author  noschinl 
Thu, 18 Aug 2011 13:37:41 +0200  
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permissions  rwrr 
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(* Title: HOL/Tools/Quickcheck/narrowing_generators.ML 
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Author: Lukas Bulwahn, TU Muenchen 
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Narrowingbased counterexample generation. 
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*) 
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signature NARROWING_GENERATORS = 
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sig 
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val allow_existentials : bool Config.T 
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val finite_functions : bool Config.T 
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val overlord : bool Config.T 
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val active : bool Config.T 
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val test_term: Proof.context > bool * bool > term * term list > Quickcheck.result 
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datatype counterexample = Universal_Counterexample of (term * counterexample) 
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 Existential_Counterexample of (term * counterexample) list 
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 Empty_Assignment 
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val put_counterexample: (unit > term list option) > Proof.context > Proof.context 
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val put_existential_counterexample : (unit > counterexample option) > Proof.context > Proof.context 
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val setup: theory > theory 
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end; 

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structure Narrowing_Generators : NARROWING_GENERATORS = 
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struct 
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(* configurations *) 
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val allow_existentials = Attrib.setup_config_bool @{binding quickcheck_allow_existentials} (K true) 
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val finite_functions = Attrib.setup_config_bool @{binding quickcheck_finite_functions} (K true) 
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val overlord = Attrib.setup_config_bool @{binding quickcheck_narrowing_overlord} (K false) 
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(* partial_term_of instances *) 
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fun mk_partial_term_of (x, T) = 
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Const (@{const_name Quickcheck_Narrowing.partial_term_of_class.partial_term_of}, 
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Term.itselfT T > @{typ narrowing_term} > @{typ Code_Evaluation.term}) 
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$ Const ("TYPE", Term.itselfT T) $ x 
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(** formal definition **) 
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fun add_partial_term_of tyco raw_vs thy = 
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let 
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val vs = map (fn (v, _) => (v, @{sort typerep})) raw_vs; 
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val ty = Type (tyco, map TFree vs); 
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val lhs = Const (@{const_name partial_term_of}, 
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Term.itselfT ty > @{typ narrowing_term} > @{typ Code_Evaluation.term}) 
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$ Free ("x", Term.itselfT ty) $ Free ("t", @{typ narrowing_term}); 
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val rhs = @{term "undefined :: Code_Evaluation.term"}; 
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val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)); 
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fun triv_name_of t = (fst o dest_Free o fst o strip_comb o fst 
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o HOLogic.dest_eq o HOLogic.dest_Trueprop) t ^ "_triv"; 
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in 
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thy 
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> Class.instantiation ([tyco], vs, @{sort partial_term_of}) 
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> `(fn lthy => Syntax.check_term lthy eq) 
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> (fn eq => Specification.definition (NONE, ((Binding.name (triv_name_of eq), []), eq))) 
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> snd 
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> Class.prove_instantiation_exit (K (Class.intro_classes_tac [])) 
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end; 
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fun ensure_partial_term_of (tyco, (raw_vs, _)) thy = 
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let 
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val need_inst = not (can (Sorts.mg_domain (Sign.classes_of thy) tyco) @{sort partial_term_of}) 
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andalso can (Sorts.mg_domain (Sign.classes_of thy) tyco) @{sort typerep}; 
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in if need_inst then add_partial_term_of tyco raw_vs thy else thy end; 
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(** code equations for datatypes **) 
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fun mk_partial_term_of_eq thy ty (i, (c, (_, tys))) = 
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let 
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val frees = map Free (Name.invent_names Name.context "a" (map (K @{typ narrowing_term}) tys)) 
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val narrowing_term = @{term "Quickcheck_Narrowing.Ctr"} $ HOLogic.mk_number @{typ code_int} i 
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$ (HOLogic.mk_list @{typ narrowing_term} (rev frees)) 
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val rhs = fold (fn u => fn t => @{term "Code_Evaluation.App"} $ t $ u) 
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(map mk_partial_term_of (frees ~~ tys)) 
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(@{term "Code_Evaluation.Const"} $ HOLogic.mk_literal c $ HOLogic.mk_typerep (tys > ty)) 
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val insts = 
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map (SOME o Thm.cterm_of thy o map_types Logic.unvarifyT_global o Logic.varify_global) 
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[Free ("ty", Term.itselfT ty), narrowing_term, rhs] 
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val cty = Thm.ctyp_of thy ty; 
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in 
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@{thm partial_term_of_anything} 
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> Drule.instantiate' [SOME cty] insts 
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> Thm.varifyT_global 
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end 
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fun add_partial_term_of_code tyco raw_vs raw_cs thy = 
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let 
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val algebra = Sign.classes_of thy; 
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val vs = map (fn (v, sort) => 
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(v, curry (Sorts.inter_sort algebra) @{sort typerep} sort)) raw_vs; 
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val ty = Type (tyco, map TFree vs); 
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val cs = (map o apsnd o apsnd o map o map_atyps) 
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(fn TFree (v, _) => TFree (v, (the o AList.lookup (op =) vs) v)) raw_cs; 
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val const = AxClass.param_of_inst thy (@{const_name partial_term_of}, tyco); 
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val var_insts = map (SOME o Thm.cterm_of thy o map_types Logic.unvarifyT_global o Logic.varify_global) 
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[Free ("ty", Term.itselfT ty), @{term "Quickcheck_Narrowing.Var p tt"}, 
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@{term "Code_Evaluation.Free (STR ''_'')"} $ HOLogic.mk_typerep ty] 
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val var_eq = 
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@{thm partial_term_of_anything} 
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> Drule.instantiate' [SOME (Thm.ctyp_of thy ty)] var_insts 
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> Thm.varifyT_global 
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val eqs = var_eq :: map_index (mk_partial_term_of_eq thy ty) cs; 
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in 
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thy 
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> Code.del_eqns const 
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> fold Code.add_eqn eqs 
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end; 
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fun ensure_partial_term_of_code (tyco, (raw_vs, cs)) thy = 
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111 
let 
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val has_inst = can (Sorts.mg_domain (Sign.classes_of thy) tyco) @{sort partial_term_of}; 
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in if has_inst then add_partial_term_of_code tyco raw_vs cs thy else thy end; 
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114 

