
bulwahn@41905: (* Author: Lukas Bulwahn, TU Muenchen *)
bulwahn@41905: 
bulwahn@43356: header {* Counterexample generator performing narrowing-based testing *}
bulwahn@41905: 
bulwahn@41930: theory Quickcheck_Narrowing
bulwahn@43312: imports Quickcheck_Exhaustive
wenzelm@46950: keywords "find_unused_assms" :: diag
bulwahn@41905: begin
bulwahn@41905: 
bulwahn@41905: subsection {* Counterexample generator *}
bulwahn@41905: 
haftmann@51143: subsubsection {* Code generation setup *}
bulwahn@43308: 
bulwahn@43308: setup {* Code_Target.extend_target ("Haskell_Quickcheck", (Code_Haskell.target, K I)) *}
bulwahn@43308: 
haftmann@52435: code_printing
haftmann@52435:   type_constructor typerep \<rightharpoonup> (Haskell_Quickcheck) "Typerep"
haftmann@52435: | constant Typerep.Typerep \<rightharpoonup> (Haskell_Quickcheck) "Typerep"
haftmann@52435: | type_constructor integer \<rightharpoonup> (Haskell_Quickcheck) "Prelude.Int"
haftmann@51143: 
bulwahn@43308: code_reserved Haskell_Quickcheck Typerep
bulwahn@41909: 
bulwahn@42021: 
bulwahn@41961: subsubsection {* Narrowing's deep representation of types and terms *}
bulwahn@41905: 
bulwahn@46758: datatype narrowing_type = Narrowing_sum_of_products "narrowing_type list list"
haftmann@51143: datatype narrowing_term = Narrowing_variable "integer list" narrowing_type | Narrowing_constructor integer "narrowing_term list"
bulwahn@46758: datatype 'a narrowing_cons = Narrowing_cons narrowing_type "(narrowing_term list => 'a) list"
bulwahn@41905: 
bulwahn@46758: primrec map_cons :: "('a => 'b) => 'a narrowing_cons => 'b narrowing_cons"
bulwahn@43356: where
bulwahn@46758:   "map_cons f (Narrowing_cons ty cs) = Narrowing_cons ty (map (%c. f o c) cs)"
bulwahn@43356: 
hoelzl@43341: subsubsection {* From narrowing's deep representation of terms to @{theory Code_Evaluation}'s terms *}
bulwahn@42980: 
bulwahn@42980: class partial_term_of = typerep +
bulwahn@43047:   fixes partial_term_of :: "'a itself => narrowing_term => Code_Evaluation.term"
bulwahn@43047: 
bulwahn@43047: lemma partial_term_of_anything: "partial_term_of x nt \<equiv> t"
bulwahn@43047:   by (rule eq_reflection) (cases "partial_term_of x nt", cases t, simp)
bulwahn@43356:  
bulwahn@41964: subsubsection {* Auxilary functions for Narrowing *}
bulwahn@41905: 
haftmann@51143: consts nth :: "'a list => integer => 'a"
bulwahn@41905: 
haftmann@52435: code_printing constant nth \<rightharpoonup> (Haskell_Quickcheck) infixl 9 "!!"
bulwahn@41905: 
bulwahn@41908: consts error :: "char list => 'a"
bulwahn@41905: 
haftmann@52435: code_printing constant error \<rightharpoonup> (Haskell_Quickcheck) "error"
bulwahn@41905: 
haftmann@51143: consts toEnum :: "integer => char"
bulwahn@41908: 
haftmann@52435: code_printing constant toEnum \<rightharpoonup> (Haskell_Quickcheck) "Prelude.toEnum"
bulwahn@41905: 
bulwahn@43316: consts marker :: "char"
bulwahn@41905: 
haftmann@52435: code_printing constant marker \<rightharpoonup> (Haskell_Quickcheck) "''\\0'"
bulwahn@43316: 
bulwahn@41961: subsubsection {* Narrowing's basic operations *}
bulwahn@41905: 
haftmann@51143: type_synonym 'a narrowing = "integer => 'a narrowing_cons"
bulwahn@41905: 
bulwahn@41961: definition empty :: "'a narrowing"
bulwahn@41905: where
bulwahn@46758:   "empty d = Narrowing_cons (Narrowing_sum_of_products []) []"
bulwahn@41905:   
bulwahn@41961: definition cons :: "'a => 'a narrowing"
bulwahn@41905: where
bulwahn@46758:   "cons a d = (Narrowing_cons (Narrowing_sum_of_products [[]]) [(%_. a)])"
bulwahn@41905: 
bulwahn@43047: fun conv :: "(narrowing_term list => 'a) list => narrowing_term => 'a"
