author | bulwahn |
Mon, 14 Mar 2011 12:34:08 +0100 | |
changeset 41961 | fdd37cfcd4a3 |
parent 41943 | 12f24ad566ea |
child 41962 | 27a61a3266d8 |
permissions | -rw-r--r-- |
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(* Author: Lukas Bulwahn, TU Muenchen *) |
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header {* Counterexample generator preforming narrowing-based testing *} |
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theory Quickcheck_Narrowing |
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imports Main "~~/src/HOL/Library/Code_Char" |
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uses ("~~/src/HOL/Tools/Quickcheck/narrowing_generators.ML") |
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begin |
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subsection {* Counterexample generator *} |
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subsubsection {* Code generation setup *} |
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code_type typerep |
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("Haskell" "Typerep") |
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code_const Typerep.Typerep |
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("Haskell" "Typerep") |
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code_reserved Haskell Typerep |
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subsubsection {* Type @{text code_int} for Haskell's Int type *} |
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typedef (open) code_int = "UNIV \<Colon> int set" |
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morphisms int_of of_int by rule |
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lemma int_of_inject [simp]: |
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"int_of k = int_of l \<longleftrightarrow> k = l" |
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by (rule int_of_inject) |
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definition nat_of :: "code_int => nat" |
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where |
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"nat_of i = nat (int_of i)" |
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instantiation code_int :: "{zero, one, minus, linorder}" |
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begin |
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definition [simp, code del]: |
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"0 = of_int 0" |
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definition [simp, code del]: |
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"1 = of_int 1" |
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definition [simp, code del]: |
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"n - m = of_int (int_of n - int_of m)" |
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definition [simp, code del]: |
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"n \<le> m \<longleftrightarrow> int_of n \<le> int_of m" |
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definition [simp, code del]: |
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"n < m \<longleftrightarrow> int_of n < int_of m" |
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instance proof qed (auto) |
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end |
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(* |
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lemma zero_code_int_code [code, code_unfold]: |
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"(0\<Colon>code_int) = Numeral0" |
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by (simp add: number_of_code_numeral_def Pls_def) |
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lemma [code_post]: "Numeral0 = (0\<Colon>code_numeral)" |
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using zero_code_numeral_code .. |
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lemma one_code_numeral_code [code, code_unfold]: |
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"(1\<Colon>code_int) = Numeral1" |
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by (simp add: number_of_code_numeral_def Pls_def Bit1_def) |
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lemma [code_post]: "Numeral1 = (1\<Colon>code_int)" |
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using one_code_numeral_code .. |
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*) |
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code_const "0 \<Colon> code_int" |
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(Haskell "0") |
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code_const "1 \<Colon> code_int" |
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(Haskell "1") |
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code_const "minus \<Colon> code_int \<Rightarrow> code_int \<Rightarrow> code_int" |
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(Haskell "(_/ -/ _)") |
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code_const "op \<le> \<Colon> code_int \<Rightarrow> code_int \<Rightarrow> bool" |
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(Haskell infix 4 "<=") |
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code_const "op < \<Colon> code_int \<Rightarrow> code_int \<Rightarrow> bool" |
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(Haskell infix 4 "<") |
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code_type code_int |
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(Haskell "Int") |
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subsubsection {* Narrowing's deep representation of types and terms *} |
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datatype type = SumOfProd "type list list" |
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datatype "term" = Var "code_int list" type | Ctr code_int "term list" |
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datatype 'a cons = C type "(term list => 'a) list" |
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subsubsection {* auxilary functions for Narrowing *} |
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consts nth :: "'a list => code_int => 'a" |
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code_const nth ("Haskell" infixl 9 "!!") |
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consts error :: "char list => 'a" |
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code_const error ("Haskell" "error") |
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consts toEnum :: "code_int => char" |
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code_const toEnum ("Haskell" "toEnum") |
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consts map_index :: "(code_int * 'a => 'b) => 'a list => 'b list" |
