author  bulwahn 
Fri, 11 Mar 2011 15:21:13 +0100  
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(* Title: HOL/Tools/Quickcheck/exhaustive_generators.ML 
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Author: Lukas Bulwahn, TU Muenchen 
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Exhaustive generators for various types. 
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*) 
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signature EXHAUSTIVE_GENERATORS = 
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sig 
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val compile_generator_expr: 
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Proof.context > term > int > term list option * Quickcheck.report option 
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val compile_generator_exprs: 
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Proof.context > term list > (int > term list option) list 
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val put_counterexample: (unit > int > term list option) 
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> Proof.context > Proof.context 
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val put_counterexample_batch: (unit > (int > term list option) list) 
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> Proof.context > Proof.context 
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val smart_quantifier : bool Config.T; 
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val quickcheck_pretty : bool Config.T; 
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val setup: theory > theory 
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end; 
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structure Exhaustive_Generators : EXHAUSTIVE_GENERATORS = 
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struct 
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(* static options *) 
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val define_foundationally = false 
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(* dynamic options *) 
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val (smart_quantifier, setup_smart_quantifier) = 

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Attrib.config_bool "quickcheck_smart_quantifier" (K true) 

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val (quickcheck_pretty, setup_quickcheck_pretty) = 
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Attrib.config_bool "quickcheck_pretty" (K true) 
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(** general term functions **) 
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fun mk_measure f = 
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let 
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val Type ("fun", [T, @{typ nat}]) = fastype_of f 
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in 
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Const (@{const_name Wellfounded.measure}, 
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(T > @{typ nat}) > HOLogic.mk_prodT (T, T) > @{typ bool}) 
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$ f 
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end 
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fun mk_sumcases rT f (Type (@{type_name Sum_Type.sum}, [TL, TR])) = 
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let 
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val lt = mk_sumcases rT f TL 
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val rt = mk_sumcases rT f TR 
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in 
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SumTree.mk_sumcase TL TR rT lt rt 
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end 
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 mk_sumcases _ f T = f T 
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fun mk_undefined T = Const(@{const_name undefined}, T) 
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(** abstract syntax **) 
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fun termifyT T = HOLogic.mk_prodT (T, @{typ "unit => Code_Evaluation.term"}); 
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val size = @{term "i :: code_numeral"} 
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val size_pred = @{term "(i :: code_numeral)  1"} 
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val size_ge_zero = @{term "(i :: code_numeral) > 0"} 
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fun test_function T = Free ("f", termifyT T > @{typ "term list option"}) 
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fun mk_none_continuation (x, y) = 
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let 
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val (T as Type(@{type_name "option"}, [T'])) = fastype_of x 
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in 
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Const (@{const_name "Quickcheck_Exhaustive.orelse"}, T > T > T) 
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$ x $ y 
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end 
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(** datatypes **) 
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(* constructing exhaustive generator instances on datatypes *) 
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exception FUNCTION_TYPE; 
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val exhaustiveN = "exhaustive"; 
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fun exhaustiveT T = (termifyT T > @{typ "Code_Evaluation.term list option"}) 
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> @{typ code_numeral} > @{typ "Code_Evaluation.term list option"} 
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fun check_allT T = (termifyT T > @{typ "Code_Evaluation.term list option"}) 
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> @{typ "Code_Evaluation.term list option"} 
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fun mk_equations thy descr vs tycos exhaustives (Ts, Us) = 
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let 
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fun mk_call T = 
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let 
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val exhaustive = Const (@{const_name "Quickcheck_Exhaustive.exhaustive_class.exhaustive"}, exhaustiveT T) 
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in 
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(T, (fn t => exhaustive $ 
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(HOLogic.split_const (T, @{typ "unit => Code_Evaluation.term"}, @{typ "Code_Evaluation.term list option"}) 
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$ absdummy (T, absdummy (@{typ "unit => Code_Evaluation.term"}, t))) $ size_pred)) 
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end 
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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, (fn t => nth exhaustives k $ 
