author  haftmann 
Thu, 13 Apr 2017 10:10:09 +0200  
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child 66148  5e60c2d0a1f1 
permissions  rwrr 
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(* Author: Lukas Bulwahn, TU Muenchen *) 
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section \<open>Counterexample generator performing narrowingbased testing\<close> 
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theory Quickcheck_Narrowing 
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imports Quickcheck_Random 
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keywords "find_unused_assms" :: diag 
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begin 
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subsection \<open>Counterexample generator\<close> 
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subsubsection \<open>Code generation setup\<close> 
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setup \<open>Code_Target.add_derived_target ("Haskell_Quickcheck", [(Code_Haskell.target, I)])\<close> 
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code_printing 
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code_module Typerep \<rightharpoonup> (Haskell_Quickcheck) \<open> 
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data Typerep = Typerep String [Typerep] 
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\<close> 
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 type_constructor typerep \<rightharpoonup> (Haskell_Quickcheck) "Typerep.Typerep" 
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 constant Typerep.Typerep \<rightharpoonup> (Haskell_Quickcheck) "Typerep.Typerep" 

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 type_constructor integer \<rightharpoonup> (Haskell_Quickcheck) "Prelude.Int" 
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code_reserved Haskell_Quickcheck Typerep 
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code_printing 
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constant "0::integer" \<rightharpoonup> 
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(Haskell_Quickcheck) "!(0/ ::/ Prelude.Int)" 
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setup \<open> 
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let 
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val target = "Haskell_Quickcheck"; 
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fun print _ = Code_Haskell.print_numeral "Prelude.Int"; 
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in 
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Numeral.add_code @{const_name Code_Numeral.Pos} I print target 
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#> Numeral.add_code @{const_name Code_Numeral.Neg} (op ~) print target 
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end 
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\<close> 
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subsubsection \<open>Narrowing's deep representation of types and terms\<close> 
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datatype (plugins only: code extraction) narrowing_type = 
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Narrowing_sum_of_products "narrowing_type list list" 
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datatype (plugins only: code extraction) narrowing_term = 
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Narrowing_variable "integer list" narrowing_type 
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 Narrowing_constructor integer "narrowing_term list" 

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datatype (plugins only: code extraction) (dead 'a) narrowing_cons = 
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Narrowing_cons narrowing_type "(narrowing_term list \<Rightarrow> 'a) list" 
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primrec map_cons :: "('a => 'b) => 'a narrowing_cons => 'b narrowing_cons" 
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where 
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"map_cons f (Narrowing_cons ty cs) = Narrowing_cons ty (map (\<lambda>c. f o c) cs)" 
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subsubsection \<open>From narrowing's deep representation of terms to @{theory Code_Evaluation}'s terms\<close> 
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class partial_term_of = typerep + 
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fixes partial_term_of :: "'a itself => narrowing_term => Code_Evaluation.term" 
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lemma partial_term_of_anything: "partial_term_of x nt \<equiv> t" 
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by (rule eq_reflection) (cases "partial_term_of x nt", cases t, simp) 
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subsubsection \<open>Auxilary functions for Narrowing\<close> 
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consts nth :: "'a list => integer => 'a" 
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code_printing constant nth \<rightharpoonup> (Haskell_Quickcheck) infixl 9 "!!" 
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consts error :: "char list => 'a" 
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code_printing constant error \<rightharpoonup> (Haskell_Quickcheck) "error" 
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consts toEnum :: "integer => char" 
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code_printing constant toEnum \<rightharpoonup> (Haskell_Quickcheck) "Prelude.toEnum" 
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consts marker :: "char" 
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code_printing constant marker \<rightharpoonup> (Haskell_Quickcheck) "''\\0'" 
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subsubsection \<open>Narrowing's basic operations\<close> 
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type_synonym 'a narrowing = "integer => 'a narrowing_cons" 
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definition cons :: "'a => 'a narrowing" 
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where 
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"cons a d = (Narrowing_cons (Narrowing_sum_of_products [[]]) [(\<lambda>_. a)])" 
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fun conv :: "(narrowing_term list => 'a) list => narrowing_term => 'a" 
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where 
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"conv cs (Narrowing_variable p _) = error (marker # map toEnum p)" 
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 "conv cs (Narrowing_constructor i xs) = (nth cs i) xs" 
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fun non_empty :: "narrowing_type => bool" 
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where 
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"non_empty (Narrowing_sum_of_products ps) = (\<not> (List.null ps))" 
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definition "apply" :: "('a => 'b) narrowing => 'a narrowing => 'b narrowing" 
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where 
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"apply f a d = (if d > 0 then 
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(case f d of Narrowing_cons (Narrowing_sum_of_products ps) cfs \<Rightarrow> 
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case a (d  1) of Narrowing_cons ta cas \<Rightarrow> 