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115 

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(* narrowing generators *) 
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117 

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(** narrowing specific names and types **) 
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120 
exception FUNCTION_TYPE; 
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121 

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val narrowingN = "narrowing"; 
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123 

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fun narrowingT T = 
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@{typ Quickcheck_Narrowing.code_int} > Type (@{type_name Quickcheck_Narrowing.cons}, [T]) 
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126 

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fun mk_empty T = Const (@{const_name Quickcheck_Narrowing.empty}, narrowingT T) 
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128 

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fun mk_cons c T = Const (@{const_name Quickcheck_Narrowing.cons}, T > narrowingT T) $ Const (c, T) 
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fun mk_apply (T, t) (U, u) = 
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let 
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val (_, U') = dest_funT U 
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in 
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(U', Const (@{const_name Quickcheck_Narrowing.apply}, 
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narrowingT U > narrowingT T > narrowingT U') $ u $ t) 
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137 
end 
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138 

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fun mk_sum (t, u) = 
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let 
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val T = fastype_of t 
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in 
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Const (@{const_name Quickcheck_Narrowing.sum}, T > T > T) $ t $ u 
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144 
end 
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(** deriving narrowing instances **) 
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147 

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fun mk_equations descr vs tycos narrowings (Ts, Us) = 
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let 
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fun mk_call T = 
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(T, Const (@{const_name "Quickcheck_Narrowing.narrowing_class.narrowing"}, narrowingT T)) 
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fun mk_aux_call fTs (k, _) (tyco, Ts) = 
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let 
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val T = Type (tyco, Ts) 
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val _ = if not (null fTs) then raise FUNCTION_TYPE else () 
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in 
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(T, nth narrowings k) 
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158 
end 
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159 
fun mk_consexpr simpleT (c, xs) = 
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let 
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val Ts = map fst xs 
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in snd (fold mk_apply xs (Ts > simpleT, mk_cons c (Ts > simpleT))) end 
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fun mk_rhs exprs = foldr1 mk_sum exprs 
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val rhss = 
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165 
Datatype_Aux.interpret_construction descr vs 
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{ atyp = mk_call, dtyp = mk_aux_call } 
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> (map o apfst) Type 
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> map (fn (T, cs) => map (mk_consexpr T) cs) 
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> map mk_rhs 
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val lhss = narrowings 
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val eqs = map (HOLogic.mk_Trueprop o HOLogic.mk_eq) (lhss ~~ rhss) 
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in 
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eqs 
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174 
end 
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175 

