1(* Author: Lukas Bulwahn, TU Muenchen *) 2 3section \<open>Counterexample generator performing narrowing-based testing\<close> 4 5theory Quickcheck_Narrowing 6imports Quickcheck_Random 7keywords "find_unused_assms" :: diag 8begin 9 10subsection \<open>Counterexample generator\<close> 11 12subsubsection \<open>Code generation setup\<close> 13 14setup \<open>Code_Target.add_derived_target ("Haskell_Quickcheck", [(Code_Haskell.target, I)])\<close> 15 16code_printing 17 code_module Typerep \<rightharpoonup> (Haskell_Quickcheck) \<open> 18data Typerep = Typerep String [Typerep] 19\<close> 20| type_constructor typerep \<rightharpoonup> (Haskell_Quickcheck) "Typerep.Typerep" 21| constant Typerep.Typerep \<rightharpoonup> (Haskell_Quickcheck) "Typerep.Typerep" 22| type_constructor integer \<rightharpoonup> (Haskell_Quickcheck) "Prelude.Int" 23 24code_reserved Haskell_Quickcheck Typerep 25 26code_printing 27 constant "0::integer" \<rightharpoonup> 28 (Haskell_Quickcheck) "!(0/ ::/ Prelude.Int)" 29 30setup \<open> 31 let 32 val target = "Haskell_Quickcheck"; 33 fun print _ = Code_Haskell.print_numeral "Prelude.Int"; 34 in 35 Numeral.add_code @{const_name Code_Numeral.Pos} I print target 36 #> Numeral.add_code @{const_name Code_Numeral.Neg} (~) print target 37 end 38\<close> 39 40 41subsubsection \<open>Narrowing's deep representation of types and terms\<close> 42 43datatype (plugins only: code extraction) narrowing_type = 44 Narrowing_sum_of_products "narrowing_type list list" 45 46datatype (plugins only: code extraction) narrowing_term = 47 Narrowing_variable "integer list" narrowing_type 48| Narrowing_constructor integer "narrowing_term list" 49 50datatype (plugins only: code extraction) (dead 'a) narrowing_cons = 51 Narrowing_cons narrowing_type "(narrowing_term list \<Rightarrow> 'a) list" 52 53primrec map_cons :: "('a => 'b) => 'a narrowing_cons => 'b narrowing_cons" 54where 55 "map_cons f (Narrowing_cons ty cs) = Narrowing_cons ty (map (\<lambda>c. f \<circ> c) cs)" 56 57subsubsection \<open>From narrowing's deep representation of terms to @{theory HOL.Code_Evaluation}'s terms\<close> 58 59class partial_term_of = typerep + 60 fixes partial_term_of :: "'a itself => narrowing_term => Code_Evaluation.term" 61 62lemma partial_term_of_anything: "partial_term_of x nt \<equiv> t" 63 by (rule eq_reflection) (cases "partial_term_of x nt", cases t, simp) 64 65subsubsection \<open>Auxilary functions for Narrowing\<close> 66 67consts nth :: "'a list => integer => 'a" 68 69code_printing constant nth \<rightharpoonup> (Haskell_Quickcheck) infixl 9 "!!" 70 71consts error :: "char list => 'a" 72 73code_printing constant error \<rightharpoonup> (Haskell_Quickcheck) "error" 74 75consts toEnum :: "integer => char" 76 77code_printing constant toEnum \<rightharpoonup> (Haskell_Quickcheck) "Prelude.toEnum" 78 79consts marker :: "char" 80 81code_printing constant marker \<rightharpoonup> (Haskell_Quickcheck) "''\\0'" 82 83subsubsection \<open>Narrowing's basic operations\<close> 84 85type_synonym 'a narrowing = "integer => 'a narrowing_cons" 86 87definition cons :: "'a => 'a narrowing" 88where 89 "cons a d = (Narrowing_cons (Narrowing_sum_of_products [[]]) [(\<lambda>_. a)])" 90 91fun conv :: "(narrowing_term list => 'a) list => narrowing_term => 'a" 92where 93 "conv cs (Narrowing_variable p _) = error (marker # map toEnum p)" 94| "conv cs (Narrowing_constructor i xs) = (nth cs i) xs" 95 96fun non_empty :: "narrowing_type => bool" 97where 98 "non_empty (Narrowing_sum_of_products ps) = (\<not> (List.null ps))" 99 100definition "apply" :: "('a => 'b) narrowing => 'a narrowing => 'b narrowing" 101where 102 "apply f a d = (if d > 0 then 103 (case f d of Narrowing_cons (Narrowing_sum_of_products ps) cfs \<Rightarrow> 104 case a (d - 1) of Narrowing_cons ta cas \<Rightarrow> 105 let 106 shallow = non_empty ta; 107 cs = [(\<lambda>(x # xs) \<Rightarrow> cf xs (conv cas x)). shallow, cf \<leftarrow> cfs] 108 in Narrowing_cons (Narrowing_sum_of_products [ta # p. shallow, p \<leftarrow> ps]) cs) 109 else Narrowing_cons (Narrowing_sum_of_products []) [])" 110 111definition sum :: "'a narrowing => 'a narrowing => 'a narrowing" 112where 113 "sum a b d = 114 (case a d of Narrowing_cons (Narrowing_sum_of_products ssa) ca \<Rightarrow> 115 case b d of Narrowing_cons (Narrowing_sum_of_products ssb) cb \<Rightarrow> 116 Narrowing_cons (Narrowing_sum_of_products (ssa @ ssb)) (ca @ cb))" 117 118lemma [fundef_cong]: 119 assumes "a d = a' d" "b d = b' d" "d = d'" 120 shows "sum a b d = sum a' b' d'" 121using assms unfolding sum_def by (auto split: narrowing_cons.split narrowing_type.split) 122 123lemma [fundef_cong]: 124 assumes "f d = f' d" "(\<And>d'. 