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consts split_At :: "code_int => 'a list => 'a list * 'a list" |
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subsubsection {* Narrowing's basic operations *} |
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type_synonym 'a narrowing = "code_int => 'a cons" |
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119 |
definition empty :: "'a narrowing" |
41905 | 120 |
where |
121 |
"empty d = C (SumOfProd []) []" |
|
122 |
||
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123 |
definition cons :: "'a => 'a narrowing" |
41905 | 124 |
where |
125 |
"cons a d = (C (SumOfProd [[]]) [(%_. a)])" |
|
126 |
||
127 |
fun conv :: "(term list => 'a) list => term => 'a" |
|
128 |
where |
|
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129 |
"conv cs (Var p _) = error (Char Nibble0 Nibble0 # map toEnum p)" |
41905 | 130 |
| "conv cs (Ctr i xs) = (nth cs i) xs" |
131 |
||
132 |
fun nonEmpty :: "type => bool" |
|
133 |
where |
|
134 |
"nonEmpty (SumOfProd ps) = (\<not> (List.null ps))" |
|
135 |
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136 |
definition "apply" :: "('a => 'b) narrowing => 'a narrowing => 'b narrowing" |
41905 | 137 |
where |
138 |
"apply f a d = |
|
139 |
(case f d of C (SumOfProd ps) cfs => |
|
140 |
case a (d - 1) of C ta cas => |
|
141 |
let |
|
142 |
shallow = (d > 0 \<and> nonEmpty ta); |
|
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cs = [(%xs'. (case xs' of [] => undefined | x # xs => cf xs (conv cas x))). shallow, cf <- cfs] |
|
144 |
in C (SumOfProd [ta # p. shallow, p <- ps]) cs)" |
|
145 |
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146 |
definition sum :: "'a narrowing => 'a narrowing => 'a narrowing" |
41905 | 147 |
where |
148 |
"sum a b d = |
|
149 |
(case a d of C (SumOfProd ssa) ca => |
|
150 |
case b d of C (SumOfProd ssb) cb => |
|
151 |
C (SumOfProd (ssa @ ssb)) (ca @ cb))" |
|
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153 |
lemma [fundef_cong]: |
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154 |
assumes "a d = a' d" "b d = b' d" "d = d'" |
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155 |
shows "sum a b d = sum a' b' d'" |
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156 |
using assms unfolding sum_def by (auto split: cons.split type.split) |
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157 |
|
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158 |
lemma [fundef_cong]: |
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159 |
assumes "f d = f' d" "(\<And>d'. 0 <= d' & d' < d ==> a d' = a' d')" |
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160 |
assumes "d = d'" |
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161 |
shows "apply f a d = apply f' a' d'" |
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162 |
proof - |
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163 |
note assms moreover |
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164 |
have "int_of (of_int 0) < int_of d' ==> int_of (of_int 0) <= int_of (of_int (int_of d' - int_of (of_int 1)))" |
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by (simp add: of_int_inverse) |
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166 |
moreover |
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167 |
have "int_of (of_int (int_of d' - int_of (of_int 1))) < int_of d'" |
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168 |
by (simp add: of_int_inverse) |
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169 |
ultimately show ?thesis |
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170 |
unfolding apply_def by (auto split: cons.split type.split simp add: Let_def) |
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171 |
qed |
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172 |
|
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173 |
definition cons0 :: "'a => 'a narrowing" |
41905 | 174 |
where |
175 |
"cons0 f = cons f" |
|
176 |
||
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177 |
type_synonym pos = "code_int list" |
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178 |
(* |
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179 |
subsubsection {* Term refinement *} |
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180 |
|
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181 |
definition new :: "pos => type list list => term list" |
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182 |
where |
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183 |
"new p ps = map_index (%(c, ts). Ctr c (map_index (%(i, t). Var (p @ [i]) t) ts)) ps" |
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184 |
|
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185 |
fun refine :: "term => pos => term list" and refineList :: "term list => pos => (term list) list" |
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|
186 |
where |
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|
187 |
"refine (Var p (SumOfProd ss)) [] = new p ss" |
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188 |
| "refine (Ctr c xs) p = map (Ctr c) (refineList xs p)" |
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189 |
| "refineList xs (i # is) = (let (ls, xrs) = split_At i xs in (case xrs of x#rs => [ls @ y # rs. y <- refine x is]))" |
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|
190 |
|
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|
191 |
text {* Find total instantiations of a partial value *} |
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|
192 |
|
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|
193 |
function total :: "term => term list" |
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194 |
where |
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|
195 |
"total (Ctr c xs) = [Ctr c ys. ys <- map total xs]" |
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196 |
| "total (Var p (SumOfProd ss)) = [y. x <- new p ss, y <- total x]" |
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197 |
by pat_completeness auto |
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|
198 |
|
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|
199 |
termination sorry |
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200 |
*) |
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201 |
subsubsection {* Narrowing generator type class *} |
41905 | 202 |
|
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203 |
class narrowing = |
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204 |
fixes narrowing :: "code_int => 'a cons" |
41905 | 205 |
|
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206 |
