author  bulwahn 
Thu, 09 Jun 2011 08:32:13 +0200  
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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/PNF_Narrowing_Engine.hs") 
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("~~/src/HOL/Tools/Quickcheck/Narrowing_Engine.hs") 
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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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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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161 

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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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subsubsection {* From narrowing's deep representation of terms to 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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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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243 
type_synonym 'a narrowing = "code_int => 'a cons" 
41905  244 

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definition empty :: "'a narrowing" 
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where 
247 
"empty d = C (SumOfProd []) []" 

248 

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definition cons :: "'a => 'a narrowing" 
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where 
251 
"cons a d = (C (SumOfProd [[]]) [(%_. a)])" 

252 

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fun conv :: "(narrowing_term list => 'a) list => narrowing_term => 'a" 
41905  254 
where 
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"conv cs (Var p _) = error (Char Nibble0 Nibble0 # map toEnum p)" 
41905  256 
 "conv cs (Ctr i xs) = (nth cs i) xs" 
257 

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fun nonEmpty :: "narrowing_type => bool" 
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260 
"nonEmpty (SumOfProd ps) = (\<not> (List.null ps))" 

261 

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definition "apply" :: "('a => 'b) narrowing => 'a narrowing => 'b narrowing" 
41905  263 
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264 
"apply f a d = 

265 
(case f d of C (SumOfProd ps) cfs => 

266 
case a (d  1) of C ta cas => 

267 
let 

268 
shallow = (d > 0 \<and> nonEmpty ta); 

269 
cs = [(%xs'. (case xs' of [] => undefined  x # xs => cf xs (conv cas x))). shallow, cf < cfs] 

270 
in C (SumOfProd [ta # p. shallow, p < ps]) cs)" 

271 

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definition sum :: "'a narrowing => 'a narrowing => 'a narrowing" 
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274 
"sum a b d = 

275 
(case a d of C (SumOfProd ssa) ca => 

276 
case b d of C (SumOfProd ssb) cb => 

277 
C (SumOfProd (ssa @ ssb)) (ca @ cb))" 

278 

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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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283 

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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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288 
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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295 
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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297 
qed 
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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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302 

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definition new :: "pos => type list list => term list" 
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304 
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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306 

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fun refine :: "term => pos => term list" and refineList :: "term list => pos => (term list) list" 
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308 
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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312 

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313 
text {* Find total instantiations of a partial value *} 
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314 

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function total :: "term => term list" 
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316 
where 
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"total (Ctr c xs) = [Ctr c ys. ys < map total xs]" 
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318 
 "total (Var p (SumOfProd ss)) = [y. x < new p ss, y < total x]" 
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319 
by pat_completeness auto 
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320 

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321 
termination sorry 
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322 
*) 
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323 
subsubsection {* Narrowing generator type class *} 
41905  324 

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325 
class narrowing = 
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326 
fixes narrowing :: "code_int => 'a cons" 
41905  327 

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328 
definition cons1 :: "('a::narrowing => 'b) => 'b narrowing" 
41905  329 
where 
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330 
"cons1 f = apply (cons f) narrowing" 
41905  331 

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332 
definition cons2 :: "('a :: narrowing => 'b :: narrowing => 'c) => 'c narrowing" 
41905  333 
where 
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334 
"cons2 f = apply (apply (cons f) narrowing) narrowing" 
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335 

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336 
definition drawn_from :: "'a list => 'a cons" 
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337 
where "drawn_from xs = C (SumOfProd (map (%_. []) xs)) (map (%x y. x) xs)" 
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338 

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339 
instantiation int :: narrowing 
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340 
begin 
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341 

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342 
definition 
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343 
"narrowing_int d = (let i = Quickcheck_Narrowing.int_of d in drawn_from [i .. i])" 
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344 

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345 
instance .. 
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346 

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347 
end 
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348 

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349 
instantiation unit :: narrowing 
41905  350 
begin 
351 

352 
definition 

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353 
"narrowing = cons ()" 
41905  354 

355 
instance .. 

356 

357 
end 

358 

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359 
instantiation bool :: narrowing 
41905  360 
begin 
361 

362 
definition 

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363 
"narrowing = sum (cons True) (cons False)" 
41905  364 

365 
instance .. 

366 

367 
end 

368 

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369 
instantiation option :: (narrowing) narrowing 
41905  370 
begin 
371 

372 
definition 

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373 
"narrowing = sum (cons None) (cons1 Some)" 
41905  374 

375 
instance .. 

376 

377 
end 

378 

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379 
instantiation sum :: (narrowing, narrowing) narrowing 
41905  380 
begin 
381 

382 
definition 

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383 
"narrowing = sum (cons1 Inl) (cons1 Inr)" 
41905  384 

385 
instance .. 