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fun contains_recursive_type_under_function_types xs = 
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exists (fn (_, (_, _, cs)) => cs > exists (snd #> exists (fn dT => 
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(case Datatype_Aux.strip_dtyp dT of (_ :: _, Datatype.DtRec _) => true  _ => false)))) xs 
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fun instantiate_narrowing_datatype config descr vs tycos prfx (names, auxnames) (Ts, Us) thy = 
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let 
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val _ = Datatype_Aux.message config "Creating narrowing generators ..."; 
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val narrowingsN = map (prefix (narrowingN ^ "_")) (names @ auxnames); 
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184 
in 
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if not (contains_recursive_type_under_function_types descr) then 
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186 
thy 
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> Class.instantiation (tycos, vs, @{sort narrowing}) 
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> Quickcheck_Common.define_functions 
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(fn narrowings => mk_equations descr vs tycos narrowings (Ts, Us), NONE) 
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prfx [] narrowingsN (map narrowingT (Ts @ Us)) 
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> Class.prove_instantiation_exit (K (Class.intro_classes_tac [])) 
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else 
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thy 
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194 
end; 
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195 

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196 
(* testing framework *) 
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197 

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val target = "Haskell_Quickcheck" 
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199 

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(** invocation of Haskell interpreter **) 
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val narrowing_engine = 
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Context.>>> (Context.map_theory_result 
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(Thy_Load.use_file (Path.explode "Tools/Quickcheck/Narrowing_Engine.hs"))) 
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205 

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val pnf_narrowing_engine = 
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Context.>>> (Context.map_theory_result 
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(Thy_Load.use_file (Path.explode "Tools/Quickcheck/PNF_Narrowing_Engine.hs"))) 
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210 
fun exec verbose code = 

211 
ML_Context.exec (fn () => Secure.use_text ML_Env.local_context (0, "generated code") verbose code) 

212 

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fun with_overlord_dir name f = 
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let 
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val path = 
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Path.append (Path.explode "$ISABELLE_HOME_USER") (Path.basic (name ^ serial_string ())) 

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val _ = Isabelle_System.mkdirs path; 
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in Exn.release (Exn.capture f path) end; 
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fun elapsed_time description e = 
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let val ({elapsed, ...}, result) = Timing.timing e () 
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in (result, (description, Time.toMilliseconds elapsed)) end 
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223 

43585  224 
fun value (contains_existentials, opts) ctxt cookie (code, value_name) = 
41905  225 
let 
43585  226 
val (limit_time, is_interactive, timeout, quiet, size) = opts 
227 
val (get, put, put_ml) = cookie 

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fun message s = if quiet then () else Output.urgent_message s 
43585  229 
val current_size = Unsynchronized.ref 0 
230 
val current_result = Unsynchronized.ref Quickcheck.empty_result 

231 
fun excipit () = 

232 
"Quickcheck ran out of time while testing at size " ^ string_of_int (!current_size) 

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val tmp_prefix = "Quickcheck_Narrowing" 
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val with_tmp_dir = 
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if Config.get ctxt overlord then with_overlord_dir else Isabelle_System.with_tmp_dir 
41905  236 
fun run in_path = 
237 
let 

238 
val code_file = Path.append in_path (Path.basic "Code.hs") 

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val narrowing_engine_file = Path.append in_path (Path.basic "Narrowing_Engine.hs") 
41905  240 
val main_file = Path.append in_path (Path.basic "Main.hs") 
241 
val main = "module Main where {\n\n" ^ 

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"import System;\n" ^ 
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"import Narrowing_Engine;\n" ^ 
41905  244 
"import Code;\n\n" ^ 
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"main = getArgs >>= \\[size] > Narrowing_Engine.depthCheck (read size) (Code.value ())\n\n" ^ 
41905  246 
"}\n" 
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val code' = prefix "module Code where {\n\ndata Typerep = Typerep String [Typerep];\n" 
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(unprefix "module Code where {" code) 
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val _ = File.write code_file code' 
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val _ = File.write narrowing_engine_file 
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(if contains_existentials then pnf_narrowing_engine else narrowing_engine) 
41905  252 
val _ = File.write main_file main 
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val executable = File.shell_path (Path.append in_path (Path.basic "isabelle_quickcheck_narrowing")) 
44321  254 
val cmd = "exec \"$ISABELLE_GHC\" XRankNTypes XScopedTypeVariables " ^ 
41946  255 
(space_implode " " (map File.shell_path [code_file, narrowing_engine_file, main_file])) ^ 
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" o " ^ executable ^ ";" 
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val (result, compilation_time) = 
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elapsed_time "Haskell compilation" (fn () => Isabelle_System.bash cmd) 
43585  259 
val _ = Quickcheck.add_timing compilation_time current_result 
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val _ = if Isabelle_System.bash cmd <> 0 then error "Compilation with GHC failed" else () 
43585  261 
fun with_size k = 
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if k > size then 
43585  263 
(NONE, !current_result) 
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else 
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let 
43910  266 
val _ = message ("[QuickcheckNarrowing] Test data size: " ^ string_of_int k) 
43585  267 
val _ = current_size := k 
268 
val ((response, _), timing) = elapsed_time ("execution of size " ^ string_of_int k) 

