author  bulwahn 
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permissions  rwrr 
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(* Author: Lukas Bulwahn, TU Muenchen *) 
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header {* A simple counterexample generator performing exhaustive testing *} 
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theory Quickcheck_Exhaustive 
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imports Quickcheck 
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uses ("Tools/Quickcheck/exhaustive_generators.ML") 
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begin 
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subsection {* basic operations for exhaustive generators *} 
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definition orelse :: "'a option => 'a option => 'a option" (infixr "orelse" 55) 
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where 
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[code_unfold]: "x orelse y = (case x of Some x' => Some x'  None => y)" 
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subsection {* exhaustive generator type classes *} 
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class exhaustive = term_of + 
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class full_exhaustive = term_of + 
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instantiation code_numeral :: full_exhaustive 
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begin 
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function full_exhaustive_code_numeral' :: "(code_numeral * (unit => term) => term list option) => code_numeral => code_numeral => term list option" 
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where "full_exhaustive_code_numeral' f d i = 

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(if d < i then None 

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else (f (i, %_. Code_Evaluation.term_of i)) orelse (full_exhaustive_code_numeral' f d (i + 1)))" 

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by pat_completeness auto 

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termination 

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by (relation "measure (%(_, d, i). Code_Numeral.nat_of (d + 1  i))") auto 

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definition "full_exhaustive f d = full_exhaustive_code_numeral' f d 0" 

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instance .. 
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end 
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instantiation code_numeral :: exhaustive 
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begin 
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function exhaustive_code_numeral' :: "(code_numeral => term list option) => code_numeral => code_numeral => term list option" 
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where "exhaustive_code_numeral' f d i = 
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(if d < i then None 
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else (f i orelse exhaustive_code_numeral' f d (i + 1)))" 
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by pat_completeness auto 
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termination 
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by (relation "measure (%(_, d, i). Code_Numeral.nat_of (d + 1  i))") auto 
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definition "exhaustive f d = exhaustive_code_numeral' f d 0" 
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instance .. 
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end 
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instantiation nat :: exhaustive 
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begin 
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definition "exhaustive f d = exhaustive (%x. f (Code_Numeral.nat_of x)) d" 
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instance .. 
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end 
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instantiation nat :: full_exhaustive 
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begin 
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definition "full_exhaustive f d = full_exhaustive (%(x, xt). f (Code_Numeral.nat_of x, %_. Code_Evaluation.term_of (Code_Numeral.nat_of x))) d" 
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instance .. 
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end 
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instantiation int :: exhaustive 
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begin 
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function exhaustive' :: "(int => term list option) => int => int => term list option" 
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where "exhaustive' f d i = (if d < i then None else (f i orelse exhaustive' f d (i + 1)))" 

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by pat_completeness auto 
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termination 
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by (relation "measure (%(_, d, i). nat (d + 1  i))") auto 
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definition "exhaustive f d = exhaustive' f (Code_Numeral.int_of d) ( (Code_Numeral.int_of d))" 
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instance .. 
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end 
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instantiation int :: full_exhaustive 
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begin 
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function full_exhaustive' :: "(int * (unit => term) => term list option) => int => int => term list option" 
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where "full_exhaustive' f d i = (if d < i then None else (case f (i, %_. Code_Evaluation.term_of i) of Some t => Some t  None => full_exhaustive' f d (i + 1)))" 

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by pat_completeness auto 

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termination 

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by (relation "measure (%(_, d, i). nat (d + 1  i))") auto 

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definition "full_exhaustive f d = full_exhaustive' f (Code_Numeral.int_of d) ( (Code_Numeral.int_of d))" 

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instance .. 
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end 
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instantiation prod :: (exhaustive, exhaustive) exhaustive 
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begin 
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definition 
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"exhaustive f d = exhaustive (%x. exhaustive (%y. f ((x, y))) d) d" 
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instance .. 
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end 
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instantiation prod :: (full_exhaustive, full_exhaustive) full_exhaustive 
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begin 
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definition 
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"full_exhaustive f d = full_exhaustive (%(x, t1). full_exhaustive (%(y, t2). f ((x, y), 