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let 
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shallow = non_empty ta; 
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cs = [(\<lambda>(x # xs) \<Rightarrow> cf xs (conv cas x)). shallow, cf \<leftarrow> cfs] 
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in Narrowing_cons (Narrowing_sum_of_products [ta # p. shallow, p \<leftarrow> ps]) cs) 
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else Narrowing_cons (Narrowing_sum_of_products []) [])" 

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definition sum :: "'a narrowing => 'a narrowing => 'a narrowing" 
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where 
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"sum a b d = 

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(case a d of Narrowing_cons (Narrowing_sum_of_products ssa) ca => 
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case b d of Narrowing_cons (Narrowing_sum_of_products ssb) cb => 
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Narrowing_cons (Narrowing_sum_of_products (ssa @ ssb)) (ca @ cb))" 
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lemma [fundef_cong]: 
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assumes "a d = a' d" "b d = b' d" "d = d'" 
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shows "sum a b d = sum a' b' d'" 
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using assms unfolding sum_def by (auto split: narrowing_cons.split narrowing_type.split) 
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lemma [fundef_cong]: 
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assumes "f d = f' d" "(\<And>d'. 0 \<le> d' \<and> d' < d \<Longrightarrow> a d' = a' d')" 
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assumes "d = d'" 
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shows "apply f a d = apply f' a' d'" 
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proof  
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note assms 
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moreover have "0 < d' \<Longrightarrow> 0 \<le> d'  1" 
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by (simp add: less_integer_def less_eq_integer_def) 
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ultimately show ?thesis 
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by (auto simp add: apply_def Let_def 
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split: narrowing_cons.split narrowing_type.split) 
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qed 
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subsubsection \<open>Narrowing generator type class\<close> 
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class narrowing = 
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fixes narrowing :: "integer => 'a narrowing_cons" 
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datatype (plugins only: code extraction) property = 
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Universal narrowing_type "(narrowing_term => property)" "narrowing_term => Code_Evaluation.term" 
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 Existential narrowing_type "(narrowing_term => property)" "narrowing_term => Code_Evaluation.term" 

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 Property bool 

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(* FIXME: hardwired maximal depth of 100 here *) 
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definition exists :: "('a :: {narrowing, partial_term_of} => property) => property" 
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where 
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"exists f = (case narrowing (100 :: integer) of Narrowing_cons ty cs => Existential ty (\<lambda> t. f (conv cs t)) (partial_term_of (TYPE('a))))" 
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definition "all" :: "('a :: {narrowing, partial_term_of} => property) => property" 
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where 
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"all f = (case narrowing (100 :: integer) of Narrowing_cons ty cs => Universal ty (\<lambda>t. f (conv cs t)) (partial_term_of (TYPE('a))))" 
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subsubsection \<open>class \<open>is_testable\<close>\<close> 
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text \<open>The class \<open>is_testable\<close> ensures that all necessary type instances are generated.\<close> 
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class is_testable 

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instance bool :: is_testable .. 

