src/HOL/Library/Quickcheck_Narrowing.thy
author bulwahn
Fri Mar 18 18:19:42 2011 +0100 (2011-03-18)
changeset 42024 51df23535105
parent 42022 101ce92333f4
child 42980 859fe9cc0838
permissions -rw-r--r--
handling a quite restricted set of functions in Quickcheck_Narrowing by an easy transformation
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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 Main "~~/src/HOL/Library/Code_Char"
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uses
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  ("~~/src/HOL/Tools/Quickcheck/narrowing_generators.ML")
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begin
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subsection {* Counterexample generator *}
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subsubsection {* Code generation setup *}
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code_type typerep
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  ("Haskell" "Typerep")
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code_const Typerep.Typerep
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  ("Haskell" "Typerep")
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code_reserved Haskell Typerep
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subsubsection {* Type @{text "code_int"} for Haskell's Int type *}
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typedef (open) code_int = "UNIV \<Colon> int set"
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  morphisms int_of of_int by rule
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lemma of_int_int_of [simp]:
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  "of_int (int_of k) = k"
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  by (rule int_of_inverse)
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lemma int_of_of_int [simp]:
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  "int_of (of_int n) = n"
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  by (rule of_int_inverse) (rule UNIV_I)
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lemma code_int:
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  "(\<And>n\<Colon>code_int. PROP P n) \<equiv> (\<And>n\<Colon>int. PROP P (of_int n))"
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proof
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  fix n :: int
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  assume "\<And>n\<Colon>code_int. PROP P n"
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  then show "PROP P (of_int n)" .
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next
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  fix n :: code_int
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  assume "\<And>n\<Colon>int. PROP P (of_int n)"
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  then have "PROP P (of_int (int_of n))" .
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  then show "PROP P n" by simp
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qed
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lemma int_of_inject [simp]:
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  "int_of k = int_of l \<longleftrightarrow> k = l"
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  by (rule int_of_inject)
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lemma of_int_inject [simp]:
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  "of_int n = of_int m \<longleftrightarrow> n = m"
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  by (rule of_int_inject) (rule UNIV_I)+
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instantiation code_int :: equal
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begin
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definition
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  "HOL.equal k l \<longleftrightarrow> HOL.equal (int_of k) (int_of l)"
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instance proof
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qed (auto simp add: equal_code_int_def equal_int_def eq_int_refl)
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end
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instantiation code_int :: number
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begin
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definition
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  "number_of = of_int"
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instance ..
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end
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lemma int_of_number [simp]:
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  "int_of (number_of k) = number_of k"
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  by (simp add: number_of_code_int_def number_of_is_id)
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definition nat_of :: "code_int => nat"
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where
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  "nat_of i = nat (int_of i)"
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instantiation code_int :: "{minus, linordered_semidom, semiring_div, linorder}"
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begin
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definition [simp, code del]:
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  "0 = of_int 0"
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definition [simp, code del]:
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  "1 = of_int 1"
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definition [simp, code del]:
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  "n + m = of_int (int_of n + int_of m)"
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definition [simp, code del]:
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  "n - m = of_int (int_of n - int_of m)"
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definition [simp, code del]:
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  "n * m = of_int (int_of n * int_of m)"
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definition [simp, code del]:
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  "n div m = of_int (int_of n div int_of m)"
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definition [simp, code del]:
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  "n mod m = of_int (int_of n mod int_of m)"
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definition [simp, code del]:
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  "n \<le> m \<longleftrightarrow> int_of n \<le> int_of m"
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definition [simp, code del]:
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  "n < m \<longleftrightarrow> int_of n < int_of m"
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instance proof
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qed (auto simp add: code_int left_distrib zmult_zless_mono2)
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end
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(*
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lemma zero_code_int_code [code, code_unfold]:
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  "(0\<Colon>code_int) = Numeral0"
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  by (simp add: number_of_code_numeral_def Pls_def)
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lemma [code_post]: "Numeral0 = (0\<Colon>code_numeral)"
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  using zero_code_numeral_code ..
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lemma one_code_numeral_code [code, code_unfold]:
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  "(1\<Colon>code_int) = Numeral1"
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  by (simp add: number_of_code_numeral_def Pls_def Bit1_def)
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lemma [code_post]: "Numeral1 = (1\<Colon>code_int)"
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  using one_code_numeral_code ..
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*)
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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 -)
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setup {* fold (Numeral.add_code @{const_name number_code_int_inst.number_of_code_int}
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  false Code_Printer.literal_numeral) ["Haskell"]  *}
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code_const "0 \<Colon> code_int"
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  (Haskell "0")
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code_const "1 \<Colon> code_int"
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  (Haskell "1")
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code_const "minus \<Colon> code_int \<Rightarrow> code_int \<Rightarrow> code_int"
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  (Haskell "(_/ -/ _)")
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code_const div_mod_code_int
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  (Haskell "divMod")
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code_const "HOL.equal \<Colon> code_int \<Rightarrow> code_int \<Rightarrow> bool"
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  (Haskell infix 4 "==")
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code_const "op \<le> \<Colon> code_int \<Rightarrow> code_int \<Rightarrow> bool"
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  (Haskell infix 4 "<=")
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code_const "op < \<Colon> code_int \<Rightarrow> code_int \<Rightarrow> bool"
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  (Haskell infix 4 "<")
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code_type code_int
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  (Haskell "Int")
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code_abort of_int
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subsubsection {* Narrowing's deep representation of types and terms *}
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datatype type = SumOfProd "type list list"
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datatype "term" = Var "code_int list" type | Ctr code_int "term list"
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datatype 'a cons = C type "(term list => 'a) list"
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subsubsection {* Auxilary functions for Narrowing *}
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consts nth :: "'a list => code_int => 'a"
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code_const nth ("Haskell" infixl 9  "!!")