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%u. let T1 = (Typerep.typerep (TYPE('a))); 
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T2 = (Typerep.typerep (TYPE('b))) 
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in Code_Evaluation.App (Code_Evaluation.App ( 
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Code_Evaluation.Const (STR ''Product_Type.Pair'') 
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(Typerep.Typerep (STR ''fun'') [T1, Typerep.Typerep (STR ''fun'') [T2, Typerep.Typerep (STR ''Product_Type.prod'') [T1, T2]]])) 
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(t1 ())) (t2 ()))) d) d" 
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instance .. 
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end 
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instantiation "fun" :: ("{equal, exhaustive}", exhaustive) exhaustive 
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begin 
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fun exhaustive_fun' :: "(('a => 'b) => term list option) => code_numeral => code_numeral => term list option" 
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"exhaustive_fun' f i d = (exhaustive (%b. f (%_. b)) d) 

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orelse (if i > 1 then 

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exhaustive_fun' (%g. exhaustive (%a. exhaustive (%b. 

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f (g(a := b))) d) d) (i  1) d else None)" 

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definition exhaustive_fun :: "(('a => 'b) => term list option) => code_numeral => term list option" 

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where 
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"exhaustive_fun f d = exhaustive_fun' f d d" 
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instance .. 
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end 
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instantiation "fun" :: ("{equal, full_exhaustive}", full_exhaustive) full_exhaustive 
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begin 
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fun full_exhaustive_fun' :: "(('a => 'b) * (unit => term) => term list option) => code_numeral => code_numeral => term list option" 

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"full_exhaustive_fun' f i d = (full_exhaustive (%(b, t). f (%_. b, %_. Code_Evaluation.Abs (STR ''x'') (Typerep.typerep TYPE('a)) (t ()))) d) 

