author  webertj 
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parent 49834  b27bbb021df1 
child 50046  0051dc4f301f 
permissions  rwrr 
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(* Author: Lukas Bulwahn, TU Muenchen *) 
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header {* Counterexample generator performing narrowingbased testing *} 
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theory Quickcheck_Narrowing 
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imports Quickcheck_Exhaustive 
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keywords "find_unused_assms" :: diag 
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begin 
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subsection {* Counterexample generator *} 

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text {* We create a new target for the necessary code generation setup. *} 
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setup {* Code_Target.extend_target ("Haskell_Quickcheck", (Code_Haskell.target, K I)) *} 
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subsubsection {* Code generation setup *} 
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code_type typerep 
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(Haskell_Quickcheck "Typerep") 
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code_const Typerep.Typerep 
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(Haskell_Quickcheck "Typerep") 
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code_reserved Haskell_Quickcheck Typerep 
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subsubsection {* Type @{text "code_int"} for Haskell Quickcheck's Int type *} 
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typedef 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 equal_int_refl) 
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end 
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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, neg_numeral, 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 = of_int ( int_of n)" 
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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 distrib_right zmult_zless_mono2) 
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end 
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lemma int_of_numeral [simp]: 
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"int_of (numeral k) = numeral k" 
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by (induct k) (simp_all only: numeral.simps plus_code_int_def 
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one_code_int_def of_int_inverse UNIV_I) 
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definition Num :: "num \<Rightarrow> code_int" 
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where [code_abbrev]: "Num = numeral" 
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lemma [code_abbrev]: 
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" numeral k = (neg_numeral k :: code_int)" 
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by (unfold neg_numeral_def) simp 
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code_datatype "0::code_int" Num 
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lemma one_code_int_code [code, code_unfold]: 
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"(1\<Colon>code_int) = Numeral1" 
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by (simp only: numeral.simps) 
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definition div_mod :: "code_int \<Rightarrow> code_int \<Rightarrow> code_int \<times> code_int" where 
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[code del]: "div_mod n m = (n div m, n mod m)" 
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134 

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lemma [code]: 
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"n div m = fst (div_mod n m)" 
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unfolding div_mod_def by simp 
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138 

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lemma [code]: 
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"n mod m = snd (div_mod n m)" 
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unfolding div_mod_def by simp 
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142 

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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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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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code_instance code_numeral :: equal 
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(Haskell_Quickcheck ) 
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setup {* fold (Numeral.add_code @{const_name Num} 
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false Code_Printer.literal_numeral) ["Haskell_Quickcheck"] *} 
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161 

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code_type code_int 
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(Haskell_Quickcheck "Prelude.Int") 
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code_const "0 \<Colon> code_int" 
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(Haskell_Quickcheck "0") 
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167 

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code_const "1 \<Colon> code_int" 
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(Haskell_Quickcheck "1") 
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170 

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code_const "minus \<Colon> code_int \<Rightarrow> code_int \<Rightarrow> code_int" 
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(Haskell_Quickcheck infixl 6 "") 
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code_const div_mod 
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(Haskell_Quickcheck "divMod") 
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176 

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code_const "HOL.equal \<Colon> code_int \<Rightarrow> code_int \<Rightarrow> bool" 
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(Haskell_Quickcheck infix 4 "==") 
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code_const "less_eq \<Colon> code_int \<Rightarrow> code_int \<Rightarrow> bool" 
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(Haskell_Quickcheck infix 4 "<=") 
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code_const "less \<Colon> code_int \<Rightarrow> code_int \<Rightarrow> bool" 
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(Haskell_Quickcheck infix 4 "<") 
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185 

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code_abort of_int 
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hide_const (open) Num div_mod 
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189 

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subsubsection {* Narrowing's deep representation of types and terms *} 
41905  191 

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datatype narrowing_type = Narrowing_sum_of_products "narrowing_type list list" 
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datatype narrowing_term = Narrowing_variable "code_int list" narrowing_type  Narrowing_constructor code_int "narrowing_term list" 
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datatype 'a narrowing_cons = Narrowing_cons narrowing_type "(narrowing_term list => 'a) list" 
41905  195 

