(* Title: HOL/Tools/exhaustive_generators.ML
Author: Lukas Bulwahn, TU Muenchen
Exhaustive generators for various types.
*)
signature EXHAUSTIVE_GENERATORS =
sig
val compile_generator_expr:
Proof.context -> term -> int -> term list option * Quickcheck.report option
val compile_generator_exprs:
Proof.context -> term list -> (int -> term list option) list
val put_counterexample: (unit -> int -> term list option)
-> Proof.context -> Proof.context
val put_counterexample_batch: (unit -> (int -> term list option) list)
-> Proof.context -> Proof.context
val smart_quantifier : bool Config.T;
val quickcheck_pretty : bool Config.T;
val setup: theory -> theory
end;
structure Exhaustive_Generators : EXHAUSTIVE_GENERATORS =
struct
(* static options *)
val define_foundationally = false
(* dynamic options *)
val (smart_quantifier, setup_smart_quantifier) =
Attrib.config_bool "quickcheck_smart_quantifier" (K true)
val (quickcheck_pretty, setup_quickcheck_pretty) =
Attrib.config_bool "quickcheck_pretty" (K true)
(** general term functions **)
fun mk_measure f =
let
val Type ("fun", [T, @{typ nat}]) = fastype_of f
in
Const (@{const_name Wellfounded.measure},
(T --> @{typ nat}) --> HOLogic.mk_prodT (T, T) --> @{typ bool})
$ f
end
fun mk_sumcases rT f (Type (@{type_name Sum_Type.sum}, [TL, TR])) =
let
val lt = mk_sumcases rT f TL
val rt = mk_sumcases rT f TR
in
SumTree.mk_sumcase TL TR rT lt rt
end
| mk_sumcases _ f T = f T
fun mk_undefined T = Const(@{const_name undefined}, T)
(** abstract syntax **)
fun termifyT T = HOLogic.mk_prodT (T, @{typ "unit => Code_Evaluation.term"});
val size = @{term "i :: code_numeral"}
val size_pred = @{term "(i :: code_numeral) - 1"}
val size_ge_zero = @{term "(i :: code_numeral) > 0"}
fun test_function T = Free ("f", termifyT T --> @{typ "term list option"})
fun mk_none_continuation (x, y) =
let
val (T as Type(@{type_name "option"}, [T'])) = fastype_of x
in
Const (@{const_name "Quickcheck_Exhaustive.orelse"}, T --> T --> T)
$ x $ y
end
(** datatypes **)
(* constructing exhaustive generator instances on datatypes *)
exception FUNCTION_TYPE;
val exhaustiveN = "exhaustive";
fun exhaustiveT T = (termifyT T --> @{typ "Code_Evaluation.term list option"})
--> @{typ code_numeral} --> @{typ "Code_Evaluation.term list option"}
fun check_allT T = (termifyT T --> @{typ "Code_Evaluation.term list option"})
--> @{typ "Code_Evaluation.term list option"}
fun mk_equations thy descr vs tycos exhaustives (Ts, Us) =
let
fun mk_call T =
let
val exhaustive = Const (@{const_name "Quickcheck_Exhaustive.exhaustive_class.exhaustive"}, exhaustiveT T)
in
(T, (fn t => exhaustive $
(HOLogic.split_const (T, @{typ "unit => Code_Evaluation.term"}, @{typ "Code_Evaluation.term list option"})
$ absdummy (T, absdummy (@{typ "unit => Code_Evaluation.term"}, t))) $ size_pred))
end
fun mk_aux_call fTs (k, _) (tyco, Ts) =
let
val T = Type (tyco, Ts)
val _ = if not (null fTs) then raise FUNCTION_TYPE else ()
in
(T, (fn t => nth exhaustives k $
(HOLogic.split_const (T, @{typ "unit => Code_Evaluation.term"}, @{typ "Code_Evaluation.term list option"})
$ absdummy (T, absdummy (@{typ "unit => Code_Evaluation.term"}, t))) $ size_pred))
end
fun mk_consexpr simpleT (c, xs) =
let
val (Ts, fns) = split_list xs
val constr = Const (c, Ts ---> simpleT)
val bounds = map (fn x => Bound (2 * x + 1)) (((length xs) - 1) downto 0)
val term_bounds = map (fn x => Bound (2 * x)) (((length xs) - 1) downto 0)
val Eval_App = Const ("Code_Evaluation.App", HOLogic.termT --> HOLogic.termT --> HOLogic.termT)
val Eval_Const = Const ("Code_Evaluation.Const", HOLogic.literalT --> @{typ typerep} --> HOLogic.termT)
val term = fold (fn u => fn t => Eval_App $ t $ (u $ @{term "()"}))
bounds (Eval_Const $ HOLogic.mk_literal c $ HOLogic.mk_typerep (Ts ---> simpleT))