bulwahn@41905: where
bulwahn@46758:   "conv cs (Narrowing_variable p _) = error (marker # map toEnum p)"
bulwahn@46758: | "conv cs (Narrowing_constructor i xs) = (nth cs i) xs"
bulwahn@41905: 
bulwahn@46758: fun non_empty :: "narrowing_type => bool"
bulwahn@41905: where
bulwahn@46758:   "non_empty (Narrowing_sum_of_products ps) = (\<not> (List.null ps))"
bulwahn@41905: 
bulwahn@41961: definition "apply" :: "('a => 'b) narrowing => 'a narrowing => 'b narrowing"
bulwahn@41905: where
bulwahn@41905:   "apply f a d =
bulwahn@46758:      (case f d of Narrowing_cons (Narrowing_sum_of_products ps) cfs =>
bulwahn@46758:        case a (d - 1) of Narrowing_cons ta cas =>
bulwahn@41905:        let
bulwahn@46758:          shallow = (d > 0 \<and> non_empty ta);
bulwahn@41905:          cs = [(%xs'. (case xs' of [] => undefined | x # xs => cf xs (conv cas x))). shallow, cf <- cfs]
bulwahn@46758:        in Narrowing_cons (Narrowing_sum_of_products [ta # p. shallow, p <- ps]) cs)"
bulwahn@41905: 
bulwahn@41961: definition sum :: "'a narrowing => 'a narrowing => 'a narrowing"
bulwahn@41905: where
bulwahn@41905:   "sum a b d =
bulwahn@46758:     (case a d of Narrowing_cons (Narrowing_sum_of_products ssa) ca => 
bulwahn@46758:       case b d of Narrowing_cons (Narrowing_sum_of_products ssb) cb =>
bulwahn@46758:       Narrowing_cons (Narrowing_sum_of_products (ssa @ ssb)) (ca @ cb))"
bulwahn@41905: 
bulwahn@41912: lemma [fundef_cong]:
bulwahn@41912:   assumes "a d = a' d" "b d = b' d" "d = d'"
bulwahn@41912:   shows "sum a b d = sum a' b' d'"
bulwahn@46758: using assms unfolding sum_def by (auto split: narrowing_cons.split narrowing_type.split)
bulwahn@41912: 
bulwahn@41912: lemma [fundef_cong]:
haftmann@51143:   assumes "f d = f' d" "(\<And>d'. 0 \<le> d' \<and> d' < d \<Longrightarrow> a d' = a' d')"
bulwahn@41912:   assumes "d = d'"
bulwahn@41912:   shows "apply f a d = apply f' a' d'"
bulwahn@41912: proof -
haftmann@51143:   note assms
haftmann@51143:   moreover have "0 < d' \<Longrightarrow> 0 \<le> d' - 1"
haftmann@51143:     by (simp add: less_integer_def less_eq_integer_def)
bulwahn@41912:   ultimately show ?thesis
haftmann@51143:     by (auto simp add: apply_def Let_def
haftmann@51143:       split: narrowing_cons.split narrowing_type.split)
bulwahn@41912: qed
bulwahn@41912: 
bulwahn@41961: subsubsection {* Narrowing generator type class *}
bulwahn@41905: 
bulwahn@41961: class narrowing =
haftmann@51143:   fixes narrowing :: "integer => 'a narrowing_cons"
bulwahn@41905: 
bulwahn@43237: 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
bulwahn@43237: 
bulwahn@43237: (* FIXME: hard-wired maximal depth of 100 here *)
bulwahn@43315: definition exists :: "('a :: {narrowing, partial_term_of} => property) => property"
bulwahn@43237: where
haftmann@51143:   "exists f = (case narrowing (100 :: integer) of Narrowing_cons ty cs => Existential ty (\<lambda> t. f (conv cs t)) (partial_term_of (TYPE('a))))"
bulwahn@43237: 
bulwahn@43315: definition "all" :: "('a :: {narrowing, partial_term_of} => property) => property"
bulwahn@43237: where
haftmann@51143:   "all f = (case narrowing (100 :: integer) of Narrowing_cons ty cs => Universal ty (\<lambda>t. f (conv cs t)) (partial_term_of (TYPE('a))))"
bulwahn@43237: 
wenzelm@41943: subsubsection {* class @{text is_testable} *}
bulwahn@41905: 
wenzelm@41943: text {* The class @{text is_testable} ensures that all necessary type instances are generated. *}
bulwahn@41905: 
bulwahn@41905: class is_testable
bulwahn@41905: 
bulwahn@41905: instance bool :: is_testable ..