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primrec map_cons :: "('a => 'b) => 'a narrowing_cons => 'b narrowing_cons" 
43356  197 
where 
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"map_cons f (Narrowing_cons ty cs) = Narrowing_cons ty (map (%c. f o c) cs)" 
43356  199 

43341  200 
subsubsection {* From narrowing's deep representation of terms to @{theory Code_Evaluation}'s terms *} 
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201 

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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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204 

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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) 
43356  207 

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subsubsection {* Auxilary functions for Narrowing *} 
41905  209 

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consts nth :: "'a list => code_int => 'a" 
41905  211 

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code_const nth (Haskell_Quickcheck infixl 9 "!!") 
41905  213 

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consts error :: "char list => 'a" 
41905  215 

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code_const error (Haskell_Quickcheck "error") 
41905  217 

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consts toEnum :: "code_int => char" 
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219 

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code_const toEnum (Haskell_Quickcheck "Prelude.toEnum") 
41905  221 

43316  222 
consts marker :: "char" 
41905  223 

43316  224 
code_const marker (Haskell_Quickcheck "''\\0'") 
225 

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subsubsection {* Narrowing's basic operations *} 
41905  227 

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type_synonym 'a narrowing = "code_int => 'a narrowing_cons" 
41905  229 

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definition empty :: "'a narrowing" 
41905  231 
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"empty d = Narrowing_cons (Narrowing_sum_of_products []) []" 
41905  233 

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definition cons :: "'a => 'a narrowing" 
41905  235 
where 
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"cons a d = (Narrowing_cons (Narrowing_sum_of_products [[]]) [(%_. a)])" 
41905  237 

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fun conv :: "(narrowing_term list => 'a) list => narrowing_term => 'a" 
41905  239 
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" 
41905  242 

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fun non_empty :: "narrowing_type => bool" 
41905  244 
where 
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"non_empty (Narrowing_sum_of_products ps) = (\<not> (List.null ps))" 
41905  246 

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definition "apply" :: "('a => 'b) narrowing => 'a narrowing => 'b narrowing" 
41905  248 
where 
249 
"apply f a d = 

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(case f d of Narrowing_cons (Narrowing_sum_of_products ps) cfs => 
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case a (d  1) of Narrowing_cons ta cas => 
41905  252 
let 
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shallow = (d > 0 \<and> non_empty ta); 
41905  254 
cs = [(%xs'. (case xs' of [] => undefined  x # xs => cf xs (conv cas x))). shallow, cf < cfs] 
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in Narrowing_cons (Narrowing_sum_of_products [ta # p. shallow, p < ps]) cs)" 
41905  256 

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definition sum :: "'a narrowing => 'a narrowing => 'a narrowing" 
41905  258 
where 
259 
"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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262 
Narrowing_cons (Narrowing_sum_of_products (ssa @ ssb)) (ca @ cb))" 
41905  263 

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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 <= d' & d' < d ==> 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 moreover 
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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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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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ultimately show ?thesis 
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unfolding apply_def by (auto split: narrowing_cons.split narrowing_type.split simp add: Let_def) 
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qed 
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subsubsection {* Narrowing generator type class *} 
41905  285 

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class narrowing = 
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fixes narrowing :: "code_int => 'a narrowing_cons" 
41905  288 

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datatype property = Universal narrowing_type "(narrowing_term => property)" "narrowing_term => Code_Evaluation.term"  Existential narrowing_type "(narrowing_term => property)" "narrowing_term => Code_Evaluation.term"  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 :: code_int) 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 :: code_int) of Narrowing_cons ty cs => Universal ty (\<lambda>t. f (conv cs t)) (partial_term_of (TYPE('a))))" 
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41943  300 
subsubsection {* class @{text is_testable} *} 
41905  301 

41943  302 
text {* The class @{text is_testable} ensures that all necessary type instances are generated. *} 
41905  303 

304 
class is_testable 

305 

306 
instance bool :: is_testable .. 

