agum (empty) → 1.0
raw patch · 7 files changed
+1365/−0 lines, 7 filesdep +basedep +containerssetup-changed
Dependencies added: base, containers
Files
- Makefile +24/−0
- Setup.hs +2/−0
- agum.cabal +34/−0
- license.txt +675/−0
- readme.txt +21/−0
- src/Algebra/AbelianGroup/Main.hs +111/−0
- src/Algebra/AbelianGroup/UnificationMatching.hs +498/−0
+ Makefile view
@@ -0,0 +1,24 @@+# Haskell/Cabal Makefile+# Requires GNU Make+# The all target creates a default configuration if need be.++PACKAGE = agum+CONFIG = dist/setup-config+SETUP = runhaskell Setup.hs++all: $(CONFIG)+ $(SETUP) build++Makefile:+ @echo make $@++$(PACKAGE).cabal:+ @echo make $@++$(CONFIG): $(PACKAGE).cabal+ $(SETUP) configure --ghc --user --prefix="${HOME}"++%: force+ $(SETUP) $@++.PHONY: all force
+ Setup.hs view
@@ -0,0 +1,2 @@+import Distribution.Simple+main = defaultMain
+ agum.cabal view
@@ -0,0 +1,34 @@+Name: agum+Version: 1.0+Maintainer: ramsdell@mitre.org+Cabal-Version: >= 1.2+License: GPL+License-File: license.txt+Synopsis: Unification and Matching in an Abelian Group+Description: The unification problem is given the problem+ statement t =? t\', find a most general+ substitution s such that s(t) = s(t\') modulo+ the axioms of an Abelian group. The matching+ problem is to find a most general substitution+ s such that s(t) = t\' modulo the axioms.+ Substitition s is more general than s\' if+ there is a substitition s\" such that s\' =+ s\" o s.+Category: Algebra.AbelianGroup+Build-Type: Simple+Extra-Source-Files: readme.txt Makefile++Library+ Build-Depends: base <= 4.1.0.0, containers+ Exposed-Modules: Algebra.AbelianGroup.UnificationMatching+ Hs-Source-Dirs: src+ GHC-Options:+ -Wall -fno-warn-name-shadowing -fwarn-unused-imports++Executable agum+ Main-Is: Algebra/AbelianGroup/Main.hs+ Build-Depends: base <= 4.1.0.0, containers+ Other-Modules: Algebra.AbelianGroup.UnificationMatching+ Hs-Source-Dirs: src+ GHC-Options:+ -Wall -fno-warn-name-shadowing -fwarn-unused-imports
+ license.txt view
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Of course, your program's commands+might be different; for a GUI interface, you would use an "about box".++ You should also get your employer (if you work as a programmer) or school,+if any, to sign a "copyright disclaimer" for the program, if necessary.+For more information on this, and how to apply and follow the GNU GPL, see+<http://www.gnu.org/licenses/>.++ The GNU General Public License does not permit incorporating your program+into proprietary programs. If your program is a subroutine library, you+may consider it more useful to permit linking proprietary applications with+the library. If this is what you want to do, use the GNU Lesser General+Public License instead of this License. But first, please read+<http://www.gnu.org/philosophy/why-not-lgpl.html>.
