--- a/src/HOL/Library/Quickcheck_Narrowing.thy Thu Jun 09 08:32:13 2011 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,523 +0,0 @@
-(* 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/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
-
-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
-
-end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Quickcheck_Narrowing.thy Thu Jun 09 08:32:14 2011 +0200
@@ -0,0 +1,523 @@
+(* 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/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
+
+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
+
+end
\ No newline at end of file