author  bulwahn 
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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 * term list) list > int list > term list option * Quickcheck.report option 
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val compile_generator_exprs: Proof.context > term list > (int > term list option) list 
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val compile_validator_exprs: Proof.context > term list > (int > bool) list 
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val put_counterexample: (unit > int > 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 put_validator_batch: (unit > (int > bool) list) > 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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(* 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) $ 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 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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(* creating the instances *) 
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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..."; 
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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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> Quickcheck_Common.define_functions 
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(fn exhaustives => mk_equations descr vs tycos exhaustives (Ts, Us), SOME termination_tac) 
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prfx ["f", "i"] exhaustivesN (map exhaustiveT (Ts @ Us)) 
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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 
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"Creation of exhaustive generators failed because the datatype contains a function type"; 
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thy) 
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(* constructing bounded_forall instances on datatypes *) 
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val bounded_forallN = "bounded_forall"; 
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fun bounded_forallT T = (T > @{typ bool}) > @{typ code_numeral} > @{typ bool} 
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fun mk_bounded_forall_equations descr vs tycos bounded_foralls (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 bounded_forall = 
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Const (@{const_name "Quickcheck_Exhaustive.bounded_forall_class.bounded_forall"}, 
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bounded_forallT T) 
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in 
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(T, (fn t => bounded_forall $ absdummy (T, 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 bounded_foralls k $ absdummy (T, 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 Bound (((length xs)  1) downto 0) 
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val start_term = Free ("P", simpleT > @{typ bool}) $ list_comb (constr, bounds) 
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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 => bool => bool => bool"} $ size_ge_zero $ 
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(foldr1 HOLogic.mk_disj exprs) $ @{term "True"} 
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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 = 
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map2 (fn t => fn T => t $ Free ("P", T > @{typ bool}) $ size) bounded_foralls (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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(* creating the bounded_forall instances *) 
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fun instantiate_bounded_forall_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 bounded universal quantifiers..."; 
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val bounded_forallsN = map (prefix (bounded_forallN ^ "_")) (names @ auxnames); 
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in 
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thy 
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> Class.instantiation (tycos, vs, @{sort bounded_forall}) 
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> Quickcheck_Common.define_functions 
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(fn bounded_foralls => 
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mk_bounded_forall_equations descr vs tycos bounded_foralls (Ts, Us), NONE) 
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prfx ["P", "i"] bounded_forallsN (map bounded_forallT (Ts @ Us)) 
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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 
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"Creation of bounded universal quantifiers 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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fun mk_generator_expr ctxt (t, eval_terms) = 
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let 
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val thy = ProofContext.theory_of ctxt 
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val ctxt' = Variable.auto_fixes t ctxt 
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val names = Term.add_free_names t [] 
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val frees = map Free (Term.add_frees t []) 
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val ([depth_name], ctxt'') = Variable.variant_fixes ["depth"] ctxt' 
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val (term_names, ctxt''') = Variable.variant_fixes (map (prefix "t_") names) ctxt'' 
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val depth = Free (depth_name, @{typ code_numeral}) 
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val term_vars = map (fn n => Free (n, @{typ "unit => term"})) term_names 
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val terms = HOLogic.mk_list @{typ term} (map (fn v => v $ @{term "()"}) term_vars) 
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val appendC = @{term "List.append :: term list => term list => term list"} 
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val return = @{term "Some :: term list => term list option"} $ (appendC $ terms $ 
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(HOLogic.mk_list @{typ "term"} (map (fn t => HOLogic.mk_term_of (fastype_of t) t) eval_terms))) 
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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 
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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 
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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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val none_t = @{term "None :: term list option"} 
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fun mk_safe_if (cond, then_t, else_t) = 
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@{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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$ cond $ then_t $ else_t) $ none_t; 
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fun lookup v = the (AList.lookup (op =) (names ~~ (frees ~~ term_vars)) v) 
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fun mk_naive_test_term t = 
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fold_rev mk_exhaustive_closure (frees ~~ term_vars) (mk_safe_if (t, none_t, return)) 
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fun mk_smart_test_term' concl bound_vars assms = 
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let 
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fun vars_of t = subtract (op =) bound_vars (Term.add_free_names t []) 
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val (vars, check) = 
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case assms of [] => (vars_of concl, (concl, none_t, return)) 
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 assm :: assms => (vars_of assm, (assm, 
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mk_smart_test_term' concl (union (op =) (vars_of assm) bound_vars) assms, none_t)) 
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in 
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fold_rev mk_exhaustive_closure (map lookup vars) (mk_safe_if check) 
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end 
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fun mk_smart_test_term t = 
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let 
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val (assms, concl) = Quickcheck_Common.strip_imp t 
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in 
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mk_smart_test_term' concl [] assms 
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end 
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val mk_test_term = 
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if Config.get ctxt smart_quantifier then mk_smart_test_term else mk_naive_test_term 
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in lambda depth (mk_test_term t) end 
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val mk_parametric_generator_expr = 
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Quickcheck_Common.gen_mk_parametric_generator_expr 
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((mk_generator_expr, absdummy (@{typ "code_numeral"}, @{term "None :: term list option"})), 
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@{typ "code_numeral => term list option"}) 
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fun mk_validator_expr ctxt t = 
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let 
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fun bounded_forallT T = (T > @{typ bool}) > @{typ code_numeral} > @{typ bool} 
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val thy = ProofContext.theory_of ctxt 
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val ctxt' = Variable.auto_fixes t ctxt 
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val ([depth_name], ctxt'') = Variable.variant_fixes ["depth"] ctxt' 
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val depth = Free (depth_name, @{typ code_numeral}) 
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fun mk_bounded_forall (s, T) t = 
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Const (@{const_name "Quickcheck_Exhaustive.bounded_forall_class.bounded_forall"}, bounded_forallT T) 
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$ lambda (Free (s, T)) t $ depth 
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fun mk_implies (cond, then_t) = 
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@{term "If :: bool => bool => bool => bool"} $ cond $ then_t $ @{term False} 
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fun mk_naive_test_term t = fold_rev mk_bounded_forall (Term.add_frees t []) t 
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fun mk_smart_test_term' concl bound_frees assms = 
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let 
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fun vars_of t = subtract (op =) bound_frees (Term.add_frees t []) 
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val (vars, check) = 
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case assms of [] => (vars_of concl, concl) 
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 assm :: assms => (vars_of assm, mk_implies (assm, 
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mk_smart_test_term' concl (union (op =) (vars_of assm) bound_frees) assms)) 
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in 
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fold_rev mk_bounded_forall vars check 
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end 
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fun mk_smart_test_term t = 
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let 
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val (assms, concl) = Quickcheck_Common.strip_imp t 
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in 
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mk_smart_test_term' concl [] assms 
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end 
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val mk_test_term = 
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if Config.get ctxt smart_quantifier then mk_smart_test_term else mk_naive_test_term 
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in lambda depth (mk_test_term t) end 
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(** generator compiliation **) 
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structure Counterexample = Proof_Data 
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( 
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type T = unit > int > 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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structure Validator_Batch = Proof_Data 
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( 
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type T = unit > (int > bool) 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_validator_batch = Validator_Batch.put; 
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val target = "Quickcheck"; 
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358 