val start_term = test_function simpleT $
(HOLogic.pair_const simpleT @{typ "unit => Code_Evaluation.term"}
$ (list_comb (constr, bounds)) $ absdummy (@{typ unit}, term))
in fold_rev (fn f => fn t => f t) fns start_term end
fun mk_rhs exprs =
@{term "If :: bool => term list option => term list option => term list option"}
$ size_ge_zero $ (foldr1 mk_none_continuation exprs) $ @{term "None :: term list option"}
val rhss =
Datatype_Aux.interpret_construction descr vs
{ atyp = mk_call, dtyp = mk_aux_call }
|> (map o apfst) Type
|> map (fn (T, cs) => map (mk_consexpr T) cs)
|> map mk_rhs
val lhss = map2 (fn t => fn T => t $ test_function T $ size) exhaustives (Ts @ Us);
val eqs = map (HOLogic.mk_Trueprop o HOLogic.mk_eq) (lhss ~~ rhss)
in
eqs
end
(* foundational definition with the function package *)
val less_int_pred = @{lemma "i > 0 ==> Code_Numeral.nat_of ((i :: code_numeral) - 1) < Code_Numeral.nat_of i" by auto}
fun mk_single_measure T = HOLogic.mk_comp (@{term "Code_Numeral.nat_of"},
Const (@{const_name "Product_Type.snd"}, T --> @{typ "code_numeral"}))
fun mk_termination_measure T =
let
val T' = fst (HOLogic.dest_prodT (HOLogic.dest_setT T))
in
mk_measure (mk_sumcases @{typ nat} mk_single_measure T')
end
fun termination_tac ctxt =
Function_Relation.relation_tac ctxt mk_termination_measure 1
THEN rtac @{thm wf_measure} 1
THEN (REPEAT_DETERM (Simplifier.asm_full_simp_tac
(HOL_basic_ss addsimps [@{thm in_measure}, @{thm o_def}, @{thm snd_conv},
@{thm nat_mono_iff}, less_int_pred] @ @{thms sum.cases}) 1))
fun pat_completeness_auto ctxt =
Pat_Completeness.pat_completeness_tac ctxt 1
THEN auto_tac (clasimpset_of ctxt)
(* creating the instances *)
fun instantiate_exhaustive_datatype config descr vs tycos prfx (names, auxnames) (Ts, Us) thy =
let
val _ = Datatype_Aux.message config "Creating exhaustive generators ...";
val exhaustivesN = map (prefix (exhaustiveN ^ "_")) (names @ auxnames);
in
thy
|> Class.instantiation (tycos, vs, @{sort exhaustive})
|> (if define_foundationally then
let
val exhaustives = map2 (fn name => fn T => Free (name, exhaustiveT T)) exhaustivesN (Ts @ Us)
val eqs = mk_equations thy descr vs tycos exhaustives (Ts, Us)
in
Function.add_function
(map (fn (name, T) =>
Syntax.no_syn (Binding.conceal (Binding.name name), SOME (exhaustiveT T)))
(exhaustivesN ~~ (Ts @ Us)))
(map (pair (apfst Binding.conceal Attrib.empty_binding)) eqs)
Function_Common.default_config pat_completeness_auto
#> snd
#> Local_Theory.restore
#> (fn lthy => Function.prove_termination NONE (termination_tac lthy) lthy)
#> snd
end
else
fold_map (fn (name, T) => Local_Theory.define
((Binding.conceal (Binding.name name), NoSyn),
(apfst Binding.conceal Attrib.empty_binding, mk_undefined (exhaustiveT T)))
#> apfst fst) (exhaustivesN ~~ (Ts @ Us))
#> (fn (exhaustives, lthy) =>
let
val eqs_t = mk_equations thy descr vs tycos exhaustives (Ts, Us)
val eqs = map (fn eq => Goal.prove lthy ["f", "i"] [] eq
(fn _ => Skip_Proof.cheat_tac (ProofContext.theory_of lthy))) eqs_t
in
fold (fn (name, eq) => Local_Theory.note
((Binding.conceal (Binding.qualify true prfx
(Binding.qualify true name (Binding.name "simps"))),
Code.add_default_eqn_attrib :: map (Attrib.internal o K)
[Simplifier.simp_add, Nitpick_Simps.add]), [eq]) #> snd) (exhaustivesN ~~ eqs) lthy
end))
|> Class.prove_instantiation_exit (K (Class.intro_classes_tac []))
end handle FUNCTION_TYPE =>
(Datatype_Aux.message config
"Creation of exhaustivevalue generators failed because the datatype contains a function type";
thy)
(** building and compiling generator expressions **)
structure Counterexample = Proof_Data
(
type T = unit -> int -> term list option
(* FIXME avoid user error with non-user text *)
fun init _ () = error "Counterexample"
);
val put_counterexample = Counterexample.put;