0 \<le> d' \<and> d' < d \<Longrightarrow> a d' = a' d')" 125 assumes "d = d'" 126 shows "apply f a d = apply f' a' d'" 127proof - 128 note assms 129 moreover have "0 < d' \<Longrightarrow> 0 \<le> d' - 1" 130 by (simp add: less_integer_def less_eq_integer_def) 131 ultimately show ?thesis 132 by (auto simp add: apply_def Let_def 133 split: narrowing_cons.split narrowing_type.split) 134qed 135 136subsubsection \<open>Narrowing generator type class\<close> 137 138class narrowing = 139 fixes narrowing :: "integer => 'a narrowing_cons" 140 141datatype (plugins only: code extraction) property = 142 Universal narrowing_type "(narrowing_term => property)" "narrowing_term => Code_Evaluation.term" 143| Existential narrowing_type "(narrowing_term => property)" "narrowing_term => Code_Evaluation.term" 144| Property bool 145 146(* FIXME: hard-wired maximal depth of 100 here *) 147definition exists :: "('a :: {narrowing, partial_term_of} => property) => property" 148where 149 "exists f = (case narrowing (100 :: integer) of Narrowing_cons ty cs \<Rightarrow> Existential ty (\<lambda> t. f (conv cs t)) (partial_term_of (TYPE('a))))" 150 151definition "all" :: "('a :: {narrowing, partial_term_of} => property) => property" 152where 153 "all f = (case narrowing (100 :: integer) of Narrowing_cons ty cs \<Rightarrow> Universal ty (\<lambda>t. f (conv cs t)) (partial_term_of (TYPE('a))))" 154 155subsubsection \<open>class \<open>is_testable\<close>\<close> 156 157text \<open>The class \<open>is_testable\<close> ensures that all necessary type instances are generated.\<close> 158 159class is_testable 160 161instance bool :: is_testable .. 162 163instance "fun" :: ("{term_of, narrowing, partial_term_of}", is_testable) is_testable .. 164 165definition ensure_testable :: "'a :: is_testable => 'a :: is_testable" 166where 167 "ensure_testable f = f" 168 169 170subsubsection \<open>Defining a simple datatype to represent functions in an incomplete and redundant way\<close> 171 172datatype (plugins only: code quickcheck_narrowing extraction) (dead 'a, dead 'b) ffun = 173 Constant 'b 174| Update 'a 'b "('a, 'b) ffun" 175 176primrec eval_ffun :: "('a, 'b) ffun => 'a => 'b" 177where 178 "eval_ffun (Constant c) x = c" 179| "eval_ffun (Update x' y f) x = (if x = x' then y else eval_ffun f x)" 180 181hide_type (open) ffun 182hide_const (open) Constant Update eval_ffun 183 184datatype (plugins only: code quickcheck_narrowing extraction) (dead 'b) cfun = Constant 'b 185 186primrec eval_cfun :: "'b cfun => 'a => 'b" 187where 188 "eval_cfun (Constant c) y = c" 189 190hide_type (open) cfun 191hide_const (open) Constant eval_cfun Abs_cfun Rep_cfun 192 193subsubsection \<open>Setting up the counterexample generator\<close> 194 195external_file "~~/src/HOL/Tools/Quickcheck/Narrowing_Engine.hs" 196external_file "~~/src/HOL/Tools/Quickcheck/PNF_Narrowing_Engine.hs" 197ML_file "Tools/Quickcheck/narrowing_generators.ML" 198 199definition narrowing_dummy_partial_term_of :: "('a :: partial_term_of) itself => narrowing_term => term" 200where 201 "narrowing_dummy_partial_term_of = partial_term_of" 202 203definition narrowing_dummy_narrowing :: "integer => ('a :: narrowing) narrowing_cons" 204where 205 "narrowing_dummy_narrowing = narrowing" 206 207lemma [code]: 208 "ensure_testable f = 209 (let 210 x = narrowing_dummy_narrowing :: integer => bool narrowing_cons; 211 y = narrowing_dummy_partial_term_of :: bool itself => narrowing_term => term; 212 z = (conv :: _ => _ => unit) in f)" 213unfolding Let_def ensure_testable_def .. 