+ readme.txt view
@@ -0,0 +1,21 @@+This package contains a library for unification and matching in+Abelian groups and a program that exercises the library.++$ agum+Abelian group unification and matching -- :? for help+agum> 2x+y=3z+Problem: 2x + y = 3z+Unifier: [x : g0,y : -2g0 + 3g2,z : g2]+Matcher: [x : g0,y : -2g0 + 3z]++agum> 2x=x+y+Problem: 2x = x + y+Unifier: [x : g1,y : g1]+Matcher: no solution++agum> 64x-41y=a+Problem: 64x - 41y = a+Unifier: [x : g0,y : g1,a : 64g0 - 41g1]+Matcher: [x : -16a - 41g6,y : -25]++agum> :quit
+ src/Algebra/AbelianGroup/Main.hs view
@@ -0,0 +1,111 @@+-- A top-level loop for unification and matching in Abelian groups+--+-- Copyright (C) 2009 John D. Ramsdell+--+-- This program is free software: you can redistribute it and/or modify+-- it under the terms of the GNU General Public License as published by+-- the Free Software Foundation, either version 3 of the License, or+-- (at your option) any later version.++-- This program is distributed in the hope that it will be useful,+-- but WITHOUT ANY WARRANTY; without even the implied warranty of+-- MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the+-- GNU General Public License for more details.++-- You should have received a copy of the GNU General Public License+-- along with this program. If not, see <http://www.gnu.org/licenses/>.++module Main (main, test) where++import System.IO (isEOF, hFlush, stdout)+import Algebra.AbelianGroup.UnificationMatching++-- Test Routine++-- Given an equation, display a unifier and a matcher.+test :: String -> IO ()+test prob =+ case readM prob of+ Err err -> putStrLn err+ Ans (Equation (t0, t1)) ->+ do+ putStr "Problem: "+ print $ Equation (t0, t1)+ subst <- unify $ Equation (t0, t1)+ putStr "Unifier: "+ print subst+ putStr "Matcher: "+ case match $ Equation (t0, t1) of+ Err err -> putStrLn err+ Ans subst -> print subst+ putStrLn ""++readM :: (Read a, Monad m) => String -> m a+readM s =+ case [ x | (x, t) <- reads s, ("", "") <- lex t ] of+ [x] -> return x+ [] -> fail "no parse"+ _ -> fail "ambiguous parse"++data AnsErr a+ = Ans a+ | Err String++instance Monad AnsErr where+ (Ans x) >>= k = k x+ (Err s) >>= _ = Err s+ return = Ans+ fail = Err++-- Main loop++main :: IO ()+main =+ do+ putStrLn "Abelian group unification and matching -- :? for help"+ loop++loop :: IO ()+loop =+ do+ putStr "agum> "+ hFlush stdout+ done <- isEOF+ case done of+ True ->+ do+ putStrLn ""+ return ()+ False ->+ do+ line <- getLine+ case () of+ _ | line == ":?" || line == ":help" ->+ do+ help+ loop+ | line == ":quit" ->+ return ()+ | otherwise ->+ do+ test line+ loop++help :: IO ()+help =+ mapM_ putStrLn mesg++mesg :: [String]+mesg =+ [ "Pose a question as an equation such as",+ " 2x + y = 3z, or",+ " 2x = x + y, or",+ " 64x - 41y = a.",+ "The agum programs shows the result of unification and matching.",+ "",+ "The unification problem is given two terms t and t', find a most",+ "general unifier s such that s(t) = s(t'). The matching problem",+ "for terms t and t' is to find a most general matcher s such that",+ "s(t) = t'.",+ "",+ ":quit quits the program, :? and :help print this message."]
+ src/Algebra/AbelianGroup/UnificationMatching.hs view
@@ -0,0 +1,498 @@+-- Unification and matching in Abelian groups+--+-- Copyright (C) 2009 John D. Ramsdell+--+-- This program is free software: you can redistribute it and/or modify+-- it under the terms of the GNU General Public License as published by+-- the Free Software Foundation, either version 3 of the License, or+-- (at your option) any later version.++-- This program is distributed in the hope that it will be useful,+-- but WITHOUT ANY WARRANTY; without even the implied warranty of+-- MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the+-- GNU General Public License for more details.