Char")
setup {*
fold String_Code.add_literal_char ["Haskell_Quickcheck"]
#> String_Code.add_literal_list_string "Haskell_Quickcheck"
*}
code_instance char :: equal
(Haskell_Quickcheck -)
code_const "HOL.equal \<Colon> char \<Rightarrow> char \<Rightarrow> bool"
(Haskell_Quickcheck infix 4 "==")
subsubsection {* Type @{text "code_int"} for Haskell_Quickcheck's Int type *}
typedef (open) code_int = "UNIV \<Colon> int set"
morphisms int_of of_int by rule
lemma of_int_int_of [simp]:
"of_int (int_of k) = k"
by (rule int_of_inverse)
lemma int_of_of_int [simp]:
"int_of (of_int n) = n"
by (rule of_int_inverse) (rule UNIV_I)
lemma code_int:
"(\<And>n\<Colon>code_int. PROP P n) \<equiv> (\<And>n\<Colon>int. PROP P (of_int n))"
proof
fix n :: int
assume "\<And>n\<Colon>code_int. PROP P n"
then show "PROP P (of_int n)" .
next
fix n :: code_int
assume "\<And>n\<Colon>int. PROP P (of_int n)"
then have "PROP P (of_int (int_of n))" .
then show "PROP P n" by simp
qed
lemma int_of_inject [simp]:
"int_of k = int_of l \<longleftrightarrow> k = l"
by (rule int_of_inject)
lemma of_int_inject [simp]:
"of_int n = of_int m \<longleftrightarrow> n = m"
by (rule of_int_inject) (rule UNIV_I)+
instantiation code_int :: equal
begin
definition
"HOL.equal k l \<longleftrightarrow> HOL.equal (int_of k) (int_of l)"
instance proof
qed (auto simp add: equal_code_int_def equal_int_def eq_int_refl)
end
instantiation code_int :: number
begin
definition
"number_of = of_int"
instance ..
end
lemma int_of_number [simp]:
"int_of (number_of k) = number_of k"
by (simp add: number_of_code_int_def number_of_is_id)
definition nat_of :: "code_int => nat"
where
"nat_of i = nat (int_of i)"
code_datatype "number_of \<Colon> int \<Rightarrow> code_int"
instantiation code_int :: "{minus, linordered_semidom, semiring_div, linorder}"
begin
definition [simp, code del]:
"0 = of_int 0"
definition [simp, code del]:
"1 = of_int 1"
definition [simp, code del]:
"n + m = of_int (int_of n + int_of m)"
definition [simp, code del]:
"n - m = of_int (int_of n - int_of m)"
definition [simp, code del]:
"n * m = of_int (int_of n * int_of m)"
definition [simp, code del]:
"n div m = of_int (int_of n div int_of m)"
definition [simp, code del]:
"n mod m = of_int (int_of n mod int_of m)"
definition [simp, code del]:
"n \<le> m \<longleftrightarrow> int_of n \<le> int_of m"
definition [simp, code del]:
"n < m \<longleftrightarrow> int_of n < int_of m"
instance proof
qed (auto simp add: code_int left_distrib zmult_zless_mono2)
end
lemma zero_code_int_code [code, code_unfold]:
"(0\<Colon>code_int) = Numeral0"
by (simp add: number_of_code_int_def Pls_def)
lemma [code_post]: "Numeral0 = (0\<Colon>code_int)"
using zero_code_int_code ..
lemma one_code_int_code [code, code_unfold]:
"(1\<Colon>code_int) = Numeral1"
by (simp add: number_of_code_int_def Pls_def Bit1_def)
lemma [code_post]: "Numeral1 = (1\<Colon>code_int)"
using one_code_int_code ..