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(fn () => Isabelle_System.bash_output (executable ^ " " ^ string_of_int k)) 
43585  270 
val _ = Quickcheck.add_timing timing current_result 
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271 
in 
43585  272 
if response = "NONE\n" then 
273 
with_size (k + 1) 

274 
else 

275 
let 

276 
val output_value = the_default "NONE" 

277 
(try (snd o split_last o filter_out (fn s => s = "") o split_lines) response) 

278 
> translate_string (fn s => if s = "\\" then "\\\\" else s) 

279 
val ml_code = "\nval _ = Context.set_thread_data (SOME (Context.map_proof (" ^ put_ml 

280 
^ " (fn () => " ^ output_value ^ ")) (ML_Context.the_generic_context ())))"; 

281 
val ctxt' = ctxt 

282 
> put (fn () => error ("Bad evaluation for " ^ quote put_ml)) 

283 
> Context.proof_map (exec false ml_code); 

284 
in (get ctxt' (), !current_result) end 

285 
end 

286 
in with_size 0 end 

287 
in 

43915  288 
(*Quickcheck.limit timeout (limit_time, is_interactive) 
289 
(fn () =>*) with_tmp_dir tmp_prefix run 

290 
(*(fn () => (message (excipit ()); (NONE, !current_result))) ()*) 

43585  291 
end; 
41905  292 

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fun dynamic_value_strict opts cookie thy postproc t = 
41905  294 
let 
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val ctxt = Proof_Context.init_global thy 
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fun evaluator naming program ((_, vs_ty), t) deps = Exn.interruptible_capture (value opts ctxt cookie) 
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(Code_Target.evaluator thy target naming program deps (vs_ty, t)); 
41905  298 
in Exn.release (Code_Thingol.dynamic_value thy (Exn.map_result o postproc) evaluator t) end; 
299 

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(** counterexample generator **) 
41905  301 

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structure Counterexample = Proof_Data 
41905  303 
( 
304 
type T = unit > term list option 

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fun init _ () = error "Counterexample" 
41905  306 
) 
307 

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datatype counterexample = Universal_Counterexample of (term * counterexample) 
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 Existential_Counterexample of (term * counterexample) list 
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 Empty_Assignment 
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311 

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fun map_counterexample f Empty_Assignment = Empty_Assignment 
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 map_counterexample f (Universal_Counterexample (t, c)) = 
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Universal_Counterexample (f t, map_counterexample f c) 
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 map_counterexample f (Existential_Counterexample cs) = 
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Existential_Counterexample (map (fn (t, c) => (f t, map_counterexample f c)) cs) 
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317 

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structure Existential_Counterexample = Proof_Data 
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( 
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type T = unit > counterexample option 
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fun init _ () = error "Counterexample" 
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322 
) 
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323 

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val put_existential_counterexample = Existential_Counterexample.put 
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325 

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val put_counterexample = Counterexample.put 
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327 

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328 
fun finitize_functions (xTs, t) = 
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let 
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val (names, boundTs) = split_list xTs 
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331 
fun mk_eval_ffun dT rT = 
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Const (@{const_name "Quickcheck_Narrowing.eval_ffun"}, 
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Type (@{type_name "Quickcheck_Narrowing.ffun"}, [dT, rT]) > dT > rT) 
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fun mk_eval_cfun dT rT = 
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Const (@{const_name "Quickcheck_Narrowing.eval_cfun"}, 
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Type (@{type_name "Quickcheck_Narrowing.cfun"}, [rT]) > dT > rT) 
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fun eval_function (T as Type (@{type_name fun}, [dT, rT])) = 
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338 
let 
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val (rt', rT') = eval_function rT 
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in 
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341 
case dT of 
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Type (@{type_name fun}, _) => 
44241  343 
(fn t => absdummy dT (rt' (mk_eval_cfun dT rT' $ incr_boundvars 1 t $ Bound 0)), 
344 
Type (@{type_name "Quickcheck_Narrowing.cfun"}, [rT'])) 