++-- You should have received a copy of the GNU General Public License+-- along with this program. If not, see <http://www.gnu.org/licenses/>.++-- |+-- Module : Algebra.AbelianGroup.UnificationMatching+-- Copyright : (C) 2009 John D. Ramsdell+-- License : GPL+--+-- This module provides unification and matching in an Abelian group.+--+-- In this module, an Abelian group is a free algebra over a signature+-- with three function symbols:+--+-- * the binary symbol +, the group operator,+--+-- * a constant 0, the identity element, and+--+-- * the unary symbol -, the inverse operator.+--+-- The algebra is generated by a set of variables. Syntactically, a+-- variable is an identifer such as x and y (see 'isVar').+--+-- The axioms associated with the algebra are:+--+-- [Communtativity] x + y = y + x+--+-- [Associativity] (x + y) + z = x + (y + z)+--+-- [Group Identity] x + 0 = x+--+-- [Cancellation] x + -x = 0+--+-- A substitution maps variables to terms. A substitution s is+-- extended to a term as follows.+--+-- * s(0) = 0+--+-- * s(-t) = -s(t)+--+-- * s(t + t\') = s(t) + s(t\')+--+-- The unification problem is given the problem statement t =? t\',+-- find a most general substitution s such that s(t) = s(t\') modulo+-- the axioms of the algebra. The matching problem is to find a most+-- general substitution s such that s(t) = t\' modulo the axioms.+-- Substitition s is more general than s\' if there is a substitition+-- s\" such that s\' = s\" o s.++module Algebra.AbelianGroup.UnificationMatching+ (+ -- * Terms+ Term, ide, isVar, var, mul, add, assocs,+ -- * Unification and Matching+ Equation(..), Maplet(..), unify, match) where++import Data.Char (isSpace, isAlpha, isAlphaNum, isDigit)+import Data.Map (Map)+import qualified Data.Map as Map++-- Chapter 8, Section 5 of the Handbook of Automated Reasoning by+-- Franz Baader and Wayne Snyder describes unification and matching in+-- communtative/monoidal theories. This module refines the described+-- algorithms for the special case of Abelian groups.++-- In this module, an Abelian group is a free algebra over a signature+-- with three function symbols:+--+-- * the binary symbol +, the group operator,+-- * a constant 0, the identity element, and+-- * the unary symbol -, the inverse operator.+--+-- The algebra is generated by a set of variables. Syntactically, a+-- variable is an identifer such as x and y.++-- The axioms associated with the algebra are:+--+-- * x + y = y + x Commutativity+-- * (x + y) + z = x + (y + z) Associativity+-- * x + 0 = x Group identity+-- * x + -x = 0 Cancellation++-- A substitution maps variables to terms. A substitution s is+-- extended to a term as follows.+--+-- s(0) = 0+-- s(-t) = -s(t)+-- s(t + t') = s(t) + s(t')++-- The unification problem is given the problem statement t =? t',+-- find a most general substitution s such that s(t) = s(t') modulo+-- the axioms of the algebra. The matching problem is to find a most+-- general substitution s such that s(t) = t' modulo the axioms.+-- Substitition s is more general than s' if there is a substitition+-- s" such that s' = s" o s.++-- A term is represented by the group identity, or as the sum of+-- factors. A factor is the product of a non-zero integer coefficient+-- and a variable. In this representation, no variable occurs twice.+-- Thus a term is represented by a finite map from variables to+-- non-negative integers.++-- | A term in an Abelian group is represented by the group identity+-- element, or as the sum of factors. A factor is the product of a+-- non-zero integer coefficient and a variable. No variable occurs+-- twice in a term. For the show and read methods, zero is the group+-- identity, the plus sign is the group operation, and the minus sign+-- is the group inverse.+newtype Term = Term (Map String Int)++-- Constructors++-- | 'ide' represents the identity element (zero).