definition div_mod_code_int :: "code_int \<Rightarrow> code_int \<Rightarrow> code_int \<times> code_int" where
[code del]: "div_mod_code_int n m = (n div m, n mod m)"
lemma [code]:
"div_mod_code_int n m = (if m = 0 then (0, n) else (n div m, n mod m))"
unfolding div_mod_code_int_def by auto
lemma [code]:
"n div m = fst (div_mod_code_int n m)"
unfolding div_mod_code_int_def by simp
lemma [code]:
"n mod m = snd (div_mod_code_int n m)"
unfolding div_mod_code_int_def by simp
lemma int_of_code [code]:
"int_of k = (if k = 0 then 0
else (if k mod 2 = 0 then 2 * int_of (k div 2) else 2 * int_of (k div 2) + 1))"
proof -
have 1: "(int_of k div 2) * 2 + int_of k mod 2 = int_of k"
by (rule mod_div_equality)
have "int_of k mod 2 = 0 \<or> int_of k mod 2 = 1" by auto
from this show ?thesis
apply auto
apply (insert 1) by (auto simp add: mult_ac)
qed
code_instance code_numeral :: equal
(Haskell_Quickcheck -)
setup {* fold (Numeral.add_code @{const_name number_code_int_inst.number_of_code_int}
false Code_Printer.literal_numeral) ["Haskell_Quickcheck"] *}
code_const "0 \<Colon> code_int"
(Haskell_Quickcheck "0")
code_const "1 \<Colon> code_int"
(Haskell_Quickcheck "1")
code_const "minus \<Colon> code_int \<Rightarrow> code_int \<Rightarrow> code_int"
(Haskell_Quickcheck "(_/ -/ _)")
code_const div_mod_code_int
(Haskell_Quickcheck "divMod")
code_const "HOL.equal \<Colon> code_int \<Rightarrow> code_int \<Rightarrow> bool"
(Haskell_Quickcheck infix 4 "==")
code_const "op \<le> \<Colon> code_int \<Rightarrow> code_int \<Rightarrow> bool"
(Haskell_Quickcheck infix 4 "<=")
code_const "op < \<Colon> code_int \<Rightarrow> code_int \<Rightarrow> bool"
(Haskell_Quickcheck infix 4 "<")
code_type code_int
(Haskell_Quickcheck "Int")
code_abort of_int
subsubsection {* Narrowing's deep representation of types and terms *}
datatype narrowing_type = SumOfProd "narrowing_type list list"
datatype narrowing_term = Var "code_int list" narrowing_type | Ctr code_int "narrowing_term list"
datatype 'a cons = C narrowing_type "(narrowing_term list => 'a) list"
subsubsection {* From narrowing's deep representation of terms to Code_Evaluation's terms *}
class partial_term_of = typerep +
fixes partial_term_of :: "'a itself => narrowing_term => Code_Evaluation.term"
lemma partial_term_of_anything: "partial_term_of x nt \<equiv> t"
by (rule eq_reflection) (cases "partial_term_of x nt", cases t, simp)
subsubsection {* Auxilary functions for Narrowing *}
consts nth :: "'a list => code_int => 'a"
code_const nth (Haskell_Quickcheck infixl 9 "!!")
consts error :: "char list => 'a"
code_const error (Haskell_Quickcheck "error")
consts toEnum :: "code_int => char"
code_const toEnum (Haskell_Quickcheck "toEnum")
consts map_index :: "(code_int * 'a => 'b) => 'a list => 'b list"
consts split_At :: "code_int => 'a list => 'a list * 'a list"
subsubsection {* Narrowing's basic operations *}
type_synonym 'a narrowing = "code_int => 'a cons"
definition empty :: "'a narrowing"
where
"empty d = C (SumOfProd []) []"
definition cons :: "'a => 'a narrowing"
where
"cons a d = (C (SumOfProd [[]]) [(%_. a)])"
fun conv :: "(narrowing_term list => 'a) list => narrowing_term => 'a"
where
"conv cs (Var p _) = error (Char Nibble0 Nibble0 # map toEnum p)"
| "conv cs (Ctr i xs) = (nth cs i) xs"
fun nonEmpty :: "narrowing_type => bool"
where
"nonEmpty (SumOfProd ps) = (\<not> (List.null ps))"
definition "apply" :: "('a => 'b) narrowing => 'a narrowing => 'b narrowing"
where
"apply f a d =
(case f d of C (SumOfProd ps) cfs =>
case a (d - 1) of C ta cas =>
let