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orelse (if i > 1 then 
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full_exhaustive_fun' (%(g, gt). full_exhaustive (%(a, at). full_exhaustive (%(b, bt). 
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f (g(a := b), 
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(%_. let A = (Typerep.typerep (TYPE('a))); 
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B = (Typerep.typerep (TYPE('b))); 
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fun = (%T U. Typerep.Typerep (STR ''fun'') [T, U]) 
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in 
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Code_Evaluation.App (Code_Evaluation.App (Code_Evaluation.App 
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(Code_Evaluation.Const (STR ''Fun.fun_upd'') (fun (fun A B) (fun A (fun B (fun A B))))) 
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(gt ())) (at ())) (bt ())))) d) d) (i  1) d else None)" 
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definition full_exhaustive_fun :: "(('a => 'b) * (unit => term) => term list option) => code_numeral => term list option" 
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where 
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"full_exhaustive_fun f d = full_exhaustive_fun' f d d" 
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instance .. 
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end 
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subsubsection {* A smarter enumeration scheme for functions over finite datatypes *} 
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class check_all = enum + term_of + 
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fixes check_all :: "('a * (unit \<Rightarrow> term) \<Rightarrow> term list option) \<Rightarrow> term list option" 
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fixes enum_term_of :: "'a itself \<Rightarrow> unit \<Rightarrow> term list" 
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fun check_all_n_lists :: "(('a :: check_all) list * (unit \<Rightarrow> term list) \<Rightarrow> term list option) \<Rightarrow> code_numeral \<Rightarrow> term list option" 
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where 
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"check_all_n_lists f n = 
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(if n = 0 then f ([], (%_. [])) else check_all (%(x, xt). check_all_n_lists (%(xs, xst). f ((x # xs), (%_. (xt () # xst ())))) (n  1)))" 
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definition mk_map_term :: " (unit \<Rightarrow> typerep) \<Rightarrow> (unit \<Rightarrow> typerep) \<Rightarrow> (unit \<Rightarrow> term list) \<Rightarrow> (unit \<Rightarrow> term list) \<Rightarrow> unit \<Rightarrow> term" 
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"mk_map_term T1 T2 domm rng = 
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(%_. let T1 = T1 (); 
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T2 = T2 (); 
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update_term = (%g (a, b). 
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Code_Evaluation.App (Code_Evaluation.App (Code_Evaluation.App 
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(Code_Evaluation.Const (STR ''Fun.fun_upd'') 
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(Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''fun'') [T1, T2], 
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Typerep.Typerep (STR ''fun'') [T1, 
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Typerep.Typerep (STR ''fun'') [T2, Typerep.Typerep (STR ''fun'') [T1, T2]]]])) 
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g) a) b) 
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in 
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List.foldl update_term (Code_Evaluation.Abs (STR ''x'') T1 (Code_Evaluation.Const (STR ''HOL.undefined'') T2)) (zip (domm ()) (rng ())))" 
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instantiation "fun" :: ("{equal, check_all}", check_all) check_all 
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begin 
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definition 
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"check_all f = 
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(let 
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mk_term = mk_map_term (%_. Typerep.typerep (TYPE('a))) (%_. Typerep.typerep (TYPE('b))) (enum_term_of (TYPE('a))); 
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enum = (Enum.enum :: 'a list) 
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in check_all_n_lists (\<lambda>(ys, yst). f (the o map_of (zip enum ys), mk_term yst)) (Code_Numeral.of_nat (length enum)))" 
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definition enum_term_of_fun :: "('a => 'b) itself => unit => term list" 
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where 
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"enum_term_of_fun = (%_ _. let 
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enum_term_of_a = enum_term_of (TYPE('a)); 
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mk_term = mk_map_term (%_. Typerep.typerep (TYPE('a))) (%_. Typerep.typerep (TYPE('b))) enum_term_of_a 
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in map (%ys. mk_term (%_. ys) ()) (Enum.n_lists (length (enum_term_of_a ())) (enum_term_of (TYPE('b)) ())))" 
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instance .. 
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end 
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instantiation unit :: check_all 
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begin 
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definition 
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"check_all f = f (Code_Evaluation.valtermify ())" 
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definition enum_term_of_unit :: "unit itself => unit => term list" 
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where 
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"enum_term_of_unit = (%_ _. [Code_Evaluation.term_of ()])" 
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instance .. 
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end 
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instantiation bool :: check_all 
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begin 
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definition 
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"check_all f = (case f (Code_Evaluation.valtermify False) of Some x' \<Rightarrow> Some x'  None \<Rightarrow> f (Code_Evaluation.valtermify True))" 
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definition enum_term_of_bool :: "bool itself => unit => term list" 
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where 
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"enum_term_of_bool = (%_ _. map Code_Evaluation.term_of (Enum.enum :: bool list))" 
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instance .. 
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end 
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instantiation prod :: (check_all, check_all) check_all 
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begin 
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definition 
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"check_all f = check_all (%(x, t1). check_all (%(y, t2). f ((x, y), 
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%u. let T1 = (Typerep.typerep (TYPE('a))); 
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T2 = (Typerep.typerep (TYPE('b))) 
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in Code_Evaluation.App (Code_Evaluation.App ( 
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Code_Evaluation.Const (STR ''Product_Type.Pair'') 
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(Typerep.Typerep (STR ''fun'') [T1, Typerep.Typerep (STR ''fun'') [T2, Typerep.Typerep (STR ''Product_Type.prod'') [T1, T2]]])) 
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(t1 ())) (t2 ()))))" 
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definition enum_term_of_prod :: "('a * 'b) itself => unit => term list" 
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where 
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"enum_term_of_prod = (%_ _. map (%(x, y). 
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let T1 = (Typerep.typerep (TYPE('a))); 
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T2 = (Typerep.typerep (TYPE('b))) 
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in Code_Evaluation.App (Code_Evaluation.App ( 
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Code_Evaluation.Const (STR ''Product_Type.Pair'') 
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(Typerep.Typerep (STR ''fun'') [T1, Typerep.Typerep (STR ''fun'') [T2, Typerep.Typerep (STR ''Product_Type.prod'') [T1, T2]]])) x) y) 
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(Enum.product (enum_term_of (TYPE('a)) ()) (enum_term_of (TYPE('b)) ()))) " 
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instance .. 
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end 
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instantiation sum :: (check_all, check_all) check_all 
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begin 
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definition 
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"check_all f = (case check_all (%(a, t). f (Inl a, %_. 
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let T1 = (Typerep.typerep (TYPE('a))); 
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T2 = (Typerep.typerep (TYPE('b))) 
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in Code_Evaluation.App (Code_Evaluation.Const (STR ''Sum_Type.Inl'') 
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(Typerep.Typerep (STR ''fun'') [T1, Typerep.Typerep (STR ''Sum_Type.sum'') [T1, T2]])) (t ()))) of Some x' => Some x' 
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 None => check_all (%(b, t). f (Inr b, %_. let 
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T1 = (Typerep.typerep (TYPE('a))); 
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T2 = (Typerep.typerep (TYPE('b))) 
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in Code_Evaluation.App (Code_Evaluation.Const (STR ''Sum_Type.Inr'') 
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(Typerep.Typerep (STR ''fun'') [T2, Typerep.Typerep (STR ''Sum_Type.sum'') [T1, T2]])) (t ()))))" 
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definition enum_term_of_sum :: "('a + 'b) itself => unit => term list" 
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where 
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"enum_term_of_sum = (%_ _. 
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let 
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T1 = (Typerep.typerep (TYPE('a))); 
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T2 = (Typerep.typerep (TYPE('b))) 
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in 
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map (Code_Evaluation.App (Code_Evaluation.Const (STR ''Sum_Type.Inl'') 
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(Typerep.Typerep (STR ''fun'') [T1, Typerep.Typerep (STR ''Sum_Type.sum'') [T1, T2]]))) 
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(enum_term_of (TYPE('a)) ()) @ 
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map (Code_Evaluation.App (Code_Evaluation.Const (STR ''Sum_Type.Inr'') 
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(Typerep.Typerep (STR ''fun'') [T2, Typerep.Typerep (STR ''Sum_Type.sum'') [T1, T2]]))) 
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(enum_term_of (TYPE('b)) ()))" 
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instance .. 
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end 
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instantiation nibble :: check_all 
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begin 
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319 