+ide :: Term+ide = Term Map.empty++-- | A variable is an alphabetic Unicode character followed by a+-- sequence of alphabetic or numeric digit Unicode characters. The+-- show method for a term works correctly when variables satisfy+-- the 'isVar' predicate.+isVar :: String -> Bool+isVar [] = False+isVar (c:s) = isAlpha c && all isAlphaNum s++-- | Return a term that consists of a single variable.+var :: String -> Term+var x = Term $ Map.singleton x 1++-- | Multiply every coefficient in a term by an integer.+mul :: Int -> Term -> Term+mul 0 (Term _) = ide+mul 1 t = t+mul n (Term t) =+ Term $ Map.map (* n) t++-- Invert a term by negating its coefficients. Same as multiplying+-- a term by -1.+neg :: Term -> Term+neg (Term t) =+ Term $ Map.map negate t++-- | Add two terms.+add :: Term -> Term -> Term+add (Term t) (Term t') =+ Term $ Map.foldWithKey f t' t -- Fold over the mappings in t+ where+ f x c t = -- Alter the mapping of+ Map.alter (g c) x t -- variable x in t+ g c Nothing = -- Variable x not currently mapped+ Just c -- so add a mapping+ g c (Just c') -- Variable x maps to c'+ | c + c' == 0 = Nothing -- Delete the mapping+ | otherwise = Just $ c + c' -- Adjust the mapping++-- | Return all variable-coefficient pairs in the term in ascending+-- variable order.+assocs :: Term -> [(String, Int)]+assocs (Term t) = Map.assocs t++-- | Convert a list of variable-coefficient pairs into a term.+term :: [(String, Int)] -> Term+term assoc =+ foldr f ide assoc+ where+ f (x, c) t = add t $ mul c $ var x++instance Eq Term where+ Term t0 == Term t1 = t0 == t1++-- Unification and Matching++-- | An equation is a pair of terms. For the show and read methods,+-- the two terms are separated by an equal sign.+newtype Equation = Equation (Term, Term) deriving Eq++-- | A maplet maps one variable into a term. For the show and read+-- methods, the variable and the term are separated by a colon. A+-- list of maplets represents a substitution.+newtype Maplet = Maplet (String, Term) deriving Eq++-- | Given 'Equation' (t0, t1), return a most general substitution s+-- such that s(t0) = s(t1) modulo the equational axioms of an Abelian+-- group.+unify :: Monad m => Equation -> m [Maplet]+unify (Equation (t0, t1)) =+ match $ Equation (add t0 (neg t1), ide)++-- Matching in Abelian groups is performed by finding integer+-- solutions to linear equations, and then using the solutions to+-- construct a most general unifier.+-- | Given 'Equation' (t0, t1), return a most general substitution s+-- such that s(t0) = t1 modulo the equational axioms of an Abelian+-- group.+match :: Monad m => Equation -> m [Maplet]+match (Equation (t0, t1)) =+ case (assocs t0, assocs t1) of+ ([], []) -> return []+ ([], _) -> fail "no solution"+ (t0, t1) ->+ do+ subst <- intLinEq (map snd t0) (map snd t1)+ return $ mgu (map fst t0) (map fst t1) subst++-- Construct a most general unifier from a solution to a linear+-- equation. The function adds the variables back into terms, and+-- generates fresh variables as needed.+mgu :: [String] -> [String] -> Subst -> [Maplet]+mgu vars syms subst =+ foldr f [] (zip vars [0..])+ where+ f (x, n) maplets =+ case lookup n subst of+ Just (factors, consts) ->+ Maplet (x, g factors consts) : maplets+ Nothing ->+ Maplet (x, var $ genSyms !! n) : maplets+ g factors consts =+ term (zip genSyms factors ++ zip syms consts)+ genSyms = genSymsAvoiding vars syms++-- Generated variables start with this character.+genChar :: Char+genChar = 'g'++-- Generated symbols are the gen start char followed by a number.+genSym :: Int -> String+genSym i = genChar : show i++-- Produce a stream of generated identifiers avoiding what's in vars and syms.