shallow = (d > 0 \<and> nonEmpty ta);
cs = [(%xs'. (case xs' of [] => undefined | x # xs => cf xs (conv cas x))). shallow, cf <- cfs]
in C (SumOfProd [ta # p. shallow, p <- ps]) cs)"
definition sum :: "'a narrowing => 'a narrowing => 'a narrowing"
where
"sum a b d =
(case a d of C (SumOfProd ssa) ca =>
case b d of C (SumOfProd ssb) cb =>
C (SumOfProd (ssa @ ssb)) (ca @ cb))"
lemma [fundef_cong]:
assumes "a d = a' d" "b d = b' d" "d = d'"
shows "sum a b d = sum a' b' d'"
using assms unfolding sum_def by (auto split: cons.split narrowing_type.split)
lemma [fundef_cong]:
assumes "f d = f' d" "(\<And>d'. 0 <= d' & d' < d ==> a d' = a' d')"
assumes "d = d'"
shows "apply f a d = apply f' a' d'"
proof -
note assms moreover
have "int_of (of_int 0) < int_of d' ==> int_of (of_int 0) <= int_of (of_int (int_of d' - int_of (of_int 1)))"
by (simp add: of_int_inverse)
moreover
have "int_of (of_int (int_of d' - int_of (of_int 1))) < int_of d'"
by (simp add: of_int_inverse)
ultimately show ?thesis
unfolding apply_def by (auto split: cons.split narrowing_type.split simp add: Let_def)
qed
type_synonym pos = "code_int list"
(*
subsubsection {* Term refinement *}
definition new :: "pos => type list list => term list"
where
"new p ps = map_index (%(c, ts). Ctr c (map_index (%(i, t). Var (p @ [i]) t) ts)) ps"
fun refine :: "term => pos => term list" and refineList :: "term list => pos => (term list) list"
where
"refine (Var p (SumOfProd ss)) [] = new p ss"
| "refine (Ctr c xs) p = map (Ctr c) (refineList xs p)"
| "refineList xs (i # is) = (let (ls, xrs) = split_At i xs in (case xrs of x#rs => [ls @ y # rs. y <- refine x is]))"
text {* Find total instantiations of a partial value *}
function total :: "term => term list"
where
"total (Ctr c xs) = [Ctr c ys. ys <- map total xs]"
| "total (Var p (SumOfProd ss)) = [y. x <- new p ss, y <- total x]"
by pat_completeness auto
termination sorry
*)
subsubsection {* Narrowing generator type class *}
class narrowing =
fixes narrowing :: "code_int => 'a cons"
definition cons1 :: "('a::narrowing => 'b) => 'b narrowing"
where
"cons1 f = apply (cons f) narrowing"
definition cons2 :: "('a :: narrowing => 'b :: narrowing => 'c) => 'c narrowing"
where
"cons2 f = apply (apply (cons f) narrowing) narrowing"
definition drawn_from :: "'a list => 'a cons"
where "drawn_from xs = C (SumOfProd (map (%_. []) xs)) (map (%x y. x) xs)"
instantiation int :: narrowing
begin
definition
"narrowing_int d = (let i = Quickcheck_Narrowing.int_of d in drawn_from [-i .. i])"
instance ..
end
instantiation unit :: narrowing
begin
definition
"narrowing = cons ()"
instance ..
end
instantiation bool :: narrowing
begin
definition
"narrowing = sum (cons True) (cons False)"
instance ..
end
instantiation option :: (narrowing) narrowing
begin
definition
"narrowing = sum (cons None) (cons1 Some)"
instance ..
end
instantiation sum :: (narrowing, narrowing) narrowing
begin
definition
"narrowing = sum (cons1 Inl) (cons1 Inr)"
instance ..
end
instantiation list :: (narrowing) narrowing
begin
function narrowing_list :: "'a list narrowing"
where
"narrowing_list d = sum (cons []) (apply (apply (cons Cons) narrowing) narrowing_list) d"
by pat_completeness auto
termination proof (relation "measure nat_of")
qed (auto simp add: of_int_inverse nat_of_def)
instance ..
end
instantiation nat :: narrowing
begin
function narrowing_nat :: "nat narrowing"
where
"narrowing_nat d = sum (cons 0) (apply (cons Suc) narrowing_nat) d"
by pat_completeness auto
termination proof (relation "measure nat_of")
qed (auto simp add: of_int_inverse nat_of_def)
instance ..