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(HOLogic.split_const (T, @{typ "unit => Code_Evaluation.term"}, @{typ "Code_Evaluation.term list option"}) 
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$ absdummy (T, absdummy (@{typ "unit => Code_Evaluation.term"}, t))) $ size_pred)) 
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end 
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fun mk_consexpr simpleT (c, xs) = 
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let 
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val (Ts, fns) = split_list xs 
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val constr = Const (c, Ts > simpleT) 
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val bounds = map (fn x => Bound (2 * x + 1)) (((length xs)  1) downto 0) 
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val term_bounds = map (fn x => Bound (2 * x)) (((length xs)  1) downto 0) 
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val Eval_App = Const ("Code_Evaluation.App", HOLogic.termT > HOLogic.termT > HOLogic.termT) 
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val Eval_Const = Const ("Code_Evaluation.Const", HOLogic.literalT > @{typ typerep} > HOLogic.termT) 
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val term = fold (fn u => fn t => Eval_App $ t $ (u $ @{term "()"})) 
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bounds (Eval_Const $ HOLogic.mk_literal c $ HOLogic.mk_typerep (Ts > simpleT)) 
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val start_term = test_function simpleT $ 
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(HOLogic.pair_const simpleT @{typ "unit => Code_Evaluation.term"} 
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$ (list_comb (constr, bounds)) $ absdummy (@{typ unit}, term)) 
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in fold_rev (fn f => fn t => f t) fns start_term end 
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fun mk_rhs exprs = 
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@{term "If :: bool => term list option => term list option => term list option"} 
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$ size_ge_zero $ (foldr1 mk_none_continuation exprs) $ @{term "None :: term list option"} 
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val rhss = 
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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 = map2 (fn t => fn T => t $ test_function T $ size) exhaustives (Ts @ Us); 
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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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end 
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(* foundational definition with the function package *) 
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val less_int_pred = @{lemma "i > 0 ==> Code_Numeral.nat_of ((i :: code_numeral)  1) < Code_Numeral.nat_of i" by auto} 
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fun mk_single_measure T = HOLogic.mk_comp (@{term "Code_Numeral.nat_of"}, 
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Const (@{const_name "Product_Type.snd"}, T > @{typ "code_numeral"})) 
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fun mk_termination_measure T = 
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let 
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val T' = fst (HOLogic.dest_prodT (HOLogic.dest_setT T)) 
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in 
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mk_measure (mk_sumcases @{typ nat} mk_single_measure T') 
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end 
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fun termination_tac ctxt = 
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Function_Relation.relation_tac ctxt mk_termination_measure 1 
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THEN rtac @{thm wf_measure} 1 
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THEN (REPEAT_DETERM (Simplifier.asm_full_simp_tac 
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(HOL_basic_ss addsimps [@{thm in_measure}, @{thm o_def}, @{thm snd_conv}, 
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@{thm nat_mono_iff}, less_int_pred] @ @{thms sum.cases}) 1)) 
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fun pat_completeness_auto ctxt = 
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Pat_Completeness.pat_completeness_tac ctxt 1 
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THEN auto_tac (clasimpset_of ctxt) 
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(* creating the instances *) 
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41918  166 
fun instantiate_exhaustive_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 exhaustive generators ..."; 
169 
val exhaustivesN = map (prefix (exhaustiveN ^ "_")) (names @ auxnames); 

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in 
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thy 
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> Class.instantiation (tycos, vs, @{sort exhaustive}) 
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> (if define_foundationally then 
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let 
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val exhaustives = map2 (fn name => fn T => Free (name, exhaustiveT T)) exhaustivesN (Ts @ Us) 
176 
val eqs = mk_equations thy descr vs tycos exhaustives (Ts, Us) 

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in 
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Function.add_function 
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(map (fn (name, T) => 
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Syntax.no_syn (Binding.conceal (Binding.name name), SOME (exhaustiveT T))) 
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(exhaustivesN ~~ (Ts @ Us))) 

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(map (pair (apfst Binding.conceal Attrib.empty_binding)) eqs) 
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Function_Common.default_config pat_completeness_auto 
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#> snd 
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#> Local_Theory.restore 
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#> (fn lthy => Function.prove_termination NONE (termination_tac lthy) lthy) 
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#> snd 
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end 
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else 
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fold_map (fn (name, T) => Local_Theory.define 
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((Binding.conceal (Binding.name name), NoSyn), 
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(apfst Binding.conceal Attrib.empty_binding, mk_undefined (exhaustiveT T))) 
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#> apfst fst) (exhaustivesN ~~ (Ts @ Us)) 