345 
 _ => 

346 
(fn t => absdummy dT (rt' (mk_eval_ffun dT rT' $ incr_boundvars 1 t $ Bound 0)), 

347 
Type (@{type_name "Quickcheck_Narrowing.ffun"}, [dT, rT'])) 

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end 
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 eval_function T = (I, T) 
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350 
val (tt, boundTs') = split_list (map eval_function boundTs) 
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val t' = subst_bounds (map2 (fn f => fn x => f x) (rev tt) (map_index (Bound o fst) boundTs), t) 
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352 
in 
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(names ~~ boundTs', t') 
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354 
end 
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355 

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356 
(** tester **) 
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357 

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358 
val rewrs = 
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359 
map (swap o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of) 
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360 
(@{thms all_simps} @ @{thms ex_simps}) 
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@ map (HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of) 
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362 
[@{thm iff_conv_conj_imp}, @{thm not_ex}, @{thm not_all}] 
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363 

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364 
fun make_pnf_term thy t = Pattern.rewrite_term thy rewrs [] t 
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365 

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366 
fun strip_quantifiers (Const (@{const_name Ex}, _) $ Abs (x, T, t)) = 
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apfst (cons (@{const_name Ex}, (x, T))) (strip_quantifiers t) 
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 strip_quantifiers (Const (@{const_name All}, _) $ Abs (x, T, t)) = 
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apfst (cons (@{const_name All}, (x, T))) (strip_quantifiers t) 
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370 
 strip_quantifiers t = ([], t) 
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371 

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fun contains_existentials t = exists (fn (Q, _) => Q = @{const_name Ex}) (fst (strip_quantifiers t)) 
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373 

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374 
fun mk_property qs t = 
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375 
let 
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fun enclose (@{const_name Ex}, (x, T)) t = 
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Const (@{const_name Quickcheck_Narrowing.exists}, (T > @{typ property}) > @{typ property}) 
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$ Abs (x, T, t) 
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 enclose (@{const_name All}, (x, T)) t = 
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380 
Const (@{const_name Quickcheck_Narrowing.all}, (T > @{typ property}) > @{typ property}) 
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381 
$ Abs (x, T, t) 
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382 
in 
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383 
fold_rev enclose qs (@{term Quickcheck_Narrowing.Property} $ 
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384 
(list_comb (t , map Bound (((length qs)  1) downto 0)))) 
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385 
end 
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386 

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387 
fun mk_case_term ctxt p ((@{const_name Ex}, (x, T)) :: qs') (Existential_Counterexample cs) = 
43317  388 
Datatype.make_case ctxt Datatype_Case.Quiet [] (Free (x, T)) (map (fn (t, c) => 
389 
(t, mk_case_term ctxt (p  1) qs' c)) cs) 

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390 
 mk_case_term ctxt p ((@{const_name All}, (x, T)) :: qs') (Universal_Counterexample (t, c)) = 
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391 
if p = 0 then t else mk_case_term ctxt (p  1) qs' c 
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392 

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393 
fun mk_terms ctxt qs result = 
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394 
let 
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395 
val 
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396 
ps = filter (fn (_, (@{const_name All}, _)) => true  _ => false) (map_index I qs) 
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397 
in 
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398 
map (fn (p, (_, (x, T))) => (x, mk_case_term ctxt p qs result)) ps 
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399 
end 
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400 

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401 
fun test_term ctxt (limit_time, is_interactive) (t, eval_terms) = 
41905  402 
let 
43585  403 
fun dest_result (Quickcheck.Result r) = r 
404 
val opts = 

