1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
|
-----------------------------------------------------------------------------
-- |
-- Module : Data.SBV.Tools.BoundedList
-- Copyright : (c) Levent Erkok
-- License : BSD3
-- Maintainer: erkokl@gmail.com
-- Stability : experimental
--
-- A collection of bounded list utilities, useful when working with symbolic lists.
-- These functions all take a concrete bound, and operate on the prefix of a symbolic
-- list that is at most that long. Due to limitations on writing recursive functions
-- over lists (the classic symbolic termination problem), we cannot write arbitrary
-- recursive programs on symbolic lists. But most of the time all we need is a
-- bounded prefix of this list, at which point these functions come in handy.
-----------------------------------------------------------------------------
{-# LANGUAGE OverloadedLists #-}
{-# LANGUAGE Rank2Types #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# OPTIONS_GHC -Wall -Werror #-}
module Data.SBV.Tools.BoundedList (
-- * General folds
bfoldr, bfoldrM, bfoldl, bfoldlM
-- * Map, filter, zipWith, elem
, bmap, bmapM, bfilter, bzipWith, belem
-- * Aggregates
, bsum, bprod, band, bor, bany, ball, bmaximum, bminimum
-- * Miscellaneous: Reverse and sort
, breverse, bsort
)
where
import Prelude hiding ((++))
import Data.SBV
import Data.SBV.List ((.:), (++))
import qualified Data.SBV.List as L
-- | Case analysis on a symbolic list. (Not exported.)
lcase :: (SymVal a, Mergeable b) => SList a -> b -> (SBV a -> SList a -> b) -> b
lcase s e c = ite (L.null s) e (c (L.head s) (L.tail s))
-- | Bounded fold from the right.
bfoldr :: (SymVal a, SymVal b) => Int -> (SBV a -> SBV b -> SBV b) -> SBV b -> SList a -> SBV b
bfoldr cnt f b = go (cnt `max` 0)
where go 0 _ = b
go i s = lcase s b (\h t -> h `f` go (i-1) t)
-- | Bounded monadic fold from the right.
bfoldrM :: forall a b m. (SymVal a, SymVal b, Monad m, Mergeable (m (SBV b)))
=> Int -> (SBV a -> SBV b -> m (SBV b)) -> SBV b -> SList a -> m (SBV b)
bfoldrM cnt f b = go (cnt `max` 0)
where go :: Int -> SList a -> m (SBV b)
go 0 _ = return b
go i s = lcase s (return b) (\h t -> f h =<< go (i-1) t)
-- | Bounded fold from the left.
bfoldl :: (SymVal a, SymVal b) => Int -> (SBV b -> SBV a -> SBV b) -> SBV b -> SList a -> SBV b
bfoldl cnt f = go (cnt `max` 0)
where go 0 b _ = b
go i b s = lcase s b (\h t -> go (i-1) (b `f` h) t)
-- | Bounded monadic fold from the left.
bfoldlM :: forall a b m. (SymVal a, SymVal b, Monad m, Mergeable (m (SBV b)))
=> Int -> (SBV b -> SBV a -> m (SBV b)) -> SBV b -> SList a -> m (SBV b)
bfoldlM cnt f = go (cnt `max` 0)
where go :: Int -> SBV b -> SList a -> m (SBV b)
go 0 b _ = return b
go i b s = lcase s (return b) (\h t -> do { fbh <- f b h; go (i-1) fbh t })
-- | Bounded sum.
bsum :: (SymVal a, Num a, Ord a) => Int -> SList a -> SBV a
bsum i = bfoldl i (+) 0
-- | Bounded product.
bprod :: (SymVal a, Num a, Ord a) => Int -> SList a -> SBV a
bprod i = bfoldl i (*) 1
-- | Bounded map.
bmap :: (SymVal a, SymVal b) => Int -> (SBV a -> SBV b) -> SList a -> SList b
bmap i f = bfoldr i (\x -> (f x .:)) []
-- | Bounded monadic map.
bmapM :: (SymVal a, SymVal b, Monad m, Mergeable (m (SBV [b])))
=> Int -> (SBV a -> m (SBV b)) -> SList a -> m (SList b)
bmapM i f = bfoldrM i (\a bs -> (.:) <$> f a <*> pure bs) []
-- | Bounded filter.
bfilter :: SymVal a => Int -> (SBV a -> SBool) -> SList a -> SList a
bfilter i f = bfoldr i (\x y -> ite (f x) (x .: y) y) []
-- | Bounded logical and
band :: Int -> SList Bool -> SBool
band i = bfoldr i (.&&) (sTrue :: SBool)
-- | Bounded logical or
bor :: Int -> SList Bool -> SBool
bor i = bfoldr i (.||) (sFalse :: SBool)
-- | Bounded any
bany :: SymVal a => Int -> (SBV a -> SBool) -> SList a -> SBool
bany i f = bor i . bmap i f
-- | Bounded all
ball :: SymVal a => Int -> (SBV a -> SBool) -> SList a -> SBool
ball i f = band i . bmap i f
-- | Bounded maximum. Undefined if list is empty.
bmaximum :: (Ord a, SymVal a) => Int -> SList a -> SBV a
bmaximum i l = bfoldl (i-1) smax (L.head l) (L.tail l)
-- | Bounded minimum. Undefined if list is empty.
bminimum :: (Ord a, SymVal a) => Int -> SList a -> SBV a
bminimum i l = bfoldl (i-1) smin (L.head l) (L.tail l)
-- | Bounded zipWith
bzipWith :: (SymVal a, SymVal b, SymVal c) => Int -> (SBV a -> SBV b -> SBV c) -> SList a -> SList b -> SList c
bzipWith cnt f = go (cnt `max` 0)
where go 0 _ _ = []
go i xs ys = ite (L.null xs .|| L.null ys)
[]
(f (L.head xs) (L.head ys) .: go (i-1) (L.tail xs) (L.tail ys))
-- | Bounded element check
belem :: (Eq a, SymVal a) => Int -> SBV a -> SList a -> SBool
belem i e = bany i (e .==)
-- | Bounded reverse
breverse :: SymVal a => Int -> SList a -> SList a
breverse cnt = bfoldr cnt (\a b -> b ++ L.singleton a) []
-- | Bounded paramorphism (not exported).
bpara :: (SymVal a, SymVal b) => Int -> (SBV a -> SBV [a] -> SBV b -> SBV b) -> SBV b -> SList a -> SBV b
bpara cnt f b = go (cnt `max` 0)
where go 0 _ = b
go i s = lcase s b (\h t -> f h t (go (i-1) t))
-- | Insert an element into a sorted list (not exported).
binsert :: (Ord a, SymVal a) => Int -> SBV a -> SList a -> SList a
binsert cnt a = bpara cnt f (L.singleton a)
where f sortedHd sortedTl sortedTl' = ite (a .< sortedHd)
(a .: sortedHd .: sortedTl)
(sortedHd .: sortedTl')
-- | Bounded insertion sort
bsort :: (Ord a, SymVal a) => Int -> SList a -> SList a
bsort cnt = bfoldr cnt (binsert cnt) []
|