194 
#> (fn (exhaustives, lthy) => 

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let 
41918  196 
val eqs_t = mk_equations thy descr vs tycos exhaustives (Ts, Us) 
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val eqs = map (fn eq => Goal.prove lthy ["f", "i"] [] eq 
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(fn _ => Skip_Proof.cheat_tac (ProofContext.theory_of lthy))) eqs_t 
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in 
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fold (fn (name, eq) => Local_Theory.note 
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((Binding.conceal (Binding.qualify true prfx 
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(Binding.qualify true name (Binding.name "simps"))), 
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Code.add_default_eqn_attrib :: map (Attrib.internal o K) 
41918  204 
[Simplifier.simp_add, Nitpick_Simps.add]), [eq]) #> snd) (exhaustivesN ~~ eqs) lthy 
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end)) 
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> Class.prove_instantiation_exit (K (Class.intro_classes_tac [])) 
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end handle FUNCTION_TYPE => 
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(Datatype_Aux.message config 
41918  209 
"Creation of exhaustivevalue generators failed because the datatype contains a function type"; 
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thy) 
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(** building and compiling generator expressions **) 
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structure Counterexample = Proof_Data 
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( 
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type T = unit > int > term list option 
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(* FIXME avoid user error with nonuser text *) 
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fun init _ () = error "Counterexample" 
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); 
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val put_counterexample = Counterexample.put; 
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structure Counterexample_Batch = Proof_Data 
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( 
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type T = unit > (int > term list option) list 
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(* FIXME avoid user error with nonuser text *) 
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fun init _ () = error "Counterexample" 
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); 
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val put_counterexample_batch = Counterexample_Batch.put; 
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val target = "Quickcheck"; 
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40907  232 
fun mk_smart_generator_expr ctxt t = 
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let 
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val thy = ProofContext.theory_of ctxt 
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val ((vnames, Ts), t') = apfst split_list (strip_abs t) 
236 
val ([depth_name], ctxt') = Variable.variant_fixes ["depth"] ctxt 

237 
val (names, ctxt'') = Variable.variant_fixes vnames ctxt' 

238 
val (term_names, ctxt''') = Variable.variant_fixes (map (prefix "t_") vnames) ctxt'' 

239 
val depth = Free (depth_name, @{typ code_numeral}) 

240 
val frees = map2 (curry Free) names Ts 

241 
val term_vars = map (fn n => Free (n, @{typ "unit => term"})) term_names 

242 
fun strip_imp (Const(@{const_name HOL.implies},_) $ A $ B) = apfst (cons A) (strip_imp B) 

243 
 strip_imp A = ([], A) 

244 
val (assms, concl) = strip_imp (subst_bounds (rev frees, t')) 

245 
val terms = HOLogic.mk_list @{typ term} (map (fn v => v $ @{term "()"}) term_vars) 

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fun mk_exhaustive_closure (free as Free (_, T), term_var) t = 
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if Sign.of_sort thy (T, @{sort enum}) then 
41918  248 
Const (@{const_name "Quickcheck_Exhaustive.check_all_class.check_all"}, check_allT T) 
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$ (HOLogic.split_const (T, @{typ "unit => term"}, @{typ "term list option"}) 
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$ lambda free (lambda term_var t)) 
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else 
41918  252 
Const (@{const_name "Quickcheck_Exhaustive.exhaustive_class.exhaustive"}, exhaustiveT T) 
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$ (HOLogic.split_const (T, @{typ "unit => term"}, @{typ "term list option"}) 
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$ lambda free (lambda term_var t)) $ depth 
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fun lookup v = the (AList.lookup (op =) (names ~~ (frees ~~ term_vars)) v) 
256 
val none_t = @{term "None :: term list option"} 