405 
(limit_time, is_interactive, seconds (Config.get ctxt Quickcheck.timeout), 

406 
Config.get ctxt Quickcheck.quiet, Config.get ctxt Quickcheck.size) 

42361  407 
val thy = Proof_Context.theory_of ctxt 
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408 
val t' = fold_rev (fn (x, T) => fn t => HOLogic.mk_all (x, T, t)) (Term.add_frees t []) t 
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409 
val pnf_t = make_pnf_term thy t' 
41905  410 
in 
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411 
if Config.get ctxt allow_existentials andalso contains_existentials pnf_t then 
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412 
let 
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413 
fun wrap f (qs, t) = 
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414 
let val (qs1, qs2) = split_list qs in 
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415 
apfst (map2 pair qs1) (f (qs2, t)) end 
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416 
val finitize = if Config.get ctxt finite_functions then wrap finitize_functions else I 
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417 
val (qs, prop_t) = finitize (strip_quantifiers pnf_t) 
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418 
val prop_term = fold_rev (fn (_, (x, T)) => fn t => Abs (x, T, t)) qs prop_t 
43619
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419 
(* FIXME proper naming convention for local_theory *) 
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420 
val ((prop_def, _), ctxt') = 
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421 
Local_Theory.define ((Binding.conceal @{binding test_property}, NoSyn), 
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422 
((Binding.conceal Binding.empty, [Code.add_default_eqn_attrib]), prop_term)) ctxt 
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423 
val (prop_def', thy') = Local_Theory.exit_result_global Morphism.term (prop_def, ctxt') 
43585  424 
val (counterexample, result) = dynamic_value_strict (true, opts) 
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425 
(Existential_Counterexample.get, Existential_Counterexample.put, 
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426 
"Narrowing_Generators.put_existential_counterexample") 
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427 
thy' (apfst o Option.map o map_counterexample) (mk_property qs prop_def') 
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428 
in 
43585  429 
Quickcheck.Result 
430 
{counterexample = Option.map (mk_terms ctxt' qs) counterexample, 

431 
evaluation_terms = Option.map (K []) counterexample, 

432 
timings = #timings (dest_result result), reports = #reports (dest_result result)} 

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433 
end 
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434 
else 
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435 
let 
44241  436 
val t' = fold_rev absfree (Term.add_frees t []) t 
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437 
fun wrap f t = list_abs (f (strip_abs t)) 
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438 
val finitize = if Config.get ctxt finite_functions then wrap finitize_functions else I 
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439 
fun ensure_testable t = 
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440 
Const (@{const_name Quickcheck_Narrowing.ensure_testable}, fastype_of t > fastype_of t) $ t 
43585  441 
val (counterexample, result) = dynamic_value_strict (false, opts) 
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442 
(Counterexample.get, Counterexample.put, "Narrowing_Generators.put_counterexample") 
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443 
thy (apfst o Option.map o map) (ensure_testable (finitize t')) 
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444 
in 
43585  445 
Quickcheck.Result 
446 
{counterexample = Option.map ((curry (op ~~)) (Term.add_free_names t [])) counterexample, 

447 
evaluation_terms = Option.map (K []) counterexample, 

448 
timings = #timings (dest_result result), reports = #reports (dest_result result)} 

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449 
end 
41905  450 
end; 
451 

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452 
fun test_goals ctxt (limit_time, is_interactive) insts goals = 
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453 
if (not (getenv "ISABELLE_GHC" = "")) then 
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454 
let 
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455 
val correct_inst_goals = Quickcheck.instantiate_goals ctxt insts goals 
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456 
in 
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457 
Quickcheck.collect_results (test_term ctxt (limit_time, is_interactive)) (maps (map snd) correct_inst_goals) [] 
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458 
end 
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459 
else 
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460 
(if Config.get ctxt Quickcheck.quiet then () else Output.urgent_message 
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461 
("Environment variable ISABELLE_GHC is not set. To use narrowingbased quickcheck, please set " 
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462 
^ "this variable to your GHC Haskell compiler in your settings file. " 
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463 
^ "To deactivate narrowingbased quickcheck, set quickcheck_narrowing_active to false."); 
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464 
[Quickcheck.empty_result]) 
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465 

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466 
(* setup *) 
41905  467 

44272  468 
val active = Attrib.setup_config_bool @{binding quickcheck_narrowing_active} (K true); 
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469 

41905  470 
val setup = 
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471 
Code.datatype_interpretation ensure_partial_term_of 
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472 
#> Code.datatype_interpretation ensure_partial_term_of_code 
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473 
#> Datatype.interpretation (Quickcheck_Common.ensure_sort_datatype 
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474 
(((@{sort typerep}, @{sort term_of}), @{sort narrowing}), instantiate_narrowing_datatype)) 
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475 
#> Context.theory_map (Quickcheck.add_tester ("narrowing", (active, test_goals))) 
41905  476 

44272  477 
end; 