307 

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instance "fun" :: ("{term_of, narrowing, partial_term_of}", is_testable) is_testable .. 
41905  309 

310 
definition ensure_testable :: "'a :: is_testable => 'a :: is_testable" 

311 
where 

312 
"ensure_testable f = f" 

313 

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subsubsection {* Defining a simple datatype to represent functions in an incomplete and redundant way *} 
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datatype ('a, 'b) ffun = Constant 'b  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 '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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subsubsection {* Setting up the counterexample generator *} 
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48891  338 
ML_file "Tools/Quickcheck/narrowing_generators.ML" 
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setup {* Narrowing_Generators.setup *} 
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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 :: "code_int => ('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 :: code_int => 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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46308  358 
subsection {* Narrowing for sets *} 
359 

360 
instantiation set :: (narrowing) narrowing 

361 
begin 

362 

363 
definition "narrowing_set = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.cons set) narrowing" 

364 

365 
instance .. 

366 

367 
end 

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43356  369 
subsection {* Narrowing for integers *} 
370 

371 

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definition drawn_from :: "'a list => 'a narrowing_cons" 
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where "drawn_from xs = Narrowing_cons (Narrowing_sum_of_products (map (%_. []) xs)) (map (%x y. x) xs)" 
43356  374 

375 
function around_zero :: "int => int list" 

376 
where 

377 
"around_zero i = (if i < 0 then [] else (if i = 0 then [0] else around_zero (i  1) @ [i, i]))" 

378 
by pat_completeness auto 

379 
termination by (relation "measure nat") auto 

380 

381 
declare around_zero.simps[simp del] 

382 

383 
lemma length_around_zero: 

384 
assumes "i >= 0" 

385 
shows "length (around_zero i) = 2 * nat i + 1" 

386 
proof (induct rule: int_ge_induct[OF assms]) 

387 
case 1 

388 
from 1 show ?case by (simp add: around_zero.simps) 

389 
next 

390 
case (2 i) 

391 
from 2 show ?case 

392 
by (simp add: around_zero.simps[of "i + 1"]) 

393 
qed 

394 

395 
instantiation int :: narrowing 

396 
begin 

397 

398 
definition 

399 
"narrowing_int d = (let (u :: _ => _ => unit) = conv; i = Quickcheck_Narrowing.int_of d in drawn_from (around_zero i))" 

400 

401 
instance .. 

402 

403 
end 

404 

405 
lemma [code, code del]: "partial_term_of (ty :: int itself) t == undefined" 

406 
by (rule partial_term_of_anything)+ 

407 

408 
lemma [code]: 

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"partial_term_of (ty :: int itself) (Narrowing_variable p t) == Code_Evaluation.Free (STR ''_'') (Typerep.Typerep (STR ''Int.int'') [])" 
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"partial_term_of (ty :: int itself) (Narrowing_constructor i []) == (if i mod 2 = 0 then 
43356  411 
Code_Evaluation.term_of ( (int_of i) div 2) else Code_Evaluation.term_of ((int_of i + 1) div 2))" 
412 
by (rule partial_term_of_anything)+ 

413 

414 
text {* Defining integers by positive and negative copy of naturals *} 

415 
(* 

416 
datatype simple_int = Positive nat  Negative nat 

417 

418 
primrec int_of_simple_int :: "simple_int => int" 

419 
where 

420 
"int_of_simple_int (Positive n) = int n" 

421 
 "int_of_simple_int (Negative n) = (1  int n)" 

422 

423 
instantiation int :: narrowing 

424 
begin 

425 

426 
definition narrowing_int :: "code_int => int cons" 

427 
where 

428 
"narrowing_int d = map_cons int_of_simple_int ((narrowing :: simple_int narrowing) d)" 

429 

430 
instance .. 

431 

432 
end 

433 

434 
text {* printing the partial terms *} 

435 

436 
lemma [code]: 

437 
"partial_term_of (ty :: int itself) t == Code_Evaluation.App (Code_Evaluation.Const (STR ''Quickcheck_Narrowing.int_of_simple_int'') 

438 
(Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''Quickcheck_Narrowing.simple_int'') [], Typerep.Typerep (STR ''Int.int'') []])) (partial_term_of (TYPE(simple_int)) t)" 

439 
by (rule partial_term_of_anything) 

440 

441 
*) 

442 

46589  443 
subsection {* The @{text find_unused_assms} command *} 
444 

48891  445 
ML_file "Tools/Quickcheck/find_unused_assms.ML" 
46589  446 

447 
subsection {* Closing up *} 

448 

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hide_type code_int narrowing_type narrowing_term narrowing_cons property 
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hide_const int_of of_int nat_of 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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end 