diff --git a/ChangeLog b/ChangeLog new file mode 100644 --- /dev/null +++ b/ChangeLog @@ -0,0 +1,3 @@ +2009-09-17 John D. Ramsdell + + * cmu.cabal (Version): Released as version 1.0. diff --git a/Makefile b/Makefile new file mode 100644 --- /dev/null +++ b/Makefile @@ -0,0 +1,24 @@ +# Haskell/Cabal Makefile +# Requires GNU Make +# The all target creates a default configuration if need be. + +PACKAGE := $(wildcard *.cabal) +CONFIG = dist/setup-config +SETUP = runhaskell Setup.hs + +all: $(CONFIG) + $(SETUP) build + +Makefile: + @echo make $@ + +$(PACKAGE): + @echo make $@ + +$(CONFIG): $(PACKAGE) + $(SETUP) configure --ghc --user --prefix="${HOME}" + +%: force + $(SETUP) $@ + +.PHONY: all force diff --git a/Setup.hs b/Setup.hs new file mode 100644 --- /dev/null +++ b/Setup.hs @@ -0,0 +1,2 @@ +import Distribution.Simple +main = defaultMain diff --git a/cmu.cabal b/cmu.cabal new file mode 100644 --- /dev/null +++ b/cmu.cabal @@ -0,0 +1,34 @@ +Name: cmu +Version: 1.0 +Maintainer: ramsdell@mitre.org +Cabal-Version: >= 1.2 +License: GPL +License-File: license.txt +Synopsis: Unification in a Commutative Monaid +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 a commutative monoid. + Substitition s is more general than s\' if + there is a substitition s\" such that s\' = + s\" o s. +Category: Algebra +Build-Type: Simple +Extra-Source-Files: readme.txt ChangeLog Makefile + +Library + Build-Depends: base <= 4.1.0.0, containers, array + Exposed-Modules: Algebra.CommutativeMonoid.Unification + Algebra.CommutativeMonoid.HomLinDiaphEq + Hs-Source-Dirs: src + GHC-Options: + -Wall -fno-warn-name-shadowing -fwarn-unused-imports + +Executable cmu + Main-Is: Algebra/CommutativeMonoid/Main.hs + Build-Depends: base <= 4.1.0.0, containers, array + Other-Modules: Algebra.CommutativeMonoid.Unification + Algebra.CommutativeMonoid.HomLinDiaphEq + Hs-Source-Dirs: src + GHC-Options: + -Wall -fno-warn-name-shadowing -fwarn-unused-imports diff --git a/license.txt b/license.txt new file mode 100644 --- /dev/null +++ b/license.txt @@ -0,0 +1,675 @@ + + GNU GENERAL PUBLIC LICENSE + Version 3, 29 June 2007 + + Copyright (C) 2007 Free Software Foundation, Inc. + Everyone is permitted to copy and distribute verbatim copies + of this license document, but changing it is not allowed. + + Preamble + + The GNU General Public License is a free, copyleft license for +software and other kinds of works. + + The licenses for most software and other practical works are designed +to take away your freedom to share and change the works. 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If not, see . + +Also add information on how to contact you by electronic and paper mail. + + If the program does terminal interaction, make it output a short +notice like this when it starts in an interactive mode: + + Copyright (C) + This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. + This is free software, and you are welcome to redistribute it + under certain conditions; type `show c' for details. + +The hypothetical commands `show w' and `show c' should show the appropriate +parts of the General Public License. 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 +. + + 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 +. diff --git a/readme.txt b/readme.txt new file mode 100644 --- /dev/null +++ b/readme.txt @@ -0,0 +1,18 @@ +This package contains a library for unification in +commutative monoid and a program that exercises the library. + +$ cmu +Commutative monoid unification -- :? for help +cmu> 2x+y=3z +Problem: 2x + y = 3z +Unifier: [x : g0 + 3g2,y : g0 + 3g1,z : g0 + g1 + 2g2] + +cmu> 2x=x+y +Problem: 2x = x + y +Unifier: [x : g0,y : g0] + +cmu> 64x=41y+a +Problem: 64x = a + 41y +Unifier: [a : 5g0 + 2g1 + 23g2 + g3 + 64g4,x : 2g0 + 9g1 + g2 + 25g3 + g4 + 41g5,y : 3g0 + 14g1 + g2 + 39g3 + 64g5] + +cmu> :quit diff --git a/src/Algebra/CommutativeMonoid/HomLinDiaphEq.hs b/src/Algebra/CommutativeMonoid/HomLinDiaphEq.hs new file mode 100644 --- /dev/null +++ b/src/Algebra/CommutativeMonoid/HomLinDiaphEq.hs @@ -0,0 +1,143 @@ +-- Homogeneous Linear Diaphantine Equation solver +-- +-- 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 . + +-- | +-- Module : Algebra.CommutativeMonoid.HomLinDiaphEq +-- Copyright : (C) 2009 John D. Ramsdell +-- License : GPL +-- +-- Homogeneous Linear Diaphantine Equation solver. +-- +-- The solver uses the algorithm of Contejean and Devie as specified +-- by David Papp and Bela Vizari in \"Effective Solutions of Linear +-- Diophantine Equation Systems with an Application to Chemistry\", +-- Rutcor Research Report RRR 28-2004, September, 2004, +-- , after +-- modification so as to ensure every basis vector is considered. +-- +-- The algorithm for systems of homogeneous linear Diophantine +-- equations follows. Let e[k] be the kth basis vector for 1 <= k <= +-- n. To find the minimal, non-negative solutions M to the system of +-- equations sum(i=1,n,a[i]*v[i]) = 0, the algorithm of Contejean and +-- Devie is: +-- +-- 1. [init] A := {e[k] | 1 <= k <= n}; M := {} +-- +-- 2. [new minimal results] M := M + {a in A | a is a solution} +-- +-- 3. [unnecessary branches] A := {a in A | all m in M : some +-- 1 <= k <= n : m[k] < a[k]} +-- +-- 4. [test] If A = {}, stop +-- +-- 5. [breadth-first search] A := {a + e[k] | a in A, 1 <= k <= n, +-- \ \< 0}; go to step 2 + +module Algebra.CommutativeMonoid.HomLinDiaphEq (homLinDiaphEq) where + +import Data.Array +import Data.Set (Set) +import qualified Data.Set as S + +{-- Debugging hack +import System.IO.Unsafe + +z :: Show a => a -> b -> b +z x y = seq (unsafePerformIO (print x)) y +--} + +type Vector a = Array Int a + +vector :: Int -> [a] -> Vector a +vector n elems = + listArray (0, n - 1) elems + +-- | The 'homLinDiaphEq' function takes a list of integers that +-- specifies a homogeneous linear Diophantine equation, and returns +-- the equation's minimal, non-negative solutions. +homLinDiaphEq :: [Int] -> [[Int]] +homLinDiaphEq [] = [] +homLinDiaphEq v = + newMinimalResults (vector n v) (basis n) S.empty + where n = length v + +-- Construct the basis vectors for an n-dimensional space +basis :: Int -> Set (Vector Int) +basis n = + foldl (flip S.insert) S.empty + [ z // [(k, 1)] | + k <- indices z ] + where z = vector n $ replicate n 0 + +-- The main loop has been reorganized to ensure every basis vector is +-- considered. The breadth-first search step is now the last step. + +-- Add elements of a that solve the equation to m and the output +newMinimalResults :: Vector Int -> Set (Vector Int) -> + Set (Vector Int) -> [[Int]] +newMinimalResults v a m = + loop m (S.toList a) -- Test each element in a + where + loop m [] = + nextSearch v a m -- Generate new a and try again + loop m (x:xs) + | prod v x == 0 && S.notMember x m = + elems x:loop (S.insert x m) xs -- Answer found + | otherwise = + loop m xs + +-- Generate the next set of test vectors--if there aren't any, your done +nextSearch :: Vector Int -> Set (Vector Int) -> + Set (Vector Int) -> [[Int]] +nextSearch v a m = + if S.null a' then + [] + else + newMinimalResults v (breadthFirstSearch v a') m + where + a' = unnecessaryBranches a m + +-- Remove unnecessary branches. A test vector is not necessary if all +-- of its elements are greater than or equal to the elements of some +-- minimal solution. +unnecessaryBranches :: Set (Vector Int) -> Set (Vector Int) -> Set (Vector Int) +unnecessaryBranches a m = + S.filter f a + where + f x = all (g x) (S.toList m) + g x y = not (lessEq y x) + +-- Compare vectors element-wise. +lessEq :: Vector Int -> Vector Int -> Bool +lessEq x y = + all (\i-> x!i <= y!i) (indices