diff --git a/LICENSE b/LICENSE new file mode 100644 --- /dev/null +++ b/LICENSE @@ -0,0 +1,31 @@ +Copyright (c) 2009-2014, Nick Smallbone +Copyright (c) 2015-2016, Chris Warburton + +All rights reserved. + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are met: + + * Redistributions of source code must retain the above copyright + notice, this list of conditions and the following disclaimer. + + * Redistributions in binary form must reproduce the above + copyright notice, this list of conditions and the following + disclaimer in the documentation and/or other materials provided + with the distribution. + + * Neither the name of Nick Smallbone nor the names of other + contributors may be used to endorse or promote products derived + from this software without specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS +"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT +LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR +A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT +OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, +SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT +LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, +DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY +THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT +(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE +OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. diff --git a/README b/README new file mode 100644 --- /dev/null +++ b/README @@ -0,0 +1,128 @@ +# Reduce Equations # + +This package provides a command `reduce-equations` which reads in a list of +equations from stdin, performs some simplification, and writes the results to +stdout. + +For example, given the equations `a = b`, `b = c` and `a = c`, one of these will +be removed as it can be inferred from the other two. Similarly, given equations +`f a = g`, `f b = g` and `a = b`, one of the first equations will be removed as +it can be recovered by subtitution. + +All of the real work is done by [QuickSpec](https://hackage.haskell.org/package/quickspec) +This package just provides stdio and machine-friendly formatting. + +## Formats ## + +All IO is encoded in JSON. Both stdin and stdout should contain a single array +of equations. The following example gives a single equation, which if written in +a more human-friendly form, would be `plus x x = times x 2`: + +``` +[ + {"relation": "~=", + "lhs": {"role": "application", + "lhs": {"role": "application", + "lhs": {"role": "constant", + "type": "Int -> Int -> Int", + "symbol": "plus"}, + "rhs": {"role": "variable", + "type": "Int", + "id": 0}}, + "rhs": {"role": "variable", + "type": "Int", + "id": 0}}, + "rhs": {"role": "application", + "lhs": {"role": "application", + "lhs": {"role": "constant", + "type": "Int -> Int -> Int", + "symbol": "times"}, + "rhs": {"role": "variable", + "type": "Int", + "id": 0}}, + "rhs": {"role": "constant", + "type": "Int", + "symbol": "two"}}} +] +``` + +### Equations ### + +An equation is an object with the following values: + + - `relation`: This is used mostly to identify that we've got an equation. In + practice, this is always `"~="` (what that means is up to you). + - `lhs`: this is a `term`, supposedly the left-hand-side of the equation, + although the only difference from `rhs` is the name. + - `rhs`: this is a `term`, just like `lhs` except it's the right-hand-side. + +Example: + +``` +{"relation": "~=", + "lhs": {"role": "application", + "lhs": {"role": "constant", + "type": "Bool -> Bool", + "symbol": "not"}, + "rhs": {"role": "application", + "lhs": {"role": "constant", + "type": "Bool -> Bool", + "symbol": "not"}, + "rhs": {"role": "variable", + "type": "Bool", + "id": 0}}}, + "rhs": {"role": "variable", + "type": "Bool", + "id": 0}} +``` + +### Terms ### + +A term is an object containing a `role`, which is one of `"constant"`, +`"variable"` or `"application"`. The other fields depend on what the term's +`role` is: + + - Constants + - `type`: The type of the constant, a string written in Haskell's type + notation. This is taken from the given function descriptions. For example + `"Int -> (Int -> Bool) -> IO Float"` + - `symbol`: The name of the constant, as a string. For example `"reverse"`. + - Variables + - `type`: The type of the variable, a string written in Haskell's type + notation. The types can be made up, but they should be consistent (e.g. + both sides of an equation should have the same type; application should be + well-typed; etc.). Unification of polymorphic types isn't supported; types + are identified syntactically. For example `"[Int]"`. + - `"id"`: A numeric ID for the variable. IDs start at `0`. Used to + distinguish between multiple variables of the same type. Variable ID only + matters within a single equation. For example, to represent three integer + variables we might use `{"role": "variable", "type": "Int", "id":0}`, + `{"role": "variable", "type": "Int", "id":1}` and + `{"role": "variable", "type": "Int", "id":2}`. + - Applications + - `lhs`: A term representing a function to apply. + - `rhs`: A term representing the argument to apply the `lhs` function to. + Functions are curried, so calling with multiple arguments should be done + via a left-leaning tree. + +## Implementation Notes ## + +We co-opt the equation-reducing machinery of the +[QuickSpec](https://hackage.haskell.org/package/quickspec-0.9.6) library to do +the actual reduction. This relies heavily on existential types and Haskell's +[Typeable](https://hackage.haskell.org/package/base-4.9.0.0/docs/Data-Typeable.html) +mechanism. + +Since the incoming equations may have arbitrary types, and GHC doesn't let us +define custom `Typeable` instances, we perform a conversion step: + + - Once an array of equations has been parsed, we recurse through the terms and + switch out each distinct type with a freshly-generated replacement, of the + form `Z`, `S Z`, `S (S Z)`, etc. (these are just Peano numerals, e.g. see + https://en.wikipedia.org/wiki/Successor_function ) + - We provide special functions `getRep` and `getVal` to plumb these Peano types + into QuickSpec's machinery, convincing it that we have a signature of + well-typed terms. + - We reduce the given equations, with their switched-out types, to get a + reduced set. + - We switch back the types for presentation purposes, pretty-printing to JSON. 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/reduce-equations.cabal b/reduce-equations.cabal new file mode 100644 --- /dev/null +++ b/reduce-equations.cabal @@ -0,0 +1,78 @@ +-- Initial reduce-equations.cabal generated by cabal init. For further +-- documentation, see http://haskell.org/cabal/users-guide/ + +name: reduce-equations +version: 0.1.1.0 +synopsis: Simplify a set of equations by removing redundancies +description: Simplify a set of equations by removing redundancies +homepage: