src/HOL/Quickcheck_Narrowing.thy
author bulwahn
Thu Jun 09 08:32:22 2011 +0200 (2011-06-09)
changeset 43316 3e274608f06b
parent 43315 893de45ac28d
child 43317 f9283eb3a4bf
permissions -rw-r--r--
removing char setup
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(* Author: Lukas Bulwahn, TU Muenchen *)
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header {* Counterexample generator preforming narrowing-based testing *}
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theory Quickcheck_Narrowing
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imports Quickcheck_Exhaustive
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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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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 marker :: "char"
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code_const marker (Haskell_Quickcheck "''\\0'")
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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 []) []"
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definition cons :: "'a => 'a narrowing"
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where
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  "cons a d = (C (SumOfProd [[]]) [(%_. a)])"
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fun conv :: "(narrowing_term list => 'a) list => narrowing_term => 'a"
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where
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  "conv cs (Var p _) = error (marker # map toEnum p)"
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| "conv cs (Ctr i xs) = (nth cs i) xs"
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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))"
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definition "apply" :: "('a => 'b) narrowing => 'a narrowing => 'b narrowing"
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where
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  "apply f a d =
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     (case f d of C (SumOfProd ps) cfs =>
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       case a (d - 1) of C ta cas =>
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       let
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         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]
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       in C (SumOfProd [ta # p. shallow, p <- ps]) cs)"
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definition sum :: "'a narrowing => 'a narrowing => 'a narrowing"
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where
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  "sum a b d =
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    (case a d of C (SumOfProd ssa) ca => 
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      case b d of C (SumOfProd ssb) cb =>
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      C (SumOfProd (ssa @ ssb)) (ca @ cb))"
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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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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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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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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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instantiation int :: narrowing
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begin
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definition
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  "narrowing_int d = (let i = Quickcheck_Narrowing.int_of d in drawn_from [-i .. i])"
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instance ..
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end
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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: hard-wired 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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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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class is_testable
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instance bool :: is_testable ..
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instance "fun" :: ("{term_of, narrowing, partial_term_of}", is_testable) is_testable ..
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definition ensure_testable :: "'a :: is_testable => 'a :: is_testable"
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where
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  "ensure_testable f = f"
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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
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subsubsection {* Setting up the counterexample generator *}
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setup {* Thy_Load.provide_file (Path.explode ("~~/src/HOL/Tools/Quickcheck/PNF_Narrowing_Engine.hs")) *}
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setup {* Thy_Load.provide_file (Path.explode ("~~/src/HOL/Tools/Quickcheck/Narrowing_Engine.hs")) *}
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use "~~/src/HOL/Tools/Quickcheck/narrowing_generators.ML"
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setup {* Narrowing_Generators.setup *}
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hide_type code_int narrowing_type narrowing_term cons property
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hide_const int_of of_int nth error toEnum marker empty
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  C cons conv nonEmpty "apply" sum ensure_testable all exists 
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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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end