(* Author: Lukas Bulwahn, TU Muenchen *)
header {* Counterexample generator preforming narrowing-based testing *}
theory Quickcheck_Narrowing
imports Main "~~/src/HOL/Library/Code_Char"
uses
("~~/src/HOL/Tools/Quickcheck/narrowing_generators.ML")
begin
subsection {* Counterexample generator *}
subsubsection {* Code generation setup *}
code_type typerep
("Haskell" "Typerep")
code_const Typerep.Typerep
("Haskell" "Typerep")
code_reserved Haskell Typerep
subsubsection {* Type @{text code_int} for Haskell's Int type *}
typedef (open) code_int = "UNIV \<Colon> int set"
morphisms int_of of_int by rule
lemma int_of_inject [simp]:
"int_of k = int_of l \<longleftrightarrow> k = l"
by (rule int_of_inject)
definition nat_of :: "code_int => nat"
where
"nat_of i = nat (int_of i)"
instantiation code_int :: "{zero, one, minus, 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 \<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)
end
(*
lemma zero_code_int_code [code, code_unfold]:
"(0\<Colon>code_int) = Numeral0"
by (simp add: number_of_code_numeral_def Pls_def)
lemma [code_post]: "Numeral0 = (0\<Colon>code_numeral)"
using zero_code_numeral_code ..
lemma one_code_numeral_code [code, code_unfold]:
"(1\<Colon>code_int) = Numeral1"
by (simp add: number_of_code_numeral_def Pls_def Bit1_def)
lemma [code_post]: "Numeral1 = (1\<Colon>code_int)"
using one_code_numeral_code ..
*)
code_const "0 \<Colon> code_int"
(Haskell "0")
code_const "1 \<Colon> code_int"
(Haskell "1")
code_const "minus \<Colon> code_int \<Rightarrow> code_int \<Rightarrow> code_int"
(Haskell "(_/ -/ _)")
code_const "op \<le> \<Colon> code_int \<Rightarrow> code_int \<Rightarrow> bool"
(Haskell infix 4 "<=")
code_const "op < \<Colon> code_int \<Rightarrow> code_int \<Rightarrow> bool"
(Haskell infix 4 "<")
code_type code_int
(Haskell "Int")
subsubsection {* Narrowing's deep representation of types and terms *}
datatype type = SumOfProd "type list list"
datatype "term" = Var "code_int list" type | Ctr code_int "term list"
datatype 'a cons = C type "(term list => 'a) list"
subsubsection {* auxilary functions for Narrowing *}
consts nth :: "'a list => code_int => 'a"
code_const nth ("Haskell" infixl 9 "!!")
consts error :: "char list => 'a"
code_const error ("Haskell" "error")
consts toEnum :: "code_int => char"
code_const toEnum ("Haskell" "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 :: "(term list => 'a) list => 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 :: "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 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 type.split simp add: Let_def)
qed
definition cons0 :: "'a => 'a narrowing"
where
"cons0 f = cons f"
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"
instantiation unit :: narrowing
begin
definition
"narrowing = cons0 ()"
instance ..
end
instantiation bool :: narrowing
begin
definition
"narrowing = sum (cons0 True) (cons0 False)"
instance ..
end
instantiation option :: (narrowing) narrowing
begin
definition
"narrowing = sum (cons0 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
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}", 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 {* Setting up the counterexample generator *}
use "~~/src/HOL/Tools/Quickcheck/narrowing_generators.ML"
setup {* Narrowing_Generators.setup *}
hide_const (open) empty
end