214 215subsection \<open>Narrowing for sets\<close> 216 217instantiation set :: (narrowing) narrowing 218begin 219 220definition "narrowing_set = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.cons set) narrowing" 221 222instance .. 223 224end 225 226subsection \<open>Narrowing for integers\<close> 227 228 229definition drawn_from :: "'a list \<Rightarrow> 'a narrowing_cons" 230where 231 "drawn_from xs = 232 Narrowing_cons (Narrowing_sum_of_products (map (\<lambda>_. []) xs)) (map (\<lambda>x _. x) xs)" 233 234function around_zero :: "int \<Rightarrow> int list" 235where 236 "around_zero i = (if i < 0 then [] else (if i = 0 then [0] else around_zero (i - 1) @ [i, -i]))" 237 by pat_completeness auto 238termination by (relation "measure nat") auto 239 240declare around_zero.simps [simp del] 241 242lemma length_around_zero: 243 assumes "i >= 0" 244 shows "length (around_zero i) = 2 * nat i + 1" 245proof (induct rule: int_ge_induct [OF assms]) 246 case 1 247 from 1 show ?case by (simp add: around_zero.simps) 248next 249 case (2 i) 250 from 2 show ?case 251 by (simp add: around_zero.simps [of "i + 1"]) 252qed 253 254instantiation int :: narrowing 255begin 256 257definition 258 "narrowing_int d = (let (u :: _ \<Rightarrow> _ \<Rightarrow> unit) = conv; i = int_of_integer d 259 in drawn_from (around_zero i))" 260 261instance .. 262 263end 264 265declare [[code drop: "partial_term_of :: int itself \<Rightarrow> _"]] 266 267lemma [code]: 268 "partial_term_of (ty :: int itself) (Narrowing_variable p t) \<equiv> 269 Code_Evaluation.Free (STR ''_'') (Typerep.Typerep (STR ''Int.int'') [])" 270 "partial_term_of (ty :: int itself) (Narrowing_constructor i []) \<equiv> 271 (if i mod 2 = 0 272 then Code_Evaluation.term_of (- (int_of_integer i) div 2) 273 else Code_Evaluation.term_of ((int_of_integer i + 1) div 2))" 274 by (rule partial_term_of_anything)+ 275 276instantiation integer :: narrowing 277begin 278 279definition 280 "narrowing_integer d = (let (u :: _ \<Rightarrow> _ \<Rightarrow> unit) = conv; i = int_of_integer d 281 in drawn_from (map integer_of_int (around_zero i)))" 282 283instance .. 284 285end 286 287declare [[code drop: "partial_term_of :: integer itself \<Rightarrow> _"]] 288 289lemma [code]: 290 "partial_term_of (ty :: integer itself) (Narrowing_variable p t) \<equiv> 291 Code_Evaluation.Free (STR ''_'') (Typerep.Typerep (STR ''Code_Numeral.integer'') [])" 292 "partial_term_of (ty :: integer itself) (Narrowing_constructor i []) \<equiv> 293 (if i mod 2 = 0 294 then Code_Evaluation.term_of (- i div 2) 295 else Code_Evaluation.term_of ((i + 1) div 2))" 296 by (rule partial_term_of_anything)+ 297 298code_printing constant "Code_Evaluation.term_of :: integer \<Rightarrow> term" \<rightharpoonup> (Haskell_Quickcheck) 299 "(let { t = Typerep.Typerep \"Code'_Numeral.integer\" []; 300 mkFunT s t = Typerep.Typerep \"fun\" [s, t]; 301 numT = Typerep.Typerep \"Num.num\" []; 302 mkBit 0 = Generated'_Code.Const \"Num.num.Bit0\" (mkFunT numT numT); 303 mkBit 1 = Generated'_Code.Const \"Num.num.Bit1\" (mkFunT numT numT); 304 mkNumeral 1 = Generated'_Code.Const \"Num.num.One\" numT; 305 mkNumeral i = let { q = i `Prelude.div` 2; r = i `Prelude.mod` 2 } 306 in Generated'_Code.App (mkBit r) (mkNumeral q); 307 mkNumber 0 = Generated'_Code.Const \"Groups.zero'_class.zero\" t; 308 mkNumber 1 = Generated'_Code.Const \"Groups.one'_class.one\" t; 309 mkNumber i = if i > 0 then 310 Generated'_Code.App 311 (Generated'_Code.Const \"Num.numeral'_class.numeral\" 312 (mkFunT numT t)) 313 (mkNumeral i) 314 else 315 Generated'_Code.App 316 (Generated'_Code.Const \"Groups.uminus'_class.uminus\" (mkFunT t t)) 317 (mkNumber (- i)); } in mkNumber)" 318 319subsection \<open>The \<open>find_unused_assms\<close> command\<close> 320 321ML_file "Tools/Quickcheck/find_unused_assms.ML" 322 323subsection \<open>Closing up\<close> 324 325hide_type narrowing_type narrowing_term narrowing_cons property 326hide_const map_cons nth error toEnum marker empty Narrowing_cons conv non_empty ensure_testable all exists drawn_from around_zero 327hide_const (open) Narrowing_variable Narrowing_constructor "apply" sum cons 328hide_fact empty_def cons_def conv.simps non_empty.simps apply_def sum_def ensure_testable_def all_def exists_def 329 330end 331