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instance "fun" :: ("{term_of, narrowing, partial_term_of}", is_testable) is_testable .. 
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definition ensure_testable :: "'a :: is_testable => 'a :: is_testable" 

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where 

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"ensure_testable f = f" 

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subsubsection \<open>Defining a simple datatype to represent functions in an incomplete and redundant way\<close> 
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datatype (plugins only: code quickcheck_narrowing extraction) (dead 'a, dead 'b) ffun = 
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Constant 'b 
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 Update 'a 'b "('a, 'b) ffun" 

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primrec eval_ffun :: "('a, 'b) ffun => 'a => 'b" 
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where 
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"eval_ffun (Constant c) x = c" 
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 "eval_ffun (Update x' y f) x = (if x = x' then y else eval_ffun f x)" 
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hide_type (open) ffun 
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hide_const (open) Constant Update eval_ffun 
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datatype (plugins only: code quickcheck_narrowing extraction) (dead 'b) cfun = Constant 'b 
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primrec eval_cfun :: "'b cfun => 'a => 'b" 
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where 
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"eval_cfun (Constant c) y = c" 
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hide_type (open) cfun 
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hide_const (open) Constant eval_cfun Abs_cfun Rep_cfun 
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60758  193 
subsubsection \<open>Setting up the counterexample generator\<close> 
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48891  195 
ML_file "Tools/Quickcheck/narrowing_generators.ML" 
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definition narrowing_dummy_partial_term_of :: "('a :: partial_term_of) itself => narrowing_term => term" 
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where 
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"narrowing_dummy_partial_term_of = partial_term_of" 
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definition narrowing_dummy_narrowing :: "integer => ('a :: narrowing) narrowing_cons" 
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where 
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"narrowing_dummy_narrowing = narrowing" 
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lemma [code]: 
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"ensure_testable f = 
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(let 
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x = narrowing_dummy_narrowing :: integer => bool narrowing_cons; 
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y = narrowing_dummy_partial_term_of :: bool itself => narrowing_term => term; 
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z = (conv :: _ => _ => unit) in f)" 
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unfolding Let_def ensure_testable_def .. 
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60758  213 
subsection \<open>Narrowing for sets\<close> 
46308  214 

215 
instantiation set :: (narrowing) narrowing 

216 
begin 

217 

218 
definition "narrowing_set = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.cons set) narrowing" 

219 

220 
instance .. 

221 

222 
end 

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60758  224 
subsection \<open>Narrowing for integers\<close> 
43356  225 

226 

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definition drawn_from :: "'a list \<Rightarrow> 'a narrowing_cons" 
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where 
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"drawn_from xs = 
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Narrowing_cons (Narrowing_sum_of_products (map (\<lambda>_. []) xs)) (map (\<lambda>x _. x) xs)" 
43356  231 

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function around_zero :: "int \<Rightarrow> int list" 
43356  233 
where 
234 
"around_zero i = (if i < 0 then [] else (if i = 0 then [0] else around_zero (i  1) @ [i, i]))" 

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by pat_completeness auto 
43356  236 
termination by (relation "measure nat") auto 
237 

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declare around_zero.simps [simp del] 
43356  239 

240 
lemma length_around_zero: 

241 
assumes "i >= 0" 

242 
shows "length (around_zero i) = 2 * nat i + 1" 

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proof (induct rule: int_ge_induct [OF assms]) 
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case 1 
245 
from 1 show ?case by (simp add: around_zero.simps) 

246 
next 

247 
case (2 i) 

248 
from 2 show ?case 

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by (simp add: around_zero.simps [of "i + 1"]) 
43356  250 
qed 
251 

252 
instantiation int :: narrowing 

253 
begin 

254 

255 
definition 

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"narrowing_int d = (let (u :: _ \<Rightarrow> _ \<Rightarrow> unit) = conv; i = int_of_integer d 
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in drawn_from (around_zero i))" 
43356  258 

259 
instance .. 