structure Counterexample_Batch = Proof_Data
(
type T = unit -> (int -> term list option) list
(* FIXME avoid user error with non-user text *)
fun init _ () = error "Counterexample"
);
val put_counterexample_batch = Counterexample_Batch.put;
val target = "Quickcheck";
fun mk_smart_generator_expr ctxt t =
let
val thy = ProofContext.theory_of ctxt
val ((vnames, Ts), t') = apfst split_list (strip_abs t)
val ([depth_name], ctxt') = Variable.variant_fixes ["depth"] ctxt
val (names, ctxt'') = Variable.variant_fixes vnames ctxt'
val (term_names, ctxt''') = Variable.variant_fixes (map (prefix "t_") vnames) ctxt''
val depth = Free (depth_name, @{typ code_numeral})
val frees = map2 (curry Free) names Ts
val term_vars = map (fn n => Free (n, @{typ "unit => term"})) term_names
fun strip_imp (Const(@{const_name HOL.implies},_) $ A $ B) = apfst (cons A) (strip_imp B)
| strip_imp A = ([], A)
val (assms, concl) = strip_imp (subst_bounds (rev frees, t'))
val terms = HOLogic.mk_list @{typ term} (map (fn v => v $ @{term "()"}) term_vars)
fun mk_exhaustive_closure (free as Free (_, T), term_var) t =
if Sign.of_sort thy (T, @{sort enum}) then
Const (@{const_name "Quickcheck_Exhaustive.check_all_class.check_all"}, check_allT T)
$ (HOLogic.split_const (T, @{typ "unit => term"}, @{typ "term list option"})
$ lambda free (lambda term_var t))
else
Const (@{const_name "Quickcheck_Exhaustive.exhaustive_class.exhaustive"}, exhaustiveT T)
$ (HOLogic.split_const (T, @{typ "unit => term"}, @{typ "term list option"})
$ lambda free (lambda term_var t)) $ depth
fun lookup v = the (AList.lookup (op =) (names ~~ (frees ~~ term_vars)) v)
val none_t = @{term "None :: term list option"}
fun mk_safe_if (cond, then_t, else_t) =
@{term "Quickcheck_Exhaustive.catch_match :: term list option => term list option => term list option"} $
(@{term "If :: bool => term list option => term list option => term list option"}
$ cond $ then_t $ else_t) $ none_t;
fun mk_test_term bound_vars assms =
let
fun vars_of t = subtract (op =) bound_vars (Term.add_free_names t [])
val (vars, check) =
case assms of [] =>
(vars_of concl, (concl, none_t, @{term "Some :: term list => term list option"} $ terms))
| assm :: assms =>
(vars_of assm, (assm, mk_test_term (union (op =) (vars_of assm) bound_vars) assms, none_t))
in
fold_rev mk_exhaustive_closure (map lookup vars) (mk_safe_if check)
end
in lambda depth (mk_test_term [] assms) end
fun mk_generator_expr ctxt t =
let
val Ts = (map snd o fst o strip_abs) t;
val thy = ProofContext.theory_of ctxt
val bound_max = length Ts - 1;
val bounds = map_index (fn (i, ty) =>
(2 * (bound_max - i) + 1, 2 * (bound_max - i), 2 * i, ty)) Ts;
val result = list_comb (t, map (fn (i, _, _, _) => Bound i) bounds);
val terms = HOLogic.mk_list @{typ term} (map (fn (_, i, _, _) => Bound i $ @{term "()"}) bounds);
val check =
@{term "Quickcheck_Exhaustive.catch_match :: term list option => term list option => term list option"} $
(@{term "If :: bool => term list option => term list option => term list option"}
$ result $ @{term "None :: term list option"} $ (@{term "Some :: term list => term list option"} $ terms))
$ @{term "None :: term list option"};
fun mk_exhaustive_closure (_, _, i, T) t =
Const (@{const_name "Quickcheck_Exhaustive.exhaustive_class.exhaustive"}, exhaustiveT T)
$ (HOLogic.split_const (T, @{typ "unit => term"}, @{typ "term list option"})
$ absdummy (T, absdummy (@{typ "unit => term"}, t))) $ Bound i
in Abs ("d", @{typ code_numeral}, fold_rev mk_exhaustive_closure bounds check) end
(** post-processing of function terms **)
fun dest_fun_upd (Const (@{const_name fun_upd}, _) $ t0 $ t1 $ t2) = (t0, (t1, t2))
| dest_fun_upd t = raise TERM ("dest_fun_upd", [t])
fun mk_fun_upd T1 T2 (t1, t2) t =
Const (@{const_name fun_upd}, (T1 --> T2) --> T1 --> T2 --> T1 --> T2) $ t $ t1 $ t2