257 
fun mk_safe_if (cond, then_t, else_t) = 

41918  258 
@{term "Quickcheck_Exhaustive.catch_match :: term list option => term list option => term list option"} $ 
40907  259 
(@{term "If :: bool => term list option => term list option => term list option"} 
260 
$ cond $ then_t $ else_t) $ none_t; 

261 
fun mk_test_term bound_vars assms = 

262 
let 

263 
fun vars_of t = subtract (op =) bound_vars (Term.add_free_names t []) 

264 
val (vars, check) = 

265 
case assms of [] => 

266 
(vars_of concl, (concl, none_t, @{term "Some :: term list => term list option"} $ terms)) 

267 
 assm :: assms => 

268 
(vars_of assm, (assm, mk_test_term (union (op =) (vars_of assm) bound_vars) assms, none_t)) 

269 
in 

41918  270 
fold_rev mk_exhaustive_closure (map lookup vars) (mk_safe_if check) 
40907  271 
end 
272 
in lambda depth (mk_test_term [] assms) end 

273 

274 
fun mk_generator_expr ctxt t = 

275 
let 

276 
val Ts = (map snd o fst o strip_abs) t; 

277 
val thy = ProofContext.theory_of ctxt 

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val bound_max = length Ts  1; 
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val bounds = map_index (fn (i, ty) => 
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(2 * (bound_max  i) + 1, 2 * (bound_max  i), 2 * i, ty)) Ts; 
40907  281 
val result = list_comb (t, map (fn (i, _, _, _) => Bound i) bounds); 
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val terms = HOLogic.mk_list @{typ term} (map (fn (_, i, _, _) => Bound i $ @{term "()"}) bounds); 
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val check = 
41918  284 
@{term "Quickcheck_Exhaustive.catch_match :: term list option => term list option => term list option"} $ 
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(@{term "If :: bool => term list option => term list option => term list option"} 
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$ result $ @{term "None :: term list option"} $ (@{term "Some :: term list => term list option"} $ terms)) 
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$ @{term "None :: term list option"}; 
41918  288 
fun mk_exhaustive_closure (_, _, i, T) t = 
289 
Const (@{const_name "Quickcheck_Exhaustive.exhaustive_class.exhaustive"}, exhaustiveT T) 

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$ (HOLogic.split_const (T, @{typ "unit => term"}, @{typ "term list option"}) 
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$ absdummy (T, absdummy (@{typ "unit => term"}, t))) $ Bound i 
41918  292 
in Abs ("d", @{typ code_numeral}, fold_rev mk_exhaustive_closure bounds check) end 
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293 

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(** postprocessing of function terms **) 
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fun dest_fun_upd (Const (@{const_name fun_upd}, _) $ t0 $ t1 $ t2) = (t0, (t1, t2)) 
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 dest_fun_upd t = raise TERM ("dest_fun_upd", [t]) 
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fun mk_fun_upd T1 T2 (t1, t2) t = 
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Const (@{const_name fun_upd}, (T1 > T2) > T1 > T2 > T1 > T2) $ t $ t1 $ t2 
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fun dest_fun_upds t = 
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case try dest_fun_upd t of 
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NONE => 
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(case t of 
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Abs (_, _, _) => ([], t) 
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 _ => raise TERM ("dest_fun_upds", [t])) 
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 SOME (t0, (t1, t2)) => apfst (cons (t1, t2)) (dest_fun_upds t0) 
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fun make_fun_upds T1 T2 (tps, t) = fold_rev (mk_fun_upd T1 T2) tps t 
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fun make_set T1 [] = Const (@{const_abbrev Set.empty}, T1 > @{typ bool}) 
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 make_set T1 ((_, @{const False}) :: tps) = make_set T1 tps 
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 make_set T1 ((t1, @{const True}) :: tps) = 
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Const (@{const_name insert}, T1 > (T1 > @{typ bool}) > T1 > @{typ bool}) 
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$ t1 $ (make_set T1 tps) 
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 make_set T1 ((_, t) :: tps) = raise TERM ("make_set", [t]) 
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318 