260 

261 
end 

262 

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lemma [code, code del]: "partial_term_of (ty :: int itself) t \<equiv> undefined" 
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by (rule partial_term_of_anything)+ 
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lemma [code]: 

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"partial_term_of (ty :: int itself) (Narrowing_variable p t) \<equiv> 
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Code_Evaluation.Free (STR ''_'') (Typerep.Typerep (STR ''Int.int'') [])" 
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"partial_term_of (ty :: int itself) (Narrowing_constructor i []) \<equiv> 
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(if i mod 2 = 0 
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then Code_Evaluation.term_of ( (int_of_integer i) div 2) 
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else Code_Evaluation.term_of ((int_of_integer i + 1) div 2))" 
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by (rule partial_term_of_anything)+ 
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instantiation integer :: narrowing 
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begin 
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definition 
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"narrowing_integer d = (let (u :: _ \<Rightarrow> _ \<Rightarrow> unit) = conv; i = int_of_integer d 
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in drawn_from (map integer_of_int (around_zero i)))" 
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instance .. 
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end 
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lemma [code, code del]: "partial_term_of (ty :: integer itself) t \<equiv> undefined" 
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by (rule partial_term_of_anything)+ 
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lemma [code]: 
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"partial_term_of (ty :: integer itself) (Narrowing_variable p t) \<equiv> 
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Code_Evaluation.Free (STR ''_'') (Typerep.Typerep (STR ''Code_Numeral.integer'') [])" 
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"partial_term_of (ty :: integer itself) (Narrowing_constructor i []) \<equiv> 
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(if i mod 2 = 0 
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then Code_Evaluation.term_of ( i div 2) 
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else Code_Evaluation.term_of ((i + 1) div 2))" 
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by (rule partial_term_of_anything)+ 
43356  297 

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code_printing constant "Code_Evaluation.term_of :: integer \<Rightarrow> term" \<rightharpoonup> (Haskell_Quickcheck) 
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"(let { t = Typerep.Typerep \"Code'_Numeral.integer\" []; 
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mkFunT s t = Typerep.Typerep \"fun\" [s, t]; 
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numT = Typerep.Typerep \"Num.num\" []; 
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mkBit 0 = Generated'_Code.Const \"Num.num.Bit0\" (mkFunT numT numT); 
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mkBit 1 = Generated'_Code.Const \"Num.num.Bit1\" (mkFunT numT numT); 
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mkNumeral 1 = Generated'_Code.Const \"Num.num.One\" numT; 
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mkNumeral i = let { q = i `Prelude.div` 2; r = i `Prelude.mod` 2 } 
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in Generated'_Code.App (mkBit r) (mkNumeral q); 
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mkNumber 0 = Generated'_Code.Const \"Groups.zero'_class.zero\" t; 
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mkNumber 1 = Generated'_Code.Const \"Groups.one'_class.one\" t; 
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mkNumber i = if i > 0 then 
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Generated'_Code.App 
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(Generated'_Code.Const \"Num.numeral'_class.numeral\" 
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(mkFunT numT t)) 
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(mkNumeral i) 
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else 
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Generated'_Code.App 
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(Generated'_Code.Const \"Groups.uminus'_class.uminus\" (mkFunT t t)) 
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(mkNumber ( i)); } in mkNumber)" 
43356  318 

61799  319 
subsection \<open>The \<open>find_unused_assms\<close> command\<close> 
46589  320 

48891  321 
ML_file "Tools/Quickcheck/find_unused_assms.ML" 
46589  322 

60758  323 
subsection \<open>Closing up\<close> 
46589  324 

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hide_type narrowing_type narrowing_term narrowing_cons property 
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hide_const map_cons nth error toEnum marker empty Narrowing_cons conv non_empty ensure_testable all exists drawn_from around_zero 
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hide_const (open) Narrowing_variable Narrowing_constructor "apply" sum cons 
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hide_fact empty_def cons_def conv.simps non_empty.simps apply_def sum_def ensure_testable_def all_def exists_def 
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330 
end 