fun dest_fun_upds t =
case try dest_fun_upd t of
NONE =>
(case t of
Abs (_, _, _) => ([], t)
| _ => raise TERM ("dest_fun_upds", [t]))
| SOME (t0, (t1, t2)) => apfst (cons (t1, t2)) (dest_fun_upds t0)
fun make_fun_upds T1 T2 (tps, t) = fold_rev (mk_fun_upd T1 T2) tps t
fun make_set T1 [] = Const (@{const_abbrev Set.empty}, T1 --> @{typ bool})
| make_set T1 ((_, @{const False}) :: tps) = make_set T1 tps
| make_set T1 ((t1, @{const True}) :: tps) =
Const (@{const_name insert}, T1 --> (T1 --> @{typ bool}) --> T1 --> @{typ bool})
$ t1 $ (make_set T1 tps)
| make_set T1 ((_, t) :: tps) = raise TERM ("make_set", [t])
fun make_coset T [] = Const (@{const_abbrev UNIV}, T --> @{typ bool})
| make_coset T tps =
let
val U = T --> @{typ bool}
fun invert @{const False} = @{const True}
| invert @{const True} = @{const False}
in
Const (@{const_name "Groups.minus_class.minus"}, U --> U --> U)
$ Const (@{const_abbrev UNIV}, U) $ make_set T (map (apsnd invert) tps)
end
fun make_map T1 T2 [] = Const (@{const_abbrev Map.empty}, T1 --> T2)
| make_map T1 T2 ((_, Const (@{const_name None}, _)) :: tps) = make_map T1 T2 tps
| make_map T1 T2 ((t1, t2) :: tps) = mk_fun_upd T1 T2 (t1, t2) (make_map T1 T2 tps)
fun post_process_term t =
let
fun map_Abs f t =
case t of Abs (x, T, t') => Abs (x, T, f t') | _ => raise TERM ("map_Abs", [t])
fun process_args t = case strip_comb t of
(c as Const (_, _), ts) => list_comb (c, map post_process_term ts)
in
case fastype_of t of
Type (@{type_name fun}, [T1, T2]) =>
(case try dest_fun_upds t of
SOME (tps, t) =>
(map (pairself post_process_term) tps, map_Abs post_process_term t)
|> (case T2 of
@{typ bool} =>
(case t of
Abs(_, _, @{const True}) => fst #> rev #> make_set T1
| Abs(_, _, @{const False}) => fst #> rev #> make_coset T1
| Abs(_, _, Const (@{const_name undefined}, _)) => fst #> rev #> make_set T1
| _ => raise TERM ("post_process_term", [t]))
| Type (@{type_name option}, _) =>
(case t of
Abs(_, _, Const(@{const_name None}, _)) => fst #> make_map T1 T2
| Abs(_, _, Const (@{const_name undefined}, _)) => fst #> make_map T1 T2
| _ => make_fun_upds T1 T2)
| _ => make_fun_upds T1 T2)
| NONE => process_args t)
| _ => process_args t
end
(** generator compiliation **)
fun compile_generator_expr ctxt t =
let
val thy = ProofContext.theory_of ctxt
val t' =
(if Config.get ctxt smart_quantifier then mk_smart_generator_expr else mk_generator_expr)
ctxt t;
val compile = Code_Runtime.dynamic_value_strict
(Counterexample.get, put_counterexample, "Exhaustive_Generators.put_counterexample")
thy (SOME target) (fn proc => fn g => g #> (Option.map o map) proc) t' [];
in
fn size => rpair NONE (compile size |>
(if Config.get ctxt quickcheck_pretty then Option.map (map post_process_term) else I))
end;
fun compile_generator_exprs ctxt ts =
let
val thy = ProofContext.theory_of ctxt
val mk_generator_expr =
if Config.get ctxt smart_quantifier then mk_smart_generator_expr else mk_generator_expr
val ts' = map (mk_generator_expr ctxt) ts;
val compiles = Code_Runtime.dynamic_value_strict
(Counterexample_Batch.get, put_counterexample_batch,
"Exhaustive_Generators.put_counterexample_batch")
thy (SOME target) (fn proc => map (fn g => g #> (Option.map o map) proc))
(HOLogic.mk_list @{typ "code_numeral => term list option"} ts') [];
in
map (fn compile => fn size => compile size |> Option.map (map post_process_term)) compiles
end;
(** setup **)
val setup =
Datatype.interpretation
(Quickcheck_Generators.ensure_sort_datatype (@{sort exhaustive}, instantiate_exhaustive_datatype))
#> setup_smart_quantifier
#> setup_quickcheck_pretty
#> Context.theory_map (Quickcheck.add_generator ("exhaustive", compile_generator_expr))
#> Context.theory_map (Quickcheck.add_batch_generator ("exhaustive", compile_generator_exprs));
end;