author  bulwahn 
Fri, 18 Mar 2011 18:19:42 +0100  
changeset 42022  101ce92333f4 
parent 42021  52551c0a3374 
child 42024  51df23535105 
permissions  rwrr 
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(* Author: Lukas Bulwahn, TU Muenchen *) 
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header {* Counterexample generator preforming narrowingbased testing *} 
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theory Quickcheck_Narrowing 
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imports Main "~~/src/HOL/Library/Code_Char" 
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uses 
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("~~/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 of_int_int_of [simp]: 
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"of_int (int_of k) = k" 
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by (rule int_of_inverse) 
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lemma int_of_of_int [simp]: 
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"int_of (of_int n) = n" 
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by (rule of_int_inverse) (rule UNIV_I) 
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lemma code_int: 
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"(\<And>n\<Colon>code_int. PROP P n) \<equiv> (\<And>n\<Colon>int. PROP P (of_int n))" 
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proof 
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fix n :: int 
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assume "\<And>n\<Colon>code_int. PROP P n" 
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then show "PROP P (of_int n)" . 
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next 
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fix n :: code_int 
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assume "\<And>n\<Colon>int. PROP P (of_int n)" 
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then have "PROP P (of_int (int_of n))" . 
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then show "PROP P n" by simp 
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qed 
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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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lemma of_int_inject [simp]: 
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"of_int n = of_int m \<longleftrightarrow> n = m" 
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by (rule of_int_inject) (rule UNIV_I)+ 
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instantiation code_int :: equal 
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begin 
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definition 
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"HOL.equal k l \<longleftrightarrow> HOL.equal (int_of k) (int_of l)" 
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instance proof 
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qed (auto simp add: equal_code_int_def equal_int_def eq_int_refl) 
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end 
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instantiation code_int :: number 
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begin 
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definition 
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"number_of = of_int" 
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instance .. 
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end 
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lemma int_of_number [simp]: 
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"int_of (number_of k) = number_of k" 
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by (simp add: number_of_code_int_def number_of_is_id) 
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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 :: "{minus, linordered_semidom, semiring_div, 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  m = of_int (int_of n  int_of m)" 
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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 div m = of_int (int_of n div int_of m)" 
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definition [simp, code del]: 
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"n mod m = of_int (int_of n mod 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 
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qed (auto simp add: code_int left_distrib zmult_zless_mono2) 
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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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129 

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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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136 

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definition div_mod_code_int :: "code_int \<Rightarrow> code_int \<Rightarrow> code_int \<times> code_int" where 
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[code del]: "div_mod_code_int n m = (n div m, n mod m)" 
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139 

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lemma [code]: 
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"div_mod_code_int n m = (if m = 0 then (0, n) else (n div m, n mod m))" 
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unfolding div_mod_code_int_def by auto 
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143 

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lemma [code]: 
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"n div m = fst (div_mod_code_int n m)" 
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unfolding div_mod_code_int_def by simp 
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147 

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lemma [code]: 
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"n mod m = snd (div_mod_code_int n m)" 
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unfolding div_mod_code_int_def by simp 
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151 

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lemma int_of_code [code]: 
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"int_of k = (if k = 0 then 0 
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else (if k mod 2 = 0 then 2 * int_of (k div 2) else 2 * int_of (k div 2) + 1))" 
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proof  
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have 1: "(int_of k div 2) * 2 + int_of k mod 2 = int_of k" 
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by (rule mod_div_equality) 
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have "int_of k mod 2 = 0 \<or> int_of k mod 2 = 1" by auto 
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159 
from this show ?thesis 
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apply auto 
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apply (insert 1) by (auto simp add: mult_ac) 
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qed 
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163 

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164 

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code_instance code_numeral :: equal 
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166 
(Haskell ) 
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167 

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setup {* fold (Numeral.add_code @{const_name number_code_int_inst.number_of_code_int} 
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false Code_Printer.literal_numeral) ["Haskell"] *} 
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170 

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code_const "0 \<Colon> code_int" 
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(Haskell "0") 
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173 

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code_const "1 \<Colon> code_int" 
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(Haskell "1") 
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176 

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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 div_mod_code_int 
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(Haskell "divMod") 
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code_const "HOL.equal \<Colon> code_int \<Rightarrow> code_int \<Rightarrow> bool" 
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(Haskell infix 4 "==") 
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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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188 

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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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191 

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code_type code_int 
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(Haskell "Int") 
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code_abort of_int 
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196 

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subsubsection {* Narrowing's deep representation of types and terms *} 
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199 
datatype type = SumOfProd "type list list" 

200 

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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" 

204 

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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") 

214 

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consts toEnum :: "code_int => char" 
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216 

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code_const toEnum ("Haskell" "toEnum") 
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consts map_index :: "(code_int * 'a => 'b) => 'a list => 'b list" 
41905  220 

