src/HOL/Library/Quickcheck_Narrowing.thy
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(* Author: Lukas Bulwahn, TU Muenchen *)
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header {* Counterexample generator based on LazySmallCheck *}
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theory LSC
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imports Main "~~/src/HOL/Library/Code_Char"
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uses ("~~/src/HOL/Tools/LSC/lazysmallcheck.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 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 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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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 :: "{zero, one, minus, 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 \<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 qed (auto)
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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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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 "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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subsubsection {* LSC's deep representation of types of 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 LSC *}
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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 {* LSC's basic operations *}
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type_synonym 'a series = "code_int => 'a cons"
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definition empty :: "'a series"
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where
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  "empty d = C (SumOfProd []) []"
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definition cons :: "'a => 'a series"
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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) series => 'a series => 'b series"
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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 series => 'a series => 'a series"
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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 (LSC.of_int 0) < int_of d' ==> int_of (LSC.of_int 0) <= int_of (LSC.of_int (int_of d' - int_of (LSC.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 (LSC.of_int (int_of d' - int_of (LSC.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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definition cons0 :: "'a => 'a series"
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where
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  "cons0 f = cons f"
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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 {* LSC's type class for enumeration *}
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class serial =
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  fixes series :: "code_int => 'a cons"
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definition cons1 :: "('a::serial => 'b) => 'b series"
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where
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  "cons1 f = apply (cons f) series"
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definition cons2 :: "('a :: serial => 'b :: serial => 'c) => 'c series"
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where
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  "cons2 f = apply (apply (cons f) series) series"
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instantiation unit :: serial
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begin
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definition
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  "series = cons0 ()"
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instance ..
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end
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instantiation bool :: serial
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begin
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definition
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  "series = sum (cons0 True) (cons0 False)" 
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instance ..
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end
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instantiation option :: (serial) serial
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begin
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definition
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  "series = sum (cons0 None) (cons1 Some)"
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instance ..
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end
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instantiation sum :: (serial, serial) serial
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begin
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definition
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  "series = sum (cons1 Inl) (cons1 Inr)"
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instance ..
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end
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instantiation list :: (serial) serial
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begin
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function series_list :: "'a list series"
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where
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  "series_list d = sum (cons []) (apply (apply (cons Cons) series) series_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 :: serial
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begin
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function series_nat :: "nat series"
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where
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  "series_nat d = sum (cons 0) (apply (cons Suc) series_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 :: serial
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begin
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definition series_finite_1 :: "Enum.finite_1 series"
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where
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  "series_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 :: serial
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begin
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definition series_finite_2 :: "Enum.finite_2 series"
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where
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  "series_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 :: serial
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begin
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definition series_finite_3 :: "Enum.finite_3 series"
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where
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  "series_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 :: serial
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begin
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definition series_finite_4 :: "Enum.finite_4 series"
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where
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  "series_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 is_testable *}
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text {* The class 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, serial}", 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 {* Setting up the counterexample generator *}
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use "~~/src/HOL/Tools/LSC/lazysmallcheck.ML"
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setup {* Lazysmallcheck.setup *}
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hide_const (open) empty
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end