bulwahn@41905: 
bulwahn@43047: instance "fun" :: ("{term_of, narrowing, partial_term_of}", is_testable) is_testable ..
bulwahn@41905: 
bulwahn@41905: definition ensure_testable :: "'a :: is_testable => 'a :: is_testable"
bulwahn@41905: where
bulwahn@41905:   "ensure_testable f = f"
bulwahn@41905: 
bulwahn@41910: 
bulwahn@42022: subsubsection {* Defining a simple datatype to represent functions in an incomplete and redundant way *}
bulwahn@42022: 
bulwahn@42022: datatype ('a, 'b) ffun = Constant 'b | Update 'a 'b "('a, 'b) ffun"
bulwahn@42022: 
bulwahn@42022: primrec eval_ffun :: "('a, 'b) ffun => 'a => 'b"
bulwahn@42022: where
bulwahn@42022:   "eval_ffun (Constant c) x = c"
bulwahn@42022: | "eval_ffun (Update x' y f) x = (if x = x' then y else eval_ffun f x)"
bulwahn@42022: 
bulwahn@42022: hide_type (open) ffun
bulwahn@42022: hide_const (open) Constant Update eval_ffun
bulwahn@42022: 
bulwahn@42024: datatype 'b cfun = Constant 'b
bulwahn@42024: 
bulwahn@42024: primrec eval_cfun :: "'b cfun => 'a => 'b"
bulwahn@42024: where
bulwahn@42024:   "eval_cfun (Constant c) y = c"
bulwahn@42024: 
bulwahn@42024: hide_type (open) cfun
huffman@45734: hide_const (open) Constant eval_cfun Abs_cfun Rep_cfun
bulwahn@42024: 
bulwahn@42024: subsubsection {* Setting up the counterexample generator *}
bulwahn@43237: 
wenzelm@48891: ML_file "Tools/Quickcheck/narrowing_generators.ML"
bulwahn@42024: 
bulwahn@42024: setup {* Narrowing_Generators.setup *}
bulwahn@42024: 
bulwahn@45001: definition narrowing_dummy_partial_term_of :: "('a :: partial_term_of) itself => narrowing_term => term"
bulwahn@45001: where
bulwahn@45001:   "narrowing_dummy_partial_term_of = partial_term_of"
bulwahn@45001: 
haftmann@51143: definition narrowing_dummy_narrowing :: "integer => ('a :: narrowing) narrowing_cons"
bulwahn@45001: where
bulwahn@45001:   "narrowing_dummy_narrowing = narrowing"
bulwahn@45001: 
bulwahn@45001: lemma [code]:
bulwahn@45001:   "ensure_testable f =
bulwahn@45001:     (let
haftmann@51143:       x = narrowing_dummy_narrowing :: integer => bool narrowing_cons;
bulwahn@45001:       y = narrowing_dummy_partial_term_of :: bool itself => narrowing_term => term;
bulwahn@45001:       z = (conv :: _ => _ => unit)  in f)"
bulwahn@45001: unfolding Let_def ensure_testable_def ..
bulwahn@45001: 
bulwahn@46308: subsection {* Narrowing for sets *}
bulwahn@46308: 
bulwahn@46308: instantiation set :: (narrowing) narrowing
bulwahn@46308: begin
bulwahn@46308: 
bulwahn@46308: definition "narrowing_set = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.cons set) narrowing"
bulwahn@46308: 
bulwahn@46308: instance ..