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consts split_At :: "code_int => 'a list => 'a list * 'a list" 
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222 

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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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definition empty :: "'a narrowing" 
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where 
229 
"empty d = C (SumOfProd []) []" 

230 

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definition cons :: "'a => 'a narrowing" 
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where 
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"cons a d = (C (SumOfProd [[]]) [(%_. a)])" 

234 

235 
fun conv :: "(term list => 'a) list => term => 'a" 

236 
where 

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"conv cs (Var p _) = error (Char Nibble0 Nibble0 # map toEnum p)" 
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 "conv cs (Ctr i xs) = (nth cs i) xs" 
239 

240 
fun nonEmpty :: "type => bool" 

241 
where 

242 
"nonEmpty (SumOfProd ps) = (\<not> (List.null ps))" 

243 

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definition "apply" :: "('a => 'b) narrowing => 'a narrowing => 'b narrowing" 
41905  245 
where 
246 
"apply f a d = 

247 
(case f d of C (SumOfProd ps) cfs => 

248 
case a (d  1) of C ta cas => 

249 
let 

250 
shallow = (d > 0 \<and> nonEmpty ta); 

251 
cs = [(%xs'. (case xs' of [] => undefined  x # xs => cf xs (conv cas x))). shallow, cf < cfs] 

252 
in C (SumOfProd [ta # p. shallow, p < ps]) cs)" 

253 

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definition sum :: "'a narrowing => 'a narrowing => 'a narrowing" 
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where 
256 
"sum a b d = 

257 
(case a d of C (SumOfProd ssa) ca => 

258 
case b d of C (SumOfProd ssb) cb => 

259 
C (SumOfProd (ssa @ ssb)) (ca @ cb))" 

260 

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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: cons.split type.split) 
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265 

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lemma [fundef_cong]: 
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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 moreover 
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272 
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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274 
moreover 
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have "int_of (of_int (int_of d'  int_of (of_int 1))) < int_of d'" 
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by (simp add: of_int_inverse) 
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277 
ultimately show ?thesis 
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unfolding apply_def by (auto split: cons.split type.split simp add: Let_def) 
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279 
qed 
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280 

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type_synonym pos = "code_int list" 
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(* 
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subsubsection {* Term refinement *} 
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284 

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285 
definition new :: "pos => type list list => term list" 
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286 
where 
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"new p ps = map_index (%(c, ts). Ctr c (map_index (%(i, t). Var (p @ [i]) t) ts)) ps" 
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288 

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fun refine :: "term => pos => term list" and refineList :: "term list => pos => (term list) list" 
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290 
where 
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"refine (Var p (SumOfProd ss)) [] = new p ss" 
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 "refine (Ctr c xs) p = map (Ctr c) (refineList xs p)" 
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 "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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294 

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295 
text {* Find total instantiations of a partial value *} 
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296 

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297 
function total :: "term => term list" 
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298 
where 
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299 
"total (Ctr c xs) = [Ctr c ys. ys < map total xs]" 
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300 
 "total (Var p (SumOfProd ss)) = [y. x < new p ss, y < total x]" 
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301 
by pat_completeness auto 
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302 

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303 
termination sorry 
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304 
*) 
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subsubsection {* Narrowing generator type class *} 
41905  306 

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class narrowing = 
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fixes narrowing :: "code_int => 'a cons" 
41905  309 

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definition cons1 :: "('a::narrowing => 'b) => 'b narrowing" 
41905  311 
where 
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"cons1 f = apply (cons f) narrowing" 
41905  313 

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definition cons2 :: "('a :: narrowing => 'b :: narrowing => 'c) => 'c narrowing" 
41905  315 
where 
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"cons2 f = apply (apply (cons f) narrowing) narrowing" 
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317 

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definition drawn_from :: "'a list => 'a cons" 
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where "drawn_from xs = C (SumOfProd (map (%_. []) xs)) (map (%x y. x) xs)" 
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320 

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321 
instantiation int :: narrowing 
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322 
begin 
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323 

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324 
definition 
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"narrowing_int d = (let i = Quickcheck_Narrowing.int_of d in drawn_from [i .. i])" 
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326 

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327 
instance .. 
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328 

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329 
end 
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330 

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331 
instantiation unit :: narrowing 
41905  332 
begin 
333 

334 
definition 

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335 
"narrowing = cons ()" 
41905  336 

337 
instance .. 

338 

339 
end 

340 

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341 
instantiation bool :: narrowing 
41905  342 
begin 
343 

344 
definition 

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"narrowing = sum (cons True) (cons False)" 
41905  346 

347 
instance .. 