386 

387 
end 

388 

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389 
instantiation list :: (narrowing) narrowing 
41905  390 
begin 
391 

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392 
function narrowing_list :: "'a list narrowing" 
41905  393 
where 
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394 
"narrowing_list d = sum (cons []) (apply (apply (cons Cons) narrowing) narrowing_list) d" 
41905  395 
by pat_completeness auto 
396 

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397 
termination proof (relation "measure nat_of") 
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398 
qed (auto simp add: of_int_inverse nat_of_def) 
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399 

41905  400 
instance .. 
401 

402 
end 

403 

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404 
instantiation nat :: narrowing 
41905  405 
begin 
406 

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407 
function narrowing_nat :: "nat narrowing" 
41905  408 
where 
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409 
"narrowing_nat d = sum (cons 0) (apply (cons Suc) narrowing_nat) d" 
41905  410 
by pat_completeness auto 
411 

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412 
termination proof (relation "measure nat_of") 
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413 
qed (auto simp add: of_int_inverse nat_of_def) 
41905  414 

415 
instance .. 

416 

417 
end 

418 

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419 
instantiation Enum.finite_1 :: narrowing 
41905  420 
begin 
421 

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422 
definition narrowing_finite_1 :: "Enum.finite_1 narrowing" 
41905  423 
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424 
"narrowing_finite_1 = cons (Enum.finite_1.a\<^isub>1 :: Enum.finite_1)" 
41905  425 

426 
instance .. 

427 

428 
end 

429 

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430 
instantiation Enum.finite_2 :: narrowing 
41905  431 
begin 
432 

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433 
definition narrowing_finite_2 :: "Enum.finite_2 narrowing" 
41905  434 
where 
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435 
"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  436 

437 
instance .. 

438 

439 
end 

440 

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441 
instantiation Enum.finite_3 :: narrowing 
41905  442 
begin 
443 

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444 
definition narrowing_finite_3 :: "Enum.finite_3 narrowing" 
41905  445 
where 
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446 
"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  447 

448 
instance .. 

449 

450 
end 

451 

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452 
instantiation Enum.finite_4 :: narrowing 
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453 
begin 
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454 

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455 
definition narrowing_finite_4 :: "Enum.finite_4 narrowing" 
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456 
where 
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457 
"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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458 

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459 
instance .. 
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460 

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461 
end 
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462 

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463 
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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464 

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465 
(* FIXME: hardwired maximal depth of 100 here *) 
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466 
fun exists :: "('a :: {narrowing, partial_term_of} => property) => property" 
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467 
where 
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468 
"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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469 

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470 
fun "all" :: "('a :: {narrowing, partial_term_of} => property) => property" 
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471 
where 
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472 
"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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473 

41943  474 
subsubsection {* class @{text is_testable} *} 
41905  475 

41943  476 
text {* The class @{text is_testable} ensures that all necessary type instances are generated. *} 
41905  477 

478 
class is_testable 

479 

480 
instance bool :: is_testable .. 

481 

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482 
instance "fun" :: ("{term_of, narrowing, partial_term_of}", is_testable) is_testable .. 
41905  483 

484 
definition ensure_testable :: "'a :: is_testable => 'a :: is_testable" 

485 
where 

486 
"ensure_testable f = f" 

487 

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488 
declare simp_thms(17,19)[code del] 
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489 

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490 
subsubsection {* Defining a simple datatype to represent functions in an incomplete and redundant way *} 
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491 

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492 
datatype ('a, 'b) ffun = Constant 'b  Update 'a 'b "('a, 'b) ffun" 
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493 

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494 
primrec eval_ffun :: "('a, 'b) ffun => 'a => 'b" 
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495 
where 
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496 
"eval_ffun (Constant c) x = c" 
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497 
 "eval_ffun (Update x' y f) x = (if x = x' then y else eval_ffun f x)" 
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498 

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499 
hide_type (open) ffun 
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500 
hide_const (open) Constant Update eval_ffun 
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501 

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502 
datatype 'b cfun = Constant 'b 
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503 

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504 
primrec eval_cfun :: "'b cfun => 'a => 'b" 
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505 
where 
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506 
"eval_cfun (Constant c) y = c" 
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507 

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508 
hide_type (open) cfun 
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509 
hide_const (open) Constant eval_cfun 
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510 

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511 
subsubsection {* Setting up the counterexample generator *} 
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512 

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513 
setup {* Thy_Load.provide_file (Path.explode ("~~/src/HOL/Tools/Quickcheck/PNF_Narrowing_Engine.hs")) *} 
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514 
setup {* Thy_Load.provide_file (Path.explode ("~~/src/HOL/Tools/Quickcheck/Narrowing_Engine.hs")) *} 
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515 
use "~~/src/HOL/Tools/Quickcheck/narrowing_generators.ML" 
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516 

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517 
setup {* Narrowing_Generators.setup *} 
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518 

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519 
hide_type (open) code_int narrowing_type narrowing_term cons 
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520 
hide_const (open) int_of of_int nth error toEnum map_index split_At empty 
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521 
C cons conv nonEmpty "apply" sum cons1 cons2 ensure_testable all exists 
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522 

41905  523 
end 