x) + +-- Breadth-first search using the algorithm of Contejean and Devie +breadthFirstSearch :: Vector Int -> Set (Vector Int) -> Set (Vector Int) +breadthFirstSearch v a = + S.fold f S.empty a + where + f x acc = + foldl (flip S.insert) acc + [ x // [(k, x!k + 1)] | + k <- indices x, + prod v x * v!k < 0 ] -- Contejean-Devie contribution + +-- Inner product +prod :: Vector Int -> Vector Int -> Int +prod x y = + sum [ x!i * y!i | i <- indices x ] diff --git a/src/Algebra/CommutativeMonoid/Main.hs b/src/Algebra/CommutativeMonoid/Main.hs new file mode 100644 --- /dev/null +++ b/src/Algebra/CommutativeMonoid/Main.hs @@ -0,0 +1,105 @@ +-- A top-level loop for unification in a Commutative Moniod +-- +-- 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 . + +module Main (main, test) where + +import System.IO (isEOF, hFlush, stdout) +import Algebra.CommutativeMonoid.Unification + +-- Test Routine + +-- Given an equation, display a unifier. +test :: String -> IO () +test prob = + case readM prob of + Err err -> putStrLn err + Ans (Equation (t0, t1)) -> + do + putStr "Problem: " + print $ Equation (t0, t1) + putStr "Unifier: " + let subst = unify $ Equation (t0, t1) + 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 "Commutative monoid unification -- :? for help" + loop + +loop :: IO () +loop = + do + putStr "cmu> " + 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 cmu programs shows the result of unification.", + "", + "The unification problem is given two terms t and t', find a most", + "general unifier s such that s(t) = s(t').", + "", + ":quit quits the program, :? and :help print this message."] diff --git a/src/Algebra/CommutativeMonoid/Unification.hs b/src/Algebra/CommutativeMonoid/Unification.hs new file mode 100644 --- /dev/null +++ b/src/Algebra/CommutativeMonoid/Unification.hs @@ -0,0 +1,355 @@ +-- Unification in a commutative monoid +-- +-- 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 . + +-- | +-- Module : Algebra.CommutativeMonoid.Unification +-- Copyright : (C) 2009 John D. Ramsdell +-- License : GPL +-- +-- This module provides unification in a commutative monoid. +-- +-- In this module, a commutative monoid is a free algebra over a +-- signature with two function symbols: +-- +-- * the binary symbol +, the group operator, +-- +-- * a constant 0, the identity element, and +-- +-- 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 +-- +-- A substitution maps variables to terms. A substitution s is +-- applied to a term as follows. +-- +-- * s(0) = 0 +-- +-- * 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. Substitition s is more general than s\' +-- if there is a substitition s\" such that s\' = s\" o s. + +module Algebra.CommutativeMonoid.Unification + ( + -- * Terms + Term, ide, isVar, var, mul, add, assocs, + -- * Equations and Substitutions + Equation(..), Substitution, subst, maplets, apply, + -- * Unification + unify) where + +import Data.Char (isSpace, isAlpha, isAlphaNum, isDigit) +import Data.List (transpose) +import Data.Map (Map) +import qualified Data.Map as Map +import Algebra.CommutativeMonoid.HomLinDiaphEq + +-- Chapter 8, Section 5 of the Handbook of Automated Reasoning by +-- Franz Baader and Wayne Snyder describes unification in +-- commutative/monoidal theories. This module refines the described +-- algorithms for the special case of a commutative monoid. + +-- In this module, a commutative monoid is a free algebra over a signature +-- with two function symbols: +-- +-- * the binary symbol +, the group operator, +-- * a constant 0, the identity element, and +-- +-- 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 + +-- A substitution maps variables to terms. A substitution s is +-- extended to a term as follows. +-- +-- s(0) = 0 +-- 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. 