bulwahn@46308: 
bulwahn@46308: end
bulwahn@45001:   
bulwahn@43356: subsection {* Narrowing for integers *}
bulwahn@43356: 
bulwahn@43356: 
haftmann@51143: definition drawn_from :: "'a list \<Rightarrow> 'a narrowing_cons"
haftmann@51143: where
haftmann@51143:   "drawn_from xs =
haftmann@51143:     Narrowing_cons (Narrowing_sum_of_products (map (\<lambda>_. []) xs)) (map (\<lambda>x _. x) xs)"
bulwahn@43356: 
haftmann@51143: function around_zero :: "int \<Rightarrow> int list"
bulwahn@43356: where
bulwahn@43356:   "around_zero i = (if i < 0 then [] else (if i = 0 then [0] else around_zero (i - 1) @ [i, -i]))"
haftmann@51143:   by pat_completeness auto
bulwahn@43356: termination by (relation "measure nat") auto
bulwahn@43356: 
haftmann@51143: declare around_zero.simps [simp del]
bulwahn@43356: 
bulwahn@43356: lemma length_around_zero:
bulwahn@43356:   assumes "i >= 0" 
bulwahn@43356:   shows "length (around_zero i) = 2 * nat i + 1"
haftmann@51143: proof (induct rule: int_ge_induct [OF assms])
bulwahn@43356:   case 1
bulwahn@43356:   from 1 show ?case by (simp add: around_zero.simps)
bulwahn@43356: next
bulwahn@43356:   case (2 i)
bulwahn@43356:   from 2 show ?case
haftmann@51143:     by (simp add: around_zero.simps [of "i + 1"])
bulwahn@43356: qed
bulwahn@43356: 
bulwahn@43356: instantiation int :: narrowing
bulwahn@43356: begin
bulwahn@43356: 
bulwahn@43356: definition
haftmann@51143:   "narrowing_int d = (let (u :: _ \<Rightarrow> _ \<Rightarrow> unit) = conv; i = int_of_integer d
haftmann@51143:     in drawn_from (around_zero i))"
bulwahn@43356: 
bulwahn@43356: instance ..
bulwahn@43356: 
bulwahn@43356: end
bulwahn@43356: 
haftmann@51143: lemma [code, code del]: "partial_term_of (ty :: int itself) t \<equiv> undefined"
haftmann@51143:   by (rule partial_term_of_anything)+
bulwahn@43356: 
bulwahn@43356: lemma [code]:
haftmann@51143:   "partial_term_of (ty :: int itself) (Narrowing_variable p t) \<equiv>
haftmann@51143:     Code_Evaluation.Free (STR ''_'') (Typerep.Typerep (STR ''Int.int'') [])"
haftmann@51143:   "partial_term_of (ty :: int itself) (Narrowing_constructor i []) \<equiv>
haftmann@51143:     (if i mod 2 = 0
haftmann@51143:      then Code_Evaluation.term_of (- (int_of_integer i) div 2)
haftmann@51143:      else Code_Evaluation.term_of ((int_of_integer i + 1) div 2))"
haftmann@51143:   by (rule partial_term_of_anything)+
haftmann@51143: 
haftmann@51143: instantiation integer :: narrowing
haftmann@51143: begin
haftmann@51143: 
haftmann@51143: definition
haftmann@51143:   "narrowing_integer d = (let (u :: _ \<Rightarrow> _ \<Rightarrow> unit) = conv; i = int_of_integer d
haftmann@51143:     in drawn_from (map integer_of_int (around_zero i)))"
haftmann@51143: 
haftmann@51143: instance ..
haftmann@51143: 
haftmann@51143: end
haftmann@51143: 
haftmann@51143: lemma [code, code del]: "partial_term_of (ty :: integer itself) t \<equiv> undefined"
haftmann@51143:   by (rule partial_term_of_anything)+
haftmann@51143: 
haftmann@51143: lemma [code]:
haftmann@51143:   "partial_term_of (ty :: integer itself) (Narrowing_variable p t) \<equiv>
haftmann@51143:     Code_Evaluation.Free (STR ''_'') (Typerep.Typerep (STR ''Code_Numeral.integer'') [])"
haftmann@51143:   "partial_term_of (ty :: integer itself) (Narrowing_constructor i []) \<equiv>
haftmann@51143:     (if i mod 2 = 0
haftmann@51143:      then Code_Evaluation.term_of (- i div 2)
haftmann@51143:      else Code_Evaluation.term_of ((i + 1) div 2))"
haftmann@51143:   by (rule partial_term_of_anything)+
bulwahn@43356: 
bulwahn@43356: 
bulwahn@46589: subsection {* The @{text find_unused_assms} command *}
bulwahn@46589: 
wenzelm@48891: ML_file "Tools/Quickcheck/find_unused_assms.ML"
bulwahn@46589: 
bulwahn@46589: subsection {* Closing up *}
bulwahn@46589: 
haftmann@51143: hide_type narrowing_type narrowing_term narrowing_cons property
haftmann@51143: hide_const map_cons nth error toEnum marker empty Narrowing_cons conv non_empty ensure_testable all exists drawn_from around_zero
bulwahn@46758: hide_const (open) Narrowing_variable Narrowing_constructor "apply" sum cons
bulwahn@46758: hide_fact empty_def cons_def conv.simps non_empty.simps apply_def sum_def ensure_testable_def all_def exists_def
bulwahn@42022: 
bulwahn@45001: end
haftmann@51143: 