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definition 
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"check_all f = 
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f (Code_Evaluation.valtermify Nibble0) orelse 
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f (Code_Evaluation.valtermify Nibble1) orelse 
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f (Code_Evaluation.valtermify Nibble2) orelse 
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f (Code_Evaluation.valtermify Nibble3) orelse 
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f (Code_Evaluation.valtermify Nibble4) orelse 
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f (Code_Evaluation.valtermify Nibble5) orelse 
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f (Code_Evaluation.valtermify Nibble6) orelse 
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f (Code_Evaluation.valtermify Nibble7) orelse 
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f (Code_Evaluation.valtermify Nibble8) orelse 
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f (Code_Evaluation.valtermify Nibble9) orelse 
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f (Code_Evaluation.valtermify NibbleA) orelse 
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f (Code_Evaluation.valtermify NibbleB) orelse 
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f (Code_Evaluation.valtermify NibbleC) orelse 
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f (Code_Evaluation.valtermify NibbleD) orelse 
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f (Code_Evaluation.valtermify NibbleE) orelse 
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f (Code_Evaluation.valtermify NibbleF)" 
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338 

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definition enum_term_of_nibble :: "nibble itself => unit => term list" 
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340 
where 
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"enum_term_of_nibble = (%_ _. map Code_Evaluation.term_of (Enum.enum :: nibble list))" 
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342 

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instance .. 
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344 

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end 
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346 

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347 

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instantiation char :: check_all 
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begin 
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350 

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351 
definition 
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"check_all f = check_all (%(x, t1). check_all (%(y, t2). f (Char x y, %_. Code_Evaluation.App (Code_Evaluation.App (Code_Evaluation.term_of Char) (t1 ())) (t2 ()))))" 
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353 

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definition enum_term_of_char :: "char itself => unit => term list" 
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355 
where 
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"enum_term_of_char = (%_ _. map Code_Evaluation.term_of (Enum.enum :: char list))" 
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357 

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358 
instance .. 
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359 

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360 
end 
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361 

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362 

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363 
instantiation option :: (check_all) check_all 
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364 
begin 
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365 

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366 
definition 
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"check_all f = f (Code_Evaluation.valtermify (None :: 'a option)) orelse check_all (%(x, t). f (Some x, %_. Code_Evaluation.App 
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368 
(Code_Evaluation.Const (STR ''Option.option.Some'') 
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(Typerep.Typerep (STR ''fun'') [Typerep.typerep TYPE('a), Typerep.Typerep (STR ''Option.option'') [Typerep.typerep TYPE('a)]])) (t ())))" 
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370 

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definition enum_term_of_option :: "'a option itself => unit => term list" 
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372 
where 
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"enum_term_of_option = (% _ _. (Code_Evaluation.term_of (None :: 'a option)) # (map (Code_Evaluation.App (Code_Evaluation.Const (STR ''Option.option.Some'') 
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(Typerep.Typerep (STR ''fun'') [Typerep.typerep TYPE('a), Typerep.Typerep (STR ''Option.option'') [Typerep.typerep TYPE('a)]]))) (enum_term_of (TYPE('a)) ())))" 
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375 