+genSymsAvoiding :: [String] -> [String] -> [String]+genSymsAvoiding vars syms =+ genSymStream 0+ where+ seen = filter genStr (syms ++ vars)+ genStr (c:_) = c == genChar+ genStr _ = False+ genSymStream n+ | elem (genSym n) seen = genSymStream (n + 1)+ | otherwise = genSym n : genSymStream (n + 1)++-- So why solve linear equations? Consider the matching problem+--+-- c[0]*x[0] + c[1]*x[1] + ... + c[n-1]*x[n-1] =?+-- d[0]*a[0] + d[1]*a[1] + ... + d[m-1]*a[m-1]+--+-- with n variables and m constants. We seek a most general unifier s+-- such that+--+-- s(c[0]*x[0] + c[1]*x[1] + ... + c[n-1]*x[n-1]) =+-- d[0]*a[0] + d[1]*a[1] + ... + d[m-1]*a[m-1]+--+-- which is the same as+--+-- c[0]*s(x[0]) + c[1]*s(x[1]) + ... + c[n-1]*s(x[n-1]) =+-- d[0]*a[0] + d[1]*a[1] + ... + d[m-1]*a[m-1]+--+-- Notice that the number of occurrences of constant a[0] in s(x[0])+-- plus s(x[1]) ... s(x[n-1]) must equal d[0]. Thus the mappings of+-- the unifier that involve constant a[0] respect integer solutions of+-- the following linear equation.+--+-- c[0]*x[0] + c[1]*x[1] + ... + c[n-1]*x[n-1] = d[0]+--+-- To compute a most general unifier, a most general integer solution+-- to a linear equation must be found.++-- Integer Solutions of Linear Inhomogeneous Equations++type LinEq = ([Int], [Int])++-- A linear equation with integer coefficients is represented as a+-- pair of lists of integers, the coefficients and the constants. If+-- there are no constants, the linear equation represented by (c, [])+-- is the homogeneous equation:+--+-- c[0]*x[0] + c[1]*x[1] + ... + c[n-1]*x[n-1] = 0+--+-- where n is the length of c. Otherwise, (c, d) represents a+-- sequence of inhomogeneous linear equations with the same+-- left-hand-side:+--+-- c[0]*x[0] + c[1]*x[1] + ... + c[n-1]*x[n-1] = d[0]+-- c[0]*x[0] + c[1]*x[1] + ... + c[n-1]*x[n-1] = d[1]+-- ...+-- c[0]*x[0] + c[1]*x[1] + ... + c[n-1]*x[n-1] = d[m-1]+--+-- where m is the length of d.++type Subst = [(Int, LinEq)]++-- A solution is a partial map from variables to terms, and a term is+-- a pair of lists of integers, the variable part of the term followed+-- by the constant part. The variable part may specify variables not+-- in the input. For example, the solution of+--+-- 64x = 41y + 1+--+-- is x = -41z - 16 and y = -64z - 25. The computed solution is read+-- off the list returned as an answer.+--+-- intLinEq [64,-41] [1] =+-- [(0,([0,0,0,0,0,0,-41],[-16])),+-- (1,([0,0,0,0,0,0,-64],[-25]))]++-- Find integer solutions to linear equations+intLinEq :: Monad m => [Int] -> [Int] -> m Subst+intLinEq coefficients constants =+ intLinEqLoop (length coefficients) (coefficients, constants) []++-- The algorithm used to find solutions is described in Vol. 2 of The+-- Art of Computer Programming / Seminumerical Alorithms, 2nd Ed.,+-- 1981, by Donald E. Knuth, pg. 327.++-- On input, n is the number of variables in the original problem, c+-- is the coefficients, d is the constants, and subst is a list of+-- eliminated variables.+intLinEqLoop :: Monad m => Int -> LinEq -> Subst -> m Subst+intLinEqLoop n (c, d) subst =+ -- Find the smallest non-zero coefficient in absolute value+ let (i, ci) = smallest c in+ case () of+ _ | ci < 0 -> intLinEqLoop n (invert c, invert d) subst+ -- Ensure the smallest coefficient is positive+ | ci == 0 -> fail "bad problem"+ -- Lack of non-zero coefficients is an error+ | ci == 1 ->+ -- A general solution of the following form has been found:+ -- x[i] = sum[j] -c'[j]*x[j] + d[k] for all k+ -- where c' is c with c'[i] = 0.+ return $ eliminate n (i, (invert (zero i c), d)) subst+ | divisible ci c ->+ -- If all the coefficients are divisible by c[i], a solution is+ -- immediate if all the constants are divisible by c[i],+ -- otherwise there is no solution.+ if divisible ci d then+ let c' = divide ci c+ d' = divide ci d in+ return $ eliminate n (i, (invert (zero i c'), d')) subst+ else+ fail "no solution"+ | otherwise ->+ -- Eliminate x[i] in favor of freshly created variable x[n],+ -- where n is the length of c.