348 

349 
end 

350 

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351 
instantiation option :: (narrowing) narrowing 
41905  352 
begin 
353 

354 
definition 

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355 
"narrowing = sum (cons None) (cons1 Some)" 
41905  356 

357 
instance .. 

358 

359 
end 

360 

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361 
instantiation sum :: (narrowing, narrowing) narrowing 
41905  362 
begin 
363 

364 
definition 

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365 
"narrowing = sum (cons1 Inl) (cons1 Inr)" 
41905  366 

367 
instance .. 

368 

369 
end 

370 

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371 
instantiation list :: (narrowing) narrowing 
41905  372 
begin 
373 

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374 
function narrowing_list :: "'a list narrowing" 
41905  375 
where 
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376 
"narrowing_list d = sum (cons []) (apply (apply (cons Cons) narrowing) narrowing_list) d" 
41905  377 
by pat_completeness auto 
378 

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379 
termination proof (relation "measure nat_of") 
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380 
qed (auto simp add: of_int_inverse nat_of_def) 
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381 

41905  382 
instance .. 
383 

384 
end 

385 

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386 
instantiation nat :: narrowing 
41905  387 
begin 
388 

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389 
function narrowing_nat :: "nat narrowing" 
41905  390 
where 
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391 
"narrowing_nat d = sum (cons 0) (apply (cons Suc) narrowing_nat) d" 
41905  392 
by pat_completeness auto 
393 

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394 
termination proof (relation "measure nat_of") 
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395 
qed (auto simp add: of_int_inverse nat_of_def) 
41905  396 

397 
instance .. 

398 

399 
end 

400 

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401 
instantiation Enum.finite_1 :: narrowing 
41905  402 
begin 
403 

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404 
definition narrowing_finite_1 :: "Enum.finite_1 narrowing" 
41905  405 
where 
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406 
"narrowing_finite_1 = cons (Enum.finite_1.a\<^isub>1 :: Enum.finite_1)" 
41905  407 

408 
instance .. 

409 

410 
end 

411 

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412 
instantiation Enum.finite_2 :: narrowing 
41905  413 
begin 
414 

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415 
definition narrowing_finite_2 :: "Enum.finite_2 narrowing" 
41905  416 
where 
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417 
"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  418 

419 
instance .. 

420 

421 
end 

422 

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423 
instantiation Enum.finite_3 :: narrowing 
41905  424 
begin 
425 

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426 
definition narrowing_finite_3 :: "Enum.finite_3 narrowing" 
41905  427 
where 
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428 
"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  429 

430 
instance .. 

431 

432 
end 

433 

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434 
instantiation Enum.finite_4 :: narrowing 
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435 
begin 
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436 

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437 
definition narrowing_finite_4 :: "Enum.finite_4 narrowing" 
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438 
where 
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439 
"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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440 

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441 
instance .. 
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442 

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443 
end 
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444 

41943  445 
subsubsection {* class @{text is_testable} *} 
41905  446 

41943  447 
text {* The class @{text is_testable} ensures that all necessary type instances are generated. *} 
41905  448 

449 
class is_testable 

450 

451 
instance bool :: is_testable .. 

452 

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453 
instance "fun" :: ("{term_of, narrowing}", is_testable) is_testable .. 
41905  454 

455 
definition ensure_testable :: "'a :: is_testable => 'a :: is_testable" 

456 
where 

457 
"ensure_testable f = f" 

458 

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459 
declare simp_thms(17,19)[code del] 
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460 

41905  461 
subsubsection {* Setting up the counterexample generator *} 
462 

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463 
use "~~/src/HOL/Tools/Quickcheck/narrowing_generators.ML" 
41905  464 

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465 
setup {* Narrowing_Generators.setup *} 
41905  466 

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467 
hide_type (open) code_int type "term" cons 
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468 
hide_const (open) int_of of_int nth error toEnum map_index split_At empty 
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469 
cons conv nonEmpty "apply" sum cons1 cons2 ensure_testable 
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470 

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471 
subsubsection {* Defining a simple datatype to represent functions in an incomplete and redundant way *} 
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472 

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473 
datatype ('a, 'b) ffun = Constant 'b  Update 'a 'b "('a, 'b) ffun" 
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474 

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475 
primrec eval_ffun :: "('a, 'b) ffun => 'a => 'b" 
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adding a simple datatype for representing functions in Quickcheck_Narrowing
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476 
where 
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adding a simple datatype for representing functions in Quickcheck_Narrowing
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477 
"eval_ffun (Constant c) x = c" 
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adding a simple datatype for representing functions in Quickcheck_Narrowing
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478 
 "eval_ffun (Update x' y f) x = (if x = x' then y else eval_ffun f x)" 
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479 

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480 
hide_type (open) ffun 
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481 
hide_const (open) Constant Update eval_ffun 
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482 

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483 

41905  484 
end 