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consts error :: "char list => 'a"
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code_const error ("Haskell" "error")
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consts toEnum :: "code_int => char"
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code_const toEnum ("Haskell" "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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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 :: "(term list => 'a) list => term => 'a"
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where
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  "conv cs (Var p _) = error (Char Nibble0 Nibble0 # map toEnum p)"
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| "conv cs (Ctr i xs) = (nth cs i) xs"
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fun nonEmpty :: "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 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 type.split simp add: Let_def)
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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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definition new :: "pos => type list list => term list"
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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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fun refine :: "term => pos => term list" and refineList :: "term list => pos => (term list) list"
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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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text {* Find total instantiations of a partial value *}
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function total :: "term => term list"
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where
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  "total (Ctr c xs) = [Ctr c ys. ys <- map total xs]"
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| "total (Var p (SumOfProd ss)) = [y. x <- new p ss, y <- total x]"
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by pat_completeness auto
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termination sorry
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*)
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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 cons1 :: "('a::narrowing => 'b) => 'b narrowing"
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where
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  "cons1 f = apply (cons f) narrowing"
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definition cons2 :: "('a :: narrowing => 'b :: narrowing => 'c) => 'c narrowing"
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where
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  "cons2 f = apply (apply (cons f) narrowing) narrowing"
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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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instantiation unit :: narrowing
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begin
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definition
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  "narrowing = cons ()"
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instance ..
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end
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instantiation bool :: narrowing
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begin
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definition
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  "narrowing = sum (cons True) (cons False)" 
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instance ..
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end
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instantiation option :: (narrowing) narrowing
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begin
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definition
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  "narrowing = sum (cons None) (cons1 Some)"
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instance ..
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end
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instantiation sum :: (narrowing, narrowing) narrowing
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begin
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definition
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  "narrowing = sum (cons1 Inl) (cons1 Inr)"
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instance ..
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end
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instantiation list :: (narrowing) narrowing
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begin
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function narrowing_list :: "'a list narrowing"
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where
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  "narrowing_list d = sum (cons []) (apply (apply (cons Cons) narrowing) narrowing_list) d"
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by pat_completeness auto
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termination proof (relation "measure nat_of")
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qed (auto simp add: of_int_inverse nat_of_def)
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instance ..
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end
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instantiation nat :: narrowing
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begin
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function narrowing_nat :: "nat narrowing"
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where
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  "narrowing_nat d = sum (cons 0) (apply (cons Suc) narrowing_nat) d"
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by pat_completeness auto
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termination proof (relation "measure nat_of")
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qed (auto simp add: of_int_inverse nat_of_def)
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instance ..
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end
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instantiation Enum.finite_1 :: narrowing
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begin
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definition narrowing_finite_1 :: "Enum.finite_1 narrowing"
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where
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  "narrowing_finite_1 = cons (Enum.finite_1.a\<^isub>1 :: Enum.finite_1)"
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instance ..
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end
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instantiation Enum.finite_2 :: narrowing
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begin
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definition narrowing_finite_2 :: "Enum.finite_2 narrowing"
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where
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  "narrowing_finite_2 = sum (cons (Enum.finite_2.a\<^isub>1 :: Enum.finite_2)) (cons (Enum.finite_2.a\<^isub>2 :: Enum.finite_2))"
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instance ..
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end
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instantiation Enum.finite_3 :: narrowing
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begin
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definition narrowing_finite_3 :: "Enum.finite_3 narrowing"
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where
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  "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)))"
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instance ..
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end
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instantiation Enum.finite_4 :: narrowing
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begin
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definition narrowing_finite_4 :: "Enum.finite_4 narrowing"
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where
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  "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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instance ..
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end
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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}", 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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declare simp_thms(17,19)[code del]
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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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   472
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datatype 'b cfun = Constant 'b
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   474
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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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   478
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hide_type (open) cfun
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hide_const (open) Constant eval_cfun
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   481
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   482
subsubsection {* Setting up the counterexample generator *}
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use "~~/src/HOL/Tools/Quickcheck/narrowing_generators.ML"
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setup {* Narrowing_Generators.setup *}
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   487
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   488
hide_type (open) code_int type "term" cons
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   489
hide_const (open) int_of of_int nth error toEnum map_index split_At empty
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  cons conv nonEmpty "apply" sum cons1 cons2 ensure_testable
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   491
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   492
end