http://chriswarbo.net/projects/repos/reduce-equations.html +license: BSD3 +license-file: LICENSE +author: Chris Warburton +maintainer: chriswarbo@gmail.com +-- copyright: +category: Math +build-type: Simple +extra-source-files: README + , test.sh + , test/data/nat-simple-expect.json + , test/data/tip.json + , test/data/nat.json + , test/data/nat-simple-raw.json + , test/data/list-extras.json +cabal-version: >=1.10 + +source-repository head + type: git + location: http://chriswarbo.net/git/reduce-equations.git + +library + exposed-modules: Algebra.Equation.Reduce + , Algebra.Equation.Internal + , Algebra.Equation.Internal.Eval + , Algebra.Equation.Internal.Types + -- other-modules: + -- other-extensions: + build-depends: base >= 4.8 && < 4.10 + , aeson + , bytestring + , containers + , QuickCheck + , quickspec == 0.9.6 + , transformers + , mtl + , haskell-src-exts >= 1.18.2 + , stringable + , text + hs-source-dirs: src + default-language: Haskell2010 + +executable reduce-equations + main-is: Main.hs + -- other-modules: + -- other-extensions: + build-depends: base >= 4.8 && < 4.10 + , reduce-equations + , aeson + hs-source-dirs: reduce-equations + default-language: Haskell2010 + +test-suite test + default-language: Haskell2010 + type: exitcode-stdio-1.0 + hs-source-dirs: test + main-is: Main.hs + build-depends: base >= 4.8 && < 4.10 + , reduce-equations + , quickspec == 0.9.6 + , directory + , aeson + , bytestring + , haskell-src-exts >= 1.18.2 + , MissingH + , stringable + , containers + , QuickCheck == 2.8.2 + , text + , tasty >= 0.7 + , tasty-quickcheck diff --git a/reduce-equations/Main.hs b/reduce-equations/Main.hs new file mode 100644 --- /dev/null +++ b/reduce-equations/Main.hs @@ -0,0 +1,6 @@ +module Main where + +import Algebra.Equation.Reduce +import System.Environment + +main = doReduce diff --git a/src/Algebra/Equation/Internal.hs b/src/Algebra/Equation/Internal.hs new file mode 100644 --- /dev/null +++ b/src/Algebra/Equation/Internal.hs @@ -0,0 +1,8 @@ +{-# LANGUAGE OverloadedStrings, RankNTypes, ExistentialQuantification, PartialTypeSignatures #-} +module Algebra.Equation.Internal ( + module Algebra.Equation.Internal.Eval + , module Algebra.Equation.Internal.Types + ) where + +import Algebra.Equation.Internal.Eval -- Used for interacting with QuickSpec +import Algebra.Equation.Internal.Types -- Our own representations diff --git a/src/Algebra/Equation/Internal/Eval.hs b/src/Algebra/Equation/Internal/Eval.hs new file mode 100644 --- /dev/null +++ b/src/Algebra/Equation/Internal/Eval.hs @@ -0,0 +1,452 @@ +{-# LANGUAGE OverloadedStrings, PartialTypeSignatures, ScopedTypeVariables, FlexibleInstances, UndecidableInstances, RankNTypes, GADTs #-} + +module Algebra.Equation.Internal.Eval where + +-- We want to use QuickSpec's `prune` function to minimise a set of equations, +-- but this is tricky since its `Symbol` type contains a `TypeRep`. We can +-- serialise a `TypeRep` easily enough with `show`, but we can't de-serialise it +-- easily. + +-- To work around this, we don't attempt to de-serialise any `TypeRep`, e.g. via +-- a function like `String -> TypeRep`; instead, since a serialised `TypeRep` is +-- just a `String` of Haskell source code corresponding to a type, we turn our +-- `prune`-invoking function into Haskell source code as well, and append the +-- `TypeRep`s as needed, as type annotations. + +import Data.Dynamic +import Data.List +import Data.Maybe +import qualified Data.Map +import qualified Data.Ord +import Data.String +import Data.Typeable +import qualified Language.Haskell.Exts.Syntax +import qualified Language.Haskell.Exts.Parser as HSE.Parser +import Algebra.Equation.Internal.Types +import System.Environment +import System.IO +import System.IO.Unsafe +import Text.Read -- Uses String as part of base, not Text + +-- Used for their types +import qualified Test.QuickCheck.Gen +import qualified Test.QuickCheck.Random +import qualified Test.QuickSpec +import qualified Test.QuickSpec.Main +import qualified Test.QuickSpec.Signature +import qualified Test.QuickSpec.Term +import qualified Test.QuickSpec.TestTree +import qualified Test.QuickSpec.Utils +import qualified Test.QuickSpec.Utils.Typeable +import qualified Test.QuickSpec.Utils.Typed +import qualified Test.QuickSpec.Utils.TypeMap +import qualified Test.QuickSpec.Utils.TypeRel (TypeRel, singleton) +import qualified Test.QuickSpec.Reasoning.NaiveEquationalReasoning +import qualified Test.QuickSpec.Equation + +-- Conversion from our representations to QuickSpec expressions + +renderTermN t sig = case (t, sigToSymN t sig) of + (App l r _, _ ) -> case (renderTermN l sig, renderTermN r sig) of + (f_l, f_r) -> app f_l f_r + (C c, f_s) -> const f_s + (V v, f_s) -> var f_s + where app = Test.QuickSpec.Term.App + const = Test.QuickSpec.Term.Const + var = Test.QuickSpec.Term.Var + +renderN (Sig cs vs) = mappend (renderQSConsts cs) (renderQSVars vs) + +renderQSConsts = foldr (mappend . renderQSConst) mempty + +renderQSVars = foldr (mappend . renderQSVarType) mempty . collectTypes + where collectTypes = groupBy eqTypes . sortBy ordTypes + eqTypes (Var t1 _ _) (Var t2 _ _) = t1 == t2 + ordTypes (Var t1 _ _) (Var t2 _ _) = compare t1 t2 + +renderQSVarType [] = Test.QuickSpec.Signature.emptySig +renderQSVarType vs@(Var t _ (Arity a):_) = case getVal (getRep t) of + MkHT x -> Test.QuickSpec.Signature.variableSig [ + Test.QuickSpec.Term.Variable + (Test.QuickSpec.Term.Atom + (Test.QuickSpec.Term.symbol (unName (varName v)) a x) + (Test.QuickSpec.Term.pgen (return x))) + | v <- vs ] + `mappend` mconcat [ Test.QuickSpec.Signature.totalSig + (Test.QuickSpec.Term.totalGen + (Test.QuickSpec.Term.pgen (return x))) + | v <- vs ] + `mappend` mconcat [ Test.QuickSpec.Signature.partialSig + (Test.QuickSpec.Term.partialGen + (Test.QuickSpec.Term.pgen (return x))) + | v <- vs ] + `mappend` Test.QuickSpec.Signature.typeSig x + +extendSig t a sig = case getVal (getRep t) of + MkHT x -> extendOrd t a (addArrowTypes sig t a) `mappend` Test.QuickSpec.Signature.typeSig x + +extendOrd t 0 sig = case getVal (getRep t) of + MkHT x -> sig `mappend` Test.QuickSpec.Signature.ord x +extendOrd (Language.Haskell.Exts.Syntax.TyFun _ i o) a sig = + extendSig o (a-1) sig +extendOrd t a sig = error ("Arity " ++ show a ++ " for non-function type " ++ show t) + +addArrowTypes sig t 0 = case getVal (getRep t) of + MkHT x -> sig `mappend` Test.QuickSpec.Signature.typeSig x +addArrowTypes sig (Language.Haskell.Exts.Syntax.TyFun _ i o) a = + case getVal (getRep i) of + MkHT x -> Test.QuickSpec.Signature.typeSig x `mappend` addArrowTypes sig o (a-1) + +-- Conceptually the same as `fun0`, `fun1`, etc. but we have to bypass those +-- wrappers as we need to supply our own TypeRep +renderQSConst :: Const -> Test.QuickSpec.Signature.Sig +renderQSConst (Const (Arity a) (Name n) t) = extendSig t a constantSig + where rawTyRep = getRep t + + tyrep = repToQSRep