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fun compile_generator_expr ctxt ts = 
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let 
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val thy = ProofContext.theory_of ctxt 
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val t' = mk_parametric_generator_expr ctxt ts; 
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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 => 
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fn card => fn size => g card size > (Option.map o map) proc) t' [] 
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in 
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fn [card, size] => rpair NONE (compile card size > 
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(if Config.get ctxt quickcheck_pretty then 
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Option.map (map Quickcheck_Common.post_process_term) else I)) 
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371 
end; 
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372 

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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 ts' = map (fn t => mk_generator_expr ctxt (t, [])) 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 
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> Option.map (map Quickcheck_Common.post_process_term)) compiles 
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end; 
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386 

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fun compile_validator_exprs ctxt ts = 
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let 
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val thy = ProofContext.theory_of ctxt 
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val ts' = map (mk_validator_expr ctxt) ts; 
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in 
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Code_Runtime.dynamic_value_strict 
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(Validator_Batch.get, put_validator_batch, "Exhaustive_Generators.put_validator_batch") 
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thy (SOME target) (K I) (HOLogic.mk_list @{typ "code_numeral => bool"} ts') [] 
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395 
end; 
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396 

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(** setup **) 
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398 

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val setup = 
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Datatype.interpretation (Quickcheck_Common.ensure_sort_datatype 
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(((@{sort typerep}, @{sort term_of}), @{sort exhaustive}), instantiate_exhaustive_datatype)) 
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#> Datatype.interpretation (Quickcheck_Common.ensure_sort_datatype 
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(((@{sort type}, @{sort type}), @{sort bounded_forall}), instantiate_bounded_forall_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)) 
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#> Context.theory_map (Quickcheck.add_batch_generator ("exhaustive", compile_generator_exprs)) 
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#> Context.theory_map (Quickcheck.add_batch_validator ("exhaustive", compile_validator_exprs)); 
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410 
end; 