author  bulwahn 
Mon, 19 Sep 2011 16:18:30 +0200  
changeset 45001  5c8d7d6db682 
parent 43887  442aceb54969 
child 45734  1024dd30da42 
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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uses 
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("Tools/Quickcheck/PNF_Narrowing_Engine.hs") 
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("Tools/Quickcheck/Narrowing_Engine.hs") 
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("Tools/Quickcheck/narrowing_generators.ML") 
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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 (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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code_datatype "number_of \<Colon> int \<Rightarrow> code_int" 
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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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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_int_def Pls_def) 
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lemma [code_post]: "Numeral0 = (0\<Colon>code_int)" 
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using zero_code_int_code .. 
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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 add: number_of_code_int_def Pls_def Bit1_def) 
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lemma [code_post]: "Numeral1 = (1\<Colon>code_int)" 
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using one_code_int_code .. 
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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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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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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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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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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 number_code_int_inst.number_of_code_int} 
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false Code_Printer.literal_numeral) ["Haskell_Quickcheck"] *} 
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code_const "0 \<Colon> code_int" 
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(Haskell_Quickcheck "0") 
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code_const "1 \<Colon> code_int" 
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(Haskell_Quickcheck "1") 
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code_const "minus \<Colon> code_int \<Rightarrow> code_int \<Rightarrow> code_int" 
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(Haskell_Quickcheck "(_/ / _)") 
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code_const div_mod_code_int 
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(Haskell_Quickcheck "divMod") 
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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 "op \<le> \<Colon> code_int \<Rightarrow> code_int \<Rightarrow> bool" 
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(Haskell_Quickcheck infix 4 "<=") 
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code_const "op < \<Colon> code_int \<Rightarrow> code_int \<Rightarrow> bool" 
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(Haskell_Quickcheck infix 4 "<") 
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code_type code_int 
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(Haskell_Quickcheck "Int") 
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code_abort of_int 
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subsubsection {* Narrowing's deep representation of types and terms *} 
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datatype narrowing_type = SumOfProd "narrowing_type list list" 
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datatype narrowing_term = Var "code_int list" narrowing_type  Ctr code_int "narrowing_term list" 
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datatype 'a cons = C narrowing_type "(narrowing_term list => 'a) list" 
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primrec map_cons :: "('a => 'b) => 'a cons => 'b cons" 
214 
where 

215 
"map_cons f (C ty cs) = C ty (map (%c. f o c) cs)" 

216 

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subsubsection {* From narrowing's deep representation of terms to @{theory Code_Evaluation}'s terms *} 
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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 {* Auxilary functions for Narrowing *} 
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consts nth :: "'a list => code_int => 'a" 
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code_const nth (Haskell_Quickcheck infixl 9 "!!") 
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consts error :: "char list => 'a" 
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code_const error (Haskell_Quickcheck "error") 
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consts toEnum :: "code_int => char" 
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code_const toEnum (Haskell_Quickcheck "toEnum") 
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43316  239 
consts marker :: "char" 
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43316  241 
code_const marker (Haskell_Quickcheck "''\\0'") 
242 

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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 
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"empty d = C (SumOfProd []) []" 

250 

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

254 

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fun conv :: "(narrowing_term list => 'a) list => narrowing_term => 'a" 
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where 
43316  257 
"conv cs (Var p _) = error (marker # map toEnum p)" 
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 "conv cs (Ctr i xs) = (nth cs i) xs" 
259 

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fun nonEmpty :: "narrowing_type => bool" 
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where 
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"nonEmpty (SumOfProd ps) = (\<not> (List.null ps))" 

263 

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definition "apply" :: "('a => 'b) narrowing => 'a narrowing => 'b narrowing" 
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where 
266 
"apply f a d = 

267 
(case f d of C (SumOfProd ps) cfs => 

268 
case a (d  1) of C ta cas => 

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let 

270 
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] 

272 
in C (SumOfProd [ta # p. shallow, p < ps]) cs)" 

273 

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

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

278 
case b d of C (SumOfProd ssb) cb => 

279 
C (SumOfProd (ssa @ ssb)) (ca @ cb))" 

280 

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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 narrowing_type.split) 
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285 

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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: cons.split narrowing_type.split simp add: Let_def) 
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qed 
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300 

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subsubsection {* Narrowing generator type class *} 
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class narrowing = 
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fixes narrowing :: "code_int => 'a cons" 
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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 C ty cs => Existential ty (\<lambda> t. f (conv cs t)) (partial_term_of (TYPE('a))))" 
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312 

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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 C ty cs => Universal ty (\<lambda>t. f (conv cs t)) (partial_term_of (TYPE('a))))" 
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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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321 
class is_testable 

322 

323 
instance bool :: is_testable .. 