end
instantiation Enum.finite_1 :: narrowing
begin
definition narrowing_finite_1 :: "Enum.finite_1 narrowing"
where
"narrowing_finite_1 = cons (Enum.finite_1.a\<^isub>1 :: Enum.finite_1)"
instance ..
end
instantiation Enum.finite_2 :: narrowing
begin
definition narrowing_finite_2 :: "Enum.finite_2 narrowing"
where
"narrowing_finite_2 = sum (cons (Enum.finite_2.a\<^isub>1 :: Enum.finite_2)) (cons (Enum.finite_2.a\<^isub>2 :: Enum.finite_2))"
instance ..
end
instantiation Enum.finite_3 :: narrowing
begin
definition narrowing_finite_3 :: "Enum.finite_3 narrowing"
where
"narrowing_finite_3 = sum (cons (Enum.finite_3.a\<^isub>1 :: Enum.finite_3)) (sum (cons (Enum.finite_3.a\<^isub>2 :: Enum.finite_3)) (cons (Enum.finite_3.a\<^isub>3 :: Enum.finite_3)))"
instance ..
end
instantiation Enum.finite_4 :: narrowing
begin
definition narrowing_finite_4 :: "Enum.finite_4 narrowing"
where
"narrowing_finite_4 = sum (cons Enum.finite_4.a\<^isub>1) (sum (cons Enum.finite_4.a\<^isub>2) (sum (cons Enum.finite_4.a\<^isub>3) (cons Enum.finite_4.a\<^isub>4)))"
instance ..
end
datatype property = Universal narrowing_type "(narrowing_term => property)" "narrowing_term => Code_Evaluation.term" | Existential narrowing_type "(narrowing_term => property)" "narrowing_term => Code_Evaluation.term" | Property bool
(* FIXME: hard-wired maximal depth of 100 here *)
fun exists :: "('a :: {narrowing, partial_term_of} => property) => property"
where
"exists f = (case narrowing (100 :: code_int) of C ty cs => Existential ty (\<lambda> t. f (conv cs t)) (partial_term_of (TYPE('a))))"
fun "all" :: "('a :: {narrowing, partial_term_of} => property) => property"
where
"all f = (case narrowing (100 :: code_int) of C ty cs => Universal ty (\<lambda>t. f (conv cs t)) (partial_term_of (TYPE('a))))"
subsubsection {* class @{text is_testable} *}
text {* The class @{text is_testable} ensures that all necessary type instances are generated. *}
class is_testable
instance bool :: is_testable ..
instance "fun" :: ("{term_of, narrowing, partial_term_of}", is_testable) is_testable ..
definition ensure_testable :: "'a :: is_testable => 'a :: is_testable"
where
"ensure_testable f = f"
declare simp_thms(17,19)[code del]
subsubsection {* Defining a simple datatype to represent functions in an incomplete and redundant way *}
datatype ('a, 'b) ffun = Constant 'b | Update 'a 'b "('a, 'b) ffun"
primrec eval_ffun :: "('a, 'b) ffun => 'a => 'b"
where
"eval_ffun (Constant c) x = c"
| "eval_ffun (Update x' y f) x = (if x = x' then y else eval_ffun f x)"
hide_type (open) ffun
hide_const (open) Constant Update eval_ffun
datatype 'b cfun = Constant 'b
primrec eval_cfun :: "'b cfun => 'a => 'b"
where
"eval_cfun (Constant c) y = c"
hide_type (open) cfun
hide_const (open) Constant eval_cfun
subsubsection {* Setting up the counterexample generator *}
setup {* Thy_Load.provide_file (Path.explode ("~~/src/HOL/Tools/Quickcheck/PNF_Narrowing_Engine.hs")) *}
setup {* Thy_Load.provide_file (Path.explode ("~~/src/HOL/Tools/Quickcheck/Narrowing_Engine.hs")) *}
use "~~/src/HOL/Tools/Quickcheck/narrowing_generators.ML"
setup {* Narrowing_Generators.setup *}
hide_type (open) code_int narrowing_type narrowing_term cons
hide_const (open) int_of of_int nth error toEnum map_index split_At empty
C cons conv nonEmpty "apply" sum cons1 cons2 ensure_testable all exists
hide_fact empty_def
end