+ -- x[n] = sum[j] (c[j] div c[i] * x[j])+ -- The new equation to be solved is:+ -- c[i]*x[n] + sum[j] (c[j] mod c[i])*x[j] = d[k] for all k+ intLinEqLoop n (map (\x -> mod x ci) c ++ [ci], d) subst'+ where+ subst' = eliminate n (i, (invert c' ++ [1], [])) subst+ c' = divide ci (zero i c)++-- Find the smallest non-zero coefficient in absolute value+smallest :: [Int] -> (Int, Int)+smallest xs =+ foldl f (-1, 0) (zip [0..] xs)+ where+ f (i, n) (j, x)+ | n == 0 = (j, x)+ | x == 0 || abs n <= abs x = (i, n)+ | otherwise = (j, x)++invert :: [Int] -> [Int]+invert t = map negate t++-- Zero the ith position in a list+zero :: Int -> [Int] -> [Int]+zero _ [] = []+zero 0 (_:xs) = 0 : xs+zero i (x:xs) = x : zero (i - 1) xs++-- Eliminate a variable from the existing substitution. If the+-- variable is in the original problem, add it to the substitution.+eliminate :: Int -> (Int, LinEq) -> Subst -> Subst+eliminate n m@(i, (c, d)) subst =+ if i < n then+ m : map f subst+ else+ map f subst+ where+ f m'@(i', (c', d')) = -- Eliminate i in c' if it occurs in c'+ case get i c' of+ 0 -> m' -- i is not in c'+ ci -> (i', (addmul ci (zero i c') c, addmul ci d' d))+ -- Find ith coefficient+ get _ [] = 0+ get 0 (x:_) = x+ get i (_:xs) = get (i - 1) xs+ -- addnum n xs ys sums xs and ys after multiplying ys by n+ addmul 1 [] ys = ys+ addmul n [] ys = map (* n) ys+ addmul _ xs [] = xs+ addmul n (x:xs) (y:ys) = (x + n * y) : addmul n xs ys++divisible :: Int -> [Int] -> Bool+divisible small t =+ all (\x -> mod x small == 0) t++divide :: Int -> [Int] -> [Int]+divide small t =+ map (\x -> div x small) t++-- Input and Output++instance Show Term where+ showsPrec _ t =+ case assocs t of+ [] -> showString "0"+ (t:ts) -> showFactor t . showl ts+ where+ showFactor (x, 1) = showString x+ showFactor (x, -1) = showChar '-' . showString x+ showFactor (x, c) = shows c . showString x+ showl [] = id+ showl ((s,n):ts)+ | n < 0 =+ showString " - " . showFactor (s, negate n) . showl ts+ showl (t:ts) = showString " + " . showFactor t . showl ts++instance Read Term where+ readsPrec _ s0 =+ [ (t1, s2) | (t0, s1) <- readSummand s0,+ (t1, s2) <- readRest t0 s1 ]+ where+ readPrimary s0 =+ [ (t0, s1) | (x, s1) <- scan s0, isVarToken x,+ let t0 = var x ] +++ [ (t0, s1) | ("0", s1) <- scan s0,+ let t0 = ide ] +++ [ (t0, s3) | ("(", s1) <- scan s0,+ (t0, s2) <- reads s1,+ (")", s3) <- scan s2 ]+ readFactor s0 =+ [ (t0, s1) | (t0, s1) <- readPrimary s0 ] +++ [ (t1, s2) | (n, s1) <- scan s0, isNumToken n,+ (t0, s2) <- readPrimary s1,+ let t1 = mul (read n) t0 ]+ readSummand s0 =+ [ (t0, s1) | (t0, s1) <- readFactor s0 ] +++ [ (t1, s2) | ("-", s1) <- scan s0,+ (t0, s2) <- readFactor s1,+ let t1 = neg t0 ]+ readRest t0 s0 =+ [ (t2, s3) | ("+", s1) <- scan s0,+ (t1, s2) <- readSummand s1,+ (t2, s3) <- readRest (add t0 t1) s2 ] +++ [ (t2, s3) | ("-", s1) <- scan s0,+ (t1, s2) <- readFactor s1,+ (t2, s3) <- readRest (add t0 (neg t1)) s2 ] +++ [ (t0, s0) | (s, _) <- scan s0, s /= "+" && s /= "-" ]++isNumToken :: String -> Bool+isNumToken (c:_) = isDigit c+isNumToken _ = False++isVarToken :: String -> Bool+isVarToken (c:_) = isAlpha c+isVarToken _ = False++scan :: ReadS String+scan "" = [("", "")]+scan (c:s)+ | isSpace c = scan s+ | isAlpha c = [ (c:part, t) | (part,t) <- [span isAlphaNum s] ]+ | isDigit c = [ (c:part, t) | (part,t) <- [span isDigit s] ]+ | otherwise = [([c], s)]++instance Show Equation where+ showsPrec _ (Equation (t0, t1)) =+ shows t0 . showString " = " . shows t1++instance Read Equation where+ readsPrec _ s0 =+ [ (Equation (t0, t1), s3) | (t0, s1) <- reads s0,+ ("=", s2) <- scan s1,+ (t1, s3) <- reads s2 ]++instance Show Maplet where+ showsPrec _ (Maplet (x, t)) =+ showString x . showString " : " . shows t++instance Read Maplet where+ readsPrec _ s0 =+ [ (Maplet (x, t), s3) | (x, s1) <- scan s0, isVarToken x,+ (":", s2) <- scan s1,+ (t, s3) <- reads s2 ]