324 

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instance "fun" :: ("{term_of, narrowing, partial_term_of}", is_testable) is_testable .. 
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327 
definition ensure_testable :: "'a :: is_testable => 'a :: is_testable" 

328 
where 

329 
"ensure_testable f = f" 

330 

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331 

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subsubsection {* Defining a simple datatype to represent functions in an incomplete and redundant way *} 
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333 

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datatype ('a, 'b) ffun = Constant 'b  Update 'a 'b "('a, 'b) ffun" 
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335 

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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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340 

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hide_type (open) ffun 
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hide_const (open) Constant Update eval_ffun 
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343 

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datatype 'b cfun = Constant 'b 
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345 

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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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349 

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hide_type (open) cfun 
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hide_const (open) Constant eval_cfun 
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352 

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subsubsection {* Setting up the counterexample generator *} 
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354 

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use "Tools/Quickcheck/narrowing_generators.ML" 
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356 

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setup {* Narrowing_Generators.setup *} 
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358 

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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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362 

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definition narrowing_dummy_narrowing :: "code_int => ('a :: narrowing) cons" 
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where 
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"narrowing_dummy_narrowing = narrowing" 
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366 

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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 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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374 

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375 

43356  376 
subsection {* Narrowing for integers *} 
377 

378 

379 
definition drawn_from :: "'a list => 'a cons" 

380 
where "drawn_from xs = C (SumOfProd (map (%_. []) xs)) (map (%x y. x) xs)" 

381 

382 
function around_zero :: "int => int list" 

383 
where 

384 
"around_zero i = (if i < 0 then [] else (if i = 0 then [0] else around_zero (i  1) @ [i, i]))" 

385 
by pat_completeness auto 

386 
termination by (relation "measure nat") auto 

387 

388 
declare around_zero.simps[simp del] 

389 

390 
lemma length_around_zero: 

391 
assumes "i >= 0" 

392 
shows "length (around_zero i) = 2 * nat i + 1" 

393 
proof (induct rule: int_ge_induct[OF assms]) 

394 
case 1 

395 
from 1 show ?case by (simp add: around_zero.simps) 

396 
next 

397 
case (2 i) 

398 
from 2 show ?case 

399 
by (simp add: around_zero.simps[of "i + 1"]) 

400 
qed 

401 

402 
instantiation int :: narrowing 

403 
begin 

404 

405 
definition 

406 
"narrowing_int d = (let (u :: _ => _ => unit) = conv; i = Quickcheck_Narrowing.int_of d in drawn_from (around_zero i))" 

407 

408 
instance .. 

409 

410 
end 

411 

412 
lemma [code, code del]: "partial_term_of (ty :: int itself) t == undefined" 

413 
by (rule partial_term_of_anything)+ 

414 

415 
lemma [code]: 

416 
"partial_term_of (ty :: int itself) (Var p t) == Code_Evaluation.Free (STR ''_'') (Typerep.Typerep (STR ''Int.int'') [])" 

417 
"partial_term_of (ty :: int itself) (Ctr i []) == (if i mod 2 = 0 then 

418 
Code_Evaluation.term_of ( (int_of i) div 2) else Code_Evaluation.term_of ((int_of i + 1) div 2))" 

419 
by (rule partial_term_of_anything)+ 

420 

421 
text {* Defining integers by positive and negative copy of naturals *} 

422 
(* 

423 
datatype simple_int = Positive nat  Negative nat 

424 

425 
primrec int_of_simple_int :: "simple_int => int" 

426 
where 

427 
"int_of_simple_int (Positive n) = int n" 

428 
 "int_of_simple_int (Negative n) = (1  int n)" 

429 

430 
instantiation int :: narrowing 

431 
begin 

432 

433 
definition narrowing_int :: "code_int => int cons" 

434 
where 

435 
"narrowing_int d = map_cons int_of_simple_int ((narrowing :: simple_int narrowing) d)" 

436 

437 
instance .. 

438 

439 
end 

440 

441 
text {* printing the partial terms *} 

442 

443 
lemma [code]: 

444 
"partial_term_of (ty :: int itself) t == Code_Evaluation.App (Code_Evaluation.Const (STR ''Quickcheck_Narrowing.int_of_simple_int'') 

445 
(Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''Quickcheck_Narrowing.simple_int'') [], Typerep.Typerep (STR ''Int.int'') []])) (partial_term_of (TYPE(simple_int)) t)" 

446 
by (rule partial_term_of_anything) 

447 

448 
*) 

449 

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hide_type code_int narrowing_type narrowing_term cons property 
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hide_const int_of of_int nth error toEnum marker empty C conv nonEmpty ensure_testable all exists 
452 
hide_const (open) Var Ctr "apply" sum cons 

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hide_fact empty_def cons_def conv.simps nonEmpty.simps apply_def sum_def ensure_testable_def all_def exists_def 
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454 

43356  455 

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456 
end 