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 positive 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 a commutative monoid is represented by the group +-- identity element, or as the sum of factors. A factor is the +-- product of a positive 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. +newtype Term = Term (Map String Int) deriving Eq + +-- 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 non-negative integer. +mul :: Int -> Term -> Term +mul 0 (Term _) = ide +mul 1 t = t +mul n (Term t) + | n < 0 = error "Negative coefficient found" + | otherwise = Term $ Map.map (* n) t + +-- Invert a term by negating its coefficients. +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 + +-- Equations and Substitutions + +-- | 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 substitution maps variables into terms. For the show and read +-- methods, the substitution is a list of maplets, and the variable +-- and the term in each element of the list are separated by a colon. +newtype Substitution = Substitution (Map String Term) deriving Eq + +-- | Construct a substitution from a list of variable-term pairs. +subst :: [(String, Term)] -> Substitution +subst assocs = + Substitution $ foldl f Map.empty assocs + where + f t (x, n) = Map.insert x n t + +-- | Return all variable-term pairs in ascending variable order. +maplets :: Substitution -> [(String, Term)] +maplets (Substitution s) = Map.assocs s + +-- | Return the result of applying a substitution to a term. +apply :: Substitution -> Term -> Term +apply (Substitution s) (Term t) = + Map.foldWithKey f ide t + where + f x n t = + add (mul n (Map.findWithDefault (var x) x s)) t + +-- Unification + +-- | Given 'Equation' (t0, t1), return a most general substitution s +-- such that s(t0) = s(t1) modulo the equational axioms of a +-- commutative monoid. +unify :: Equation -> Substitution +unify (Equation (t0, t1)) = + case assocs (add t0 (neg t1)) of + [] -> Substitution Map.empty + t -> + let basis = homLinDiaphEq (map snd t) in + mgu (map fst t) basis + +-- Construct a most general unifier the minimal non-negative solutions +-- to a linear equation. The function adds the variables back into +-- terms, and generates fresh variables as needed. +mgu :: [String] -> [[Int]] -> Substitution +mgu vars basis = + subst (zip vars terms) + where + terms = map (term . zip genSyms) (transpose basis) + genSyms = genSymsAvoiding vars + +-- 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. +genSymsAvoiding :: [String] -> [String] +genSymsAvoiding vars = + genSymStream 0 + where + seen = filter genStr 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 non-negative 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, the set of minimal non-negative +-- integer solutions to a linear equation must be found. See module +-- Algebra.CommutativeMonoid.HomLinDiaphEq. + +-- 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, c) = shows c . showString x + showl [] = id + showl (t:ts) = showString " + " . showFactor t . showl ts + +instance Read Term where + readsPrec _ s0 = + [ (t1, s2) | (t0, s1) <- readFactor 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 ] + readRest t0 s0 = + [ (t2, s3) | ("+", s1) <- scan s0, + (t1, s2) <- readFactor s1, + (t2, s3) <- readRest (add t0 t1) s2 ] ++ + [ (t0, s0) | (s, _) <- scan s0, 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 ] + +-- This datatype is used only in the read and show methods for +-- substitutions. +newtype Maplet = Maplet (String, Term) deriving Eq + +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 ] + +instance Show Substitution where + showsPrec _ s = + shows $ map Maplet $ maplets s + +instance Read Substitution where + readsPrec _ s0 = + [ (subst $ map pair ms, s1) | (ms, s1) <- reads s0 ] + where + pair (Maplet (x, t)) = (x, t)