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fun make_coset T [] = Const (@{const_abbrev UNIV}, T > @{typ bool}) 
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 make_coset T tps = 
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let 
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val U = T > @{typ bool} 
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fun invert @{const False} = @{const True} 
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 invert @{const True} = @{const False} 
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in 
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Const (@{const_name "Groups.minus_class.minus"}, U > U > U) 
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$ Const (@{const_abbrev UNIV}, U) $ make_set T (map (apsnd invert) tps) 
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end 
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329 

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fun make_map T1 T2 [] = Const (@{const_abbrev Map.empty}, T1 > T2) 
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 make_map T1 T2 ((_, Const (@{const_name None}, _)) :: tps) = make_map T1 T2 tps 
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 make_map T1 T2 ((t1, t2) :: tps) = mk_fun_upd T1 T2 (t1, t2) (make_map T1 T2 tps) 
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333 

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fun post_process_term t = 
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let 
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fun map_Abs f t = 
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case t of Abs (x, T, t') => Abs (x, T, f t')  _ => raise TERM ("map_Abs", [t]) 
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fun process_args t = case strip_comb t of 
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(c as Const (_, _), ts) => list_comb (c, map post_process_term ts) 
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in 
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case fastype_of t of 
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Type (@{type_name fun}, [T1, T2]) => 
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(case try dest_fun_upds t of 
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SOME (tps, t) => 
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(map (pairself post_process_term) tps, map_Abs post_process_term t) 
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> (case T2 of 
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@{typ bool} => 
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(case t of 
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Abs(_, _, @{const True}) => fst #> rev #> make_set T1 
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 Abs(_, _, @{const False}) => fst #> rev #> make_coset T1 
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 Abs(_, _, Const (@{const_name undefined}, _)) => fst #> rev #> make_set T1 
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 _ => raise TERM ("post_process_term", [t])) 
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 Type (@{type_name option}, _) => 
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(case t of 
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Abs(_, _, Const(@{const_name None}, _)) => fst #> make_map T1 T2 
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 Abs(_, _, Const (@{const_name undefined}, _)) => fst #> make_map T1 T2 
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 _ => make_fun_upds T1 T2) 
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 _ => make_fun_upds T1 T2) 
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 NONE => process_args t) 
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 _ => process_args t 
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end 
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362 

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(** generator compiliation **) 
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364 

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fun compile_generator_expr ctxt t = 
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let 
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val thy = ProofContext.theory_of ctxt 
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val t' = 
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(if Config.get ctxt smart_quantifier then mk_smart_generator_expr else mk_generator_expr) 
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ctxt t; 
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val compile = Code_Runtime.dynamic_value_strict 
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(Counterexample.get, put_counterexample, "Exhaustive_Generators.put_counterexample") 
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thy (SOME target) (fn proc => fn g => g #> (Option.map o map) proc) t' []; 
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in 
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fn size => rpair NONE (compile size > 
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(if Config.get ctxt quickcheck_pretty then Option.map (map post_process_term) else I)) 
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end; 
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378 

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fun compile_generator_exprs ctxt ts = 
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let 
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val thy = ProofContext.theory_of ctxt 
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val mk_generator_expr = 
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if Config.get ctxt smart_quantifier then mk_smart_generator_expr else mk_generator_expr 
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val ts' = map (mk_generator_expr ctxt) ts; 
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val compiles = Code_Runtime.dynamic_value_strict 
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(Counterexample_Batch.get, put_counterexample_batch, 
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"Exhaustive_Generators.put_counterexample_batch") 
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thy (SOME target) (fn proc => map (fn g => g #> (Option.map o map) proc)) 
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(HOLogic.mk_list @{typ "code_numeral => term list option"} ts') []; 
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in 
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map (fn compile => fn size => compile size > Option.map (map post_process_term)) compiles 
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end; 
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393 

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394 

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(** setup **) 
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396 

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val setup = 
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Datatype.interpretation 
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(Quickcheck_Generators.ensure_sort_datatype (@{sort exhaustive}, instantiate_exhaustive_datatype)) 
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#> setup_smart_quantifier 
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#> setup_quickcheck_pretty 
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#> Context.theory_map (Quickcheck.add_generator ("exhaustive", compile_generator_expr)) 
403 
#> Context.theory_map (Quickcheck.add_batch_generator ("exhaustive", compile_generator_exprs)); 

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end; 