definition cons1 :: "('a::narrowing => 'b) => 'b narrowing" |
41905 | 207 |
where |
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208 |
"cons1 f = apply (cons f) narrowing" |
41905 | 209 |
|
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210 |
definition cons2 :: "('a :: narrowing => 'b :: narrowing => 'c) => 'c narrowing" |
41905 | 211 |
where |
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212 |
"cons2 f = apply (apply (cons f) narrowing) narrowing" |
41905 | 213 |
|
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214 |
instantiation unit :: narrowing |
41905 | 215 |
begin |
216 |
||
217 |
definition |
|
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218 |
"narrowing = cons0 ()" |
41905 | 219 |
|
220 |
instance .. |
|
221 |
||
222 |
end |
|
223 |
||
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224 |
instantiation bool :: narrowing |
41905 | 225 |
begin |
226 |
||
227 |
definition |
|
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|
228 |
"narrowing = sum (cons0 True) (cons0 False)" |
41905 | 229 |
|
230 |
instance .. |
|
231 |
||
232 |
end |
|
233 |
||
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234 |
instantiation option :: (narrowing) narrowing |
41905 | 235 |
begin |
236 |
||
237 |
definition |
|
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238 |
"narrowing = sum (cons0 None) (cons1 Some)" |
41905 | 239 |
|
240 |
instance .. |
|
241 |
||
242 |
end |
|
243 |
||
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244 |
instantiation sum :: (narrowing, narrowing) narrowing |
41905 | 245 |
begin |
246 |
||
247 |
definition |
|
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248 |
"narrowing = sum (cons1 Inl) (cons1 Inr)" |
41905 | 249 |
|
250 |
instance .. |
|
251 |
||
252 |
end |
|
253 |
||
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|
254 |
instantiation list :: (narrowing) narrowing |
41905 | 255 |
begin |
256 |
||
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|
257 |
function narrowing_list :: "'a list narrowing" |
41905 | 258 |
where |
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259 |
"narrowing_list d = sum (cons []) (apply (apply (cons Cons) narrowing) narrowing_list) d" |
41905 | 260 |
by pat_completeness auto |
261 |
||
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262 |
termination proof (relation "measure nat_of") |
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263 |
qed (auto simp add: of_int_inverse nat_of_def) |
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264 |
|
41905 | 265 |
instance .. |
266 |
||
267 |
end |
|
268 |
||
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269 |
instantiation nat :: narrowing |
41905 | 270 |
begin |
271 |
||
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272 |
function narrowing_nat :: "nat narrowing" |
41905 | 273 |
where |
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274 |
"narrowing_nat d = sum (cons 0) (apply (cons Suc) narrowing_nat) d" |
41905 | 275 |
by pat_completeness auto |
276 |
||
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277 |
termination proof (relation "measure nat_of") |
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278 |
qed (auto simp add: of_int_inverse nat_of_def) |
41905 | 279 |
|
280 |
instance .. |
|
281 |
||
282 |
end |
|
283 |
||
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|
284 |
instantiation Enum.finite_1 :: narrowing |
41905 | 285 |
begin |
286 |
||
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|
287 |
definition narrowing_finite_1 :: "Enum.finite_1 narrowing" |
41905 | 288 |
where |
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|
289 |
"narrowing_finite_1 = cons (Enum.finite_1.a\<^isub>1 :: Enum.finite_1)" |
41905 | 290 |
|
291 |
instance .. |
|
292 |
||
293 |
end |
|
294 |
||
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|
295 |
instantiation Enum.finite_2 :: narrowing |
41905 | 296 |
begin |
297 |
||
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298 |
definition narrowing_finite_2 :: "Enum.finite_2 narrowing" |
41905 | 299 |
where |
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|
300 |
"narrowing_finite_2 = sum (cons (Enum.finite_2.a\<^isub>1 :: Enum.finite_2)) (cons (Enum.finite_2.a\<^isub>2 :: Enum.finite_2))" |
41905 | 301 |
|
302 |
instance .. |
|
303 |
||
304 |
end |
|
305 |
||
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|
306 |
instantiation Enum.finite_3 :: narrowing |
41905 | 307 |
begin |
308 |
||
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|
309 |
definition narrowing_finite_3 :: "Enum.finite_3 narrowing" |
41905 | 310 |
where |
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|
311 |
"narrowing_finite_3 = sum (cons (Enum.finite_3.a\<^isub>1 :: Enum.finite_3)) (sum (cons (Enum.finite_3.a\<^isub>2 :: Enum.finite_3)) (cons (Enum.finite_3.a\<^isub>3 :: Enum.finite_3)))" |
41905 | 312 |
|
313 |
instance .. |
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end |
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instantiation Enum.finite_4 :: narrowing |
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begin |
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definition narrowing_finite_4 :: "Enum.finite_4 narrowing" |
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where |
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"narrowing_finite_4 = sum (cons Enum.finite_4.a\<^isub>1) (sum (cons Enum.finite_4.a\<^isub>2) (sum (cons Enum.finite_4.a\<^isub>3) (cons Enum.finite_4.a\<^isub>4)))" |
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instance .. |
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end |
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subsubsection {* class @{text is_testable} *} |
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text {* The class @{text is_testable} ensures that all necessary type instances are generated. *} |
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class is_testable |
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instance bool :: is_testable .. |
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instance "fun" :: ("{term_of, narrowing}", 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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declare simp_thms(17,19)[code del] |
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subsubsection {* Setting up the counterexample generator *} |
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use "~~/src/HOL/Tools/Quickcheck/narrowing_generators.ML" |
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setup {* Narrowing_Generators.setup *} |
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hide_const (open) empty |
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end |