The generation of algebraic data structures is split into two steps. First, structure generators create instances of the structure type where the element nodes are filled with a constant value, or a constant label for multi-sorted structures. These can be considered templates'' with holes'' for data elements. New generators are built by substituting elements, extracted from a generator, into these templates to create one or more copies of each structure. These new generators use the substitution method from the Structure class to integrate the elements into the structures; the structures must be an instance of the Structure class, or Structure2, etc. for multi-sorted structures. The substitution functions use a structure generator and an element generator, with each node (or hole'') in the template filled with a single value. Elements are assumed to be base type generators, meaning they have no rank. The different substitution functions produce differing numbers of copies of each structure template'' to provide some flexibility in building generators. Note that structures can be composed with other structures using the Composition module, which takes into account the rank of both structure types. These compound structures can themselves be filled using the substitution functions. \begin{code}

module Test.GenCheck.Generator.Substitution
( Structure(..)
, subst
, substN
, substAll
-- , subPerm
-- , subComb
, substStdGenN
, substStdGenAll
-- , subStdGenPerm
-- , subStdGenComb
, Structure2(..)
, subst2
, subst2N
, subst2StdGen
-- , subst2StdGenN
-- , subst2StdGenAll
-- , subst2StdGenPerm
-- , subst2StdGenComb
, Structure3(..)
) where

import Data.Maybe (catMaybes)
import Test.GenCheck.Generator.Generator (Generator, StandardGens(..))

\end{code} The Structure class defines the \emph{substitution} method for a structure, the mechanical details of replacing holes'' in the data structure with elements. This is an order based operation with nodes populated from a list of elements. There are two methods provided, one that assumes that enough elements are available and throws an error if that is not so, the other returns a Maybe encapsulated value and any leftover elements from the input list. Instances of Structure can be generated mechanically from the data constructor definitions via Template Haskell. \begin{code}
class Structure c where
substitute    :: c a -> [b] -> (Maybe (c b), [b])

class Structure2 c where
substitute2   :: c a b -> [a'] -> [b'] -> (Maybe (c a' b'), [a'], [b'])

class Structure3 c where
substitute3 :: c a1 a2 a3 -> [a1'] -> [a2'] -> [a3']
-> (Maybe (c a1' a2' a3'), [a1'], [a2'], [a3'])


\end{code} \begin{description} \item[subst] populate one copy of each generated structure \item[substN] populate n copies without reusing the generated elements (does not guarantee unique values of those elements) \item[substAll] populates each instance of the structure of a given rank with the same list of elements, and then all shapes with the next list until exhausting the element generator (non-terminating for infinite generators) \end{description} These three combinators all assume that the elements are of rank 1, regardless of the rank of the structure. This is typical of a structure with base type nodes. \begin{code}
subst :: Structure c => Generator (c a) -> Generator b -> Generator (c b)
subst gfx gy r =
let fxs = (gfx r)
ys  = gy 1
in gsub fxs ys

gsub :: Structure c => [c a] -> [b] -> [c b]
gsub [] _       = []
gsub (fx:fxs) ys =
let (mfy, ys') = substitute fx ys
in case mfy of
Nothing -> []
Just fy -> fy : (gsub fxs ys')

substN :: Structure c => Int -> Generator (c a) -> Generator b -> Generator (c b)
substN n gfx gy r = gsubN n n (gfx r) (gy 1)

gsubN :: Structure c => Int -> Int -> [c a] -> [b] -> [c b]
gsubN _ 0 _  _       = []
gsubN _ _ [] _       = []
gsubN n k fxs@(fx:fxss) ys =
let (mfy, ys') = substitute fx ys
in if k > 1 then maybe [] (\fy -> fy : (gsubN n (k-1) fxs ys')) mfy
else maybe [] (\fy -> fy : (gsubN n n fxss ys')) mfy

substAll :: Structure c => Generator (c a) -> Generator b -> Generator (c b)
substAll gfx gy r =
let ys = gy 1
fxs = gfx r
in gsubAll fxs ys

gsubAll :: Structure c => [c a] -> [b] -> [c b]
gsubAll [] _ = []
gsubAll l@(fx:fxs) ys =
let (mfy, ys') = substitute fx ys  -- subst elements into first structure
-- sub same ys into remaining list of structures, guaranteed finite
fys = catMaybes \$  map (fst.((flip substitute) ys)) fxs
in maybe [] (\x -> x:(fys ++ (gsubAll l ys'))) mfy

\end{code} The standard'' list of generators is exhaustive, extreme (boundary), uniform and random. Using the data type instead of the StandardGens class avoids all sorts of type hassles, and locks the random seed and uniform sampling size into the record. \begin{code}
substStdGenN :: Structure c => Int -> StandardGens (c a) -> Generator b -> StandardGens (c b)
substStdGenN n (StdGens g1 g2 g3 g4) g =
let ga = substN n g1 g
gx = substN n g2 g
gu k = substN n (g3 k) g
gr s = substN n (g4 s) g
in StdGens ga gx gu gr
substStdGenN _ (UnrankedGen _) _ = error "Can only substituted into ranked"

substStdGenAll :: Structure c => StandardGens (c a) -> Generator b -> StandardGens (c b)
substStdGenAll (StdGens g1 g2 g3 g4) g =
let ga = substAll g1 g
gx = substAll g2 g
gu k = substAll (g3 k) g
gr s = substAll (g4 s) g
in StdGens ga gx gu gr
substStdGenAll (UnrankedGen _) _ = error "Can only substitute into ranked"

\end{code} Multi-sort structure substitution. This could be accomplished with multiple steps of single sort substitution, but simultaneous substitution allows more options. Again, the elements are assumed to be of rank 1. \begin{code}
subst2 :: Structure2 c => Generator (c a b) -> Generator a' -> Generator b'
-> Generator (c a' b')
subst2 gfxy gx' gy' r =
let fxs = gfxy r
xs  = gx' 1
ys  = gy' 1
in gsub2 fxs xs ys

gsub2 :: Structure2 c => [c a b] -> [a'] -> [b'] -> [c a' b']
gsub2 [] _ _       = []
gsub2 (fx:fxs) ys zs =
let (mfy, ys', zs') = substitute2 fx ys zs
in maybe [] (\fy -> fy : (gsub2 fxs ys' zs')) mfy

subst2N :: Structure2 c => Int -> Generator (c a b) -> Generator a'
-> Generator b' -> Generator (c a' b')
subst2N n gfx gy gz r = gsub2N n 1 (gfx r) (gy 1) (gz 1)

gsub2N :: Structure2 c => Int -> Int -> [c a b] -> [a'] -> [b'] -> [c a' b']
gsub2N _ 0 _  _ _      = []
gsub2N _ _ [] _ _      = []
gsub2N n k fxs@(fx:fxss) ys zs =
let (mfy, ys', zs') = substitute2 fx ys zs
in if n > k then maybe [] (\fy -> fy : (gsub2N n (k+1) fxs ys' zs')) mfy
else maybe [] (\fy -> fy : (gsub2N n 1 fxss ys' zs')) mfy

subst2StdGen :: Structure2 c => StandardGens (c a b) -> Generator a' -> Generator b'
-> StandardGens (c a' b')
subst2StdGen (StdGens g1 g2 g3 g4) ga' gb' =
let ga = subst2 g1 ga' gb'
gx = subst2 g2 ga' gb'
gu k = subst2 (g3 k) ga' gb'
gr s = subst2 (g4 s) ga' gb'
in StdGens ga gx gu gr
subst2StdGen (UnrankedGen _) _ _ = error "Can only substitute into ranked"

\end{code}
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