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376 
instance .. 
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377 

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378 
end 
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379 

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380 

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instantiation Enum.finite_1 :: check_all 
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382 
begin 
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383 

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384 
definition 
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385 
"check_all f = f (Code_Evaluation.valtermify Enum.finite_1.a\<^isub>1)" 
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386 

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387 
definition enum_term_of_finite_1 :: "Enum.finite_1 itself => unit => term list" 
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388 
where 
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389 
"enum_term_of_finite_1 = (%_ _. [Code_Evaluation.term_of Enum.finite_1.a\<^isub>1])" 
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390 

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391 
instance .. 
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392 

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393 
end 
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394 

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395 
instantiation Enum.finite_2 :: check_all 
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396 
begin 
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397 

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398 
definition 
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399 
"check_all f = (case f (Code_Evaluation.valtermify Enum.finite_2.a\<^isub>1) of Some x' \<Rightarrow> Some x'  None \<Rightarrow> f (Code_Evaluation.valtermify Enum.finite_2.a\<^isub>2))" 
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400 

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401 
definition enum_term_of_finite_2 :: "Enum.finite_2 itself => unit => term list" 
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402 
where 
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403 
"enum_term_of_finite_2 = (%_ _. map Code_Evaluation.term_of (Enum.enum :: Enum.finite_2 list))" 
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404 

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405 
instance .. 
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406 

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407 
end 
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408 

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409 
instantiation Enum.finite_3 :: check_all 
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410 
begin 
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411 

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412 
definition 
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413 
"check_all f = (case f (Code_Evaluation.valtermify Enum.finite_3.a\<^isub>1) of Some x' \<Rightarrow> Some x'  None \<Rightarrow> (case f (Code_Evaluation.valtermify Enum.finite_3.a\<^isub>2) of Some x' \<Rightarrow> Some x'  None \<Rightarrow> f (Code_Evaluation.valtermify Enum.finite_3.a\<^isub>3)))" 
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414 

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415 
definition enum_term_of_finite_3 :: "Enum.finite_3 itself => unit => term list" 
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416 
where 
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417 
"enum_term_of_finite_3 = (%_ _. map Code_Evaluation.term_of (Enum.enum :: Enum.finite_3 list))" 
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418 

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419 
instance .. 
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420 

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421 
end 
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422 

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423 
subsection {* Bounded universal quantifiers *} 
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424 

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425 
class bounded_forall = 
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426 
fixes bounded_forall :: "('a \<Rightarrow> bool) \<Rightarrow> code_numeral \<Rightarrow> bool" 
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427 

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428 
subsection {* Fast exhaustive combinators *} 
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429 

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430 

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431 
class fast_exhaustive = term_of + 
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432 
fixes fast_exhaustive :: "('a \<Rightarrow> unit) \<Rightarrow> code_numeral \<Rightarrow> unit" 
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433 

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434 
consts throw_Counterexample :: "term list => unit" 
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435 
consts catch_Counterexample :: "unit => term list option" 
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436 

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437 
code_const throw_Counterexample 
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438 
(Quickcheck "raise (Exhaustive'_Generators.Counterexample _)") 
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439 
code_const catch_Counterexample 
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440 
(Quickcheck "(((_); NONE) handle Exhaustive'_Generators.Counterexample ts => SOME ts)") 
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441 

40620  442 
subsection {* Defining combinators for any firstorder data type *} 
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443 

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definition catch_match :: "term list option => term list option => term list option" 
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445 
where 
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446 
[code del]: "catch_match t1 t2 = (SOME t. t = t1 \<or> t = t2)" 
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447 

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448 
code_const catch_match 
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(Quickcheck "(_) handle Match => _") 
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450 

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use "Tools/Quickcheck/exhaustive_generators.ML" 
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452 

41918  453 
setup {* Exhaustive_Generators.setup *} 
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454 

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declare [[quickcheck_tester = exhaustive]] 
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456 

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hide_fact orelse_def catch_match_def 
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no_notation orelse (infixr "orelse" 55) 
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hide_const (open) orelse catch_match mk_map_term check_all_n_lists 
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460 

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461 
end 