rawTyRep + + typerelsingleton :: Typeable a => Test.QuickSpec.Term.Constant a -> Test.QuickSpec.Utils.TypeRel.TypeRel Test.QuickSpec.Term.Constant + typerelsingleton x = typeMapfromList (Test.QuickSpec.Utils.Typed.O [x]) + + typeMapfromList :: Typeable a => f a -> Test.QuickSpec.Utils.TypeMap.TypeMap f + typeMapfromList x = Data.Map.fromList [ (tyrep, Test.QuickSpec.Utils.Typed.Some x) ] + + constantSig = case getVal rawTyRep of + MkHT x -> Test.QuickSpec.Signature.emptySig { + Test.QuickSpec.Signature.constants = typerelsingleton + (Test.QuickSpec.Term.Constant + (Test.QuickSpec.Term.Atom + (Test.QuickSpec.Term.Symbol 0 n a False False tyrep) + x)) -- Pruning equations shouldn't force any values + } + +type ListOfConstants = Test.QuickSpec.Utils.Typed.O + Test.QuickSpec.Utils.Typed.List + Test.QuickSpec.Term.Constant + +repToQSRep tr = (Test.QuickSpec.Utils.Typeable.typeOf [()]) { + Test.QuickSpec.Utils.Typeable.unTypeRep = tr + } + +-- Turn a normalised Type (e.g. `S (S Z) -> S Z`) into a TypeRep +getRep :: Type -> TypeRep +getRep t = case t of + Language.Haskell.Exts.Syntax.TyFun _ l r -> mkFunTy (getRep l) (getRep r) + + Language.Haskell.Exts.Syntax.TyApp _ + (Language.Haskell.Exts.Syntax.TyCon _ + (Language.Haskell.Exts.Syntax.UnQual _ + (Language.Haskell.Exts.Syntax.Ident _ "S"))) + x -> mkTyConApp sCon [getRep x] + + Language.Haskell.Exts.Syntax.TyCon _ + (Language.Haskell.Exts.Syntax.UnQual _ + (Language.Haskell.Exts.Syntax.Ident _ "Z")) -> typeRep [Z ()] + + _ -> error ("Unknown type '" ++ show t ++ "'. Did you forget replaceTypes?") + + where sCon = typeRepTyCon (typeRep [S (Z ())]) -- The Z gets discarded + +-- Construct a value of some normalised type. We use an existential so that +-- getVal can return a (wrapped up) value of the correct type. Since we should +-- only need these values in order to match up their TypeReps, this should be +-- sufficient. +getVal :: TypeRep -> HasType +getVal tr = case () of + _ | thisCon == zCon -> MkHT (Z ()) + _ | thisCon == sCon -> case typeRepArgs tr of + [] -> error ("No args in " ++ show tr) + x:xs -> case getVal x of + MkHT x -> MkHT (S x) + _ | thisCon == funCon -> case map getVal (typeRepArgs tr) of + [MkHT i, MkHT o] -> MkHT (\x -> case asTypeOf i x of + _ -> o) + where thisCon = typeRepTyCon tr + funCon = typeRepTyCon (typeRep [not]) -- Simple monomorphic function + zCon = typeRepTyCon (typeRep [Z ()]) + sCon = typeRepTyCon (typeRep [S (Z ())]) -- The Z gets discarded + +data HasType = forall a. (Ord a, Typeable a) => MkHT a + +instance Show HasType where + show (MkHT x) = "HasType(" ++ show (typeRep [x]) ++ ")" + +sigToSymN :: Term -> Test.QuickSpec.Signature.Sig -> Test.QuickSpec.Term.Symbol +sigToSymN t sig = case filter pred symbols of + [] -> error . show $ + (("Error", "No symbol found"), + ("term t", t), + ("Name n", n), + ("symbols", symbols), + ("sig", sig)) + x:_ -> x + where pred x = name x == n + Name n = case t of + C c -> constName c + V v -> varName v + App{} -> error ("Tried to get sym for " ++ show t) + + symbols = Test.QuickSpec.Signature.symbols sig + name = Test.QuickSpec.Term.name + +-- Type synonyms for verbose QuickSpec types + +type QSSig = Test.QuickSpec.Signature.Sig + +type Ctx = Test.QuickSpec.Reasoning.NaiveEquationalReasoning.Context + +type Eqs = [Test.QuickSpec.Equation.Equation] + +-- Pruning algorithm adapted from Test.QuickSpec.quickSpec + +checkNames :: [Name] -> [Test.QuickSpec.Term.Symbol] -> Bool +checkNames ns syms = all (isIn syms) (map unName ns) + +isIn :: [Test.QuickSpec.Term.Symbol] -> String -> Bool +isIn syms n = any f syms + where f = (n ==) . Test.QuickSpec.Term.name + +classesFromEqs :: [Equation] -> [[Term]] +classesFromEqs eqs = combine [] clss' + where clss = foldl addToClasses [] eqs + clss' = map nub (foldl extend clss terms) + terms = concatMap (\(Eq l r) -> l : r : subTerms l ++ subTerms r) eqs + + subTerms (C _) = [] + subTerms (V _) = [] + subTerms (App l r _) = let l' = if termArity l == Arity 0 + then [l] + else [] + r' = if termArity l == Arity 0 + then [l] + else [] + ls = if termArity l > Arity 0 + then subTerms l + else [] + rs = if termArity r > Arity 0 + then subTerms r + else [] + in l' ++ r' ++ ls ++ rs + + extend [] t = [[t]] + extend (c:cs) t = if t `elem` c + then c:cs + else c : extend cs t + +addToClasses cs (Eq l r) = case cs of + [] -> [[l, r]] + x:xs -> if l `elem` x || r `elem` x + then (l:r:x):xs + else x : addToClasses xs (Eq l r) + +combine acc [] = acc +combine [] (c:cs) = combine [c] cs +combine acc (c:cs) = case nub (overlaps c) of + [] -> combine (c:acc) cs + xs -> combine [c ++ concat xs] (without xs) + where classesWith t = filter (t `elem`) (acc ++ cs) + overlaps = foldr ((++) . classesWith) [] + without xs = filter (`notElem` xs) (acc ++ cs) + +unSomeClassesN2 :: [Equation] + -> Test.QuickSpec.Signature.Sig + -> [Test.QuickSpec.Utils.Typed.Several Test.QuickSpec.Term.Expr] +unSomeClassesN2 eqs sig = collectExprs result + where classes = classesFromEqs eqs + result = unSomeSortedClasses classes sig + +unSomeSortedClasses classes sig = + unSomeSortedQSClasses (map (mkUnSomeClassN2 sig) classes) + +unSomeSortedQSClasses classes = sortBy multi classes + where multi (x:_) (y:_) = compareTerms x y + +compareTerms (Test.QuickSpec.Utils.Typed.Some x) (Test.QuickSpec.Utils.Typed.Some y) = + compare (Test.QuickSpec.Term.term x) (Test.QuickSpec.Term.term y) + +collectExprs :: [[Test.QuickSpec.Utils.Typed.Some Test.QuickSpec.Term.Expr]] + -> [Test.QuickSpec.Utils.Typed.Several Test.QuickSpec.Term.Expr] +collectExprs arg@[] = [] +collectExprs (xs:xss) = collectOne xs : collectExprs xss + where collectOne :: [Test.QuickSpec.Utils.Typed.Some Test.QuickSpec.Term.Expr] + -> Test.QuickSpec.Utils.Typed.Several Test.QuickSpec.Term.Expr + collectOne xs = case xs of + [] -> Test.QuickSpec.Utils.Typed.Some + (Test.QuickSpec.Utils.Typed.O + ([] :: [Test.QuickSpec.Term.Expr ()])) + [Test.QuickSpec.Utils.Typed.Some x] -> Test.QuickSpec.Utils.Typed.Some + (Test.QuickSpec.Utils.Typed.O + [x]) + (Test.QuickSpec.Utils.Typed.Some x:xs) -> case collectOne xs of + Test.QuickSpec.Utils.Typed.Some + (Test.QuickSpec.Utils.Typed.O (y:xs')) -> case cast x `asTypeOf` Just y of + Just x' -> Test.QuickSpec.Utils.Typed.Some + (Test.QuickSpec.Utils.Typed.O (x':y:xs')) + +mkUnSomeClassN :: Test.QuickSpec.Signature.Sig -> [Term] -> [Test.QuickSpec.Term.Expr a] +mkUnSomeClassN sig [] = [] +mkUnSomeClassN sig (x:xs) = termToExprN x sig : mkUnSomeClassN sig xs + +mkUnSomeClassN2 :: Test.QuickSpec.Signature.Sig + -> [Term] + -> [Test.QuickSpec.Utils.Typed.Some Test.QuickSpec.Term.Expr] +mkUnSomeClassN2 sig trms = sortBy compareTerms (mkUnSomeClassN2' sig trms) + +mkUnSomeClassN2' sig [] = [] +mkUnSomeClassN2' sig (x:xs) = + case termType (setForTerm x) of + Nothing -> error ("No type for " ++ show x) + Just t -> case getVal (getRep t) of + MkHT v -> (Test.QuickSpec.Utils.Typed.Some + (Test.QuickSpec.Term.Expr term + arity + (const v))) : xs' + where term = renderTermN x sig + Arity arity = termArity x + xs' = mkUnSomeClassN2' sig xs + +unSomePruneN :: [[Test.QuickSpec.Term.Expr Term]] + -> Test.QuickSpec.Signature.Sig + -> [Test.QuickSpec.Equation.Equation] +unSomePruneN clss sig = result + where result = Test.QuickSpec.Main.prune ctx reps id eqs + + ctx = mkCxt clss sig + + reps = classesToReps clss + + eqs = sort (mkEqs2N clss) + +mkCxt :: (Typeable a) => [[Test.QuickSpec.Term.Expr a]] + -> Test.QuickSpec.Signature.Sig + -> Test.QuickSpec.Reasoning.NaiveEquationalReasoning.Context +mkCxt clss sig = Test.QuickSpec.Reasoning.NaiveEquationalReasoning.initial + (Test.QuickSpec.Signature.maxDepth sig) + (Test.QuickSpec.Signature.symbols sig) + (mkUniv2N clss) + +classesToReps clss = filter (not . Test.QuickSpec.Term.isUndefined) + (map (term . head) clss) + +mkUniv2N :: Typeable a => [[Test.QuickSpec.Term.Expr a]] + -> [Test.QuickSpec.Utils.Typed.Tagged Test.QuickSpec.Term.Term] +mkUniv2N = concatMap (map univ2N) + +univ2N :: Data.Typeable.Typeable a => Test.QuickSpec.Term.Expr a + -> Test.QuickSpec.Utils.Typed.Tagged Test.QuickSpec.Term.Term +univ2N y = Test.QuickSpec.Utils.Typed.Tagged + (Test.QuickSpec.Utils.Typed.Some + (Test.QuickSpec.Utils.Typed.Witness (stripN y))) + (term y) + +stripN :: Test.QuickSpec.Term.Expr a -> a +stripN x = Test.QuickSpec.Term.eval x (Test.QuickSpec.Term.Valuation unvar) + where unvar = runGen . + Test.QuickSpec.Term.totalGen . + Test.QuickSpec.Term.value . + Test.QuickSpec.Term.unVariable + qcgen = Test.QuickCheck.Random.mkQCGen 0 + runGen g = Test.QuickCheck.Gen.unGen g qcgen 0 + +term = Test.QuickSpec.Term.term + +mkEqs2N cs = sort (concatMap f cs) + where f (z:zs) = [term y Test.QuickSpec.Equation.:=: term z | y <- zs] + +termToExprN :: Term + -> Test.QuickSpec.Signature.Sig + -> (Test.QuickSpec.Term.Expr a) +termToExprN t sig = Test.QuickSpec.Term.Expr term arity eval + where term = renderTermN t sig + Arity arity = termArity t + eval = undefined + +newtype Z = Z () deriving (Eq, Show, Typeable) +newtype S a = S a deriving (Eq, Show, Typeable, Ord) +instance Ord Z where compare _ _ = EQ +instance Ord (a -> b) where + compare _ _ = EQ +instance Eq (a -> b) where + _ == _ = True + +showEqsOnLinesN :: [Test.QuickSpec.Equation.Equation] + -> [Equation] +showEqsOnLinesN = map qsEqToEq + +qsEqToEq (l Test.QuickSpec.Equation.:=: r) = Eq (qsTermToTerm l) (qsTermToTerm r) + +qsTermToTerm t = case t of + Test.QuickSpec.Term.Var s -> V (symToVar s) + Test.QuickSpec.Term.Const s -> C (symToConst s) + Test.QuickSpec.Term.App l r -> App (qsTermToTerm l) (qsTermToTerm r) Nothing + +symToVar s = + let n = Test.QuickSpec.Term.name s + [_, rawT, idx] = splitCommas n + in Var (case HSE.Parser.parseType rawT of + HSE.Parser.ParseOk t' -> unwrapParens (const () <$> t') + HSE.Parser.ParseFailed _ err -> error (concat [ + "Failed to parse var type: ", + err, + ". Type was: ", + rawT])) + (read (init idx) :: Int) + (Arity (Test.QuickSpec.Term.symbolArity s)) + +symToConst s = + let t = HSE.Parser.parseType (show (Test.QuickSpec.Term.symbolType s)) + in Const (Arity (Test.QuickSpec.Term.symbolArity s)) + (Name (Test.QuickSpec.Term.name s)) + (case t of + HSE.Parser.ParseOk t' -> unwrapParens (const () <$> t') + HSE.Parser.ParseFailed _ err -> error (concat [ + "Failed to parse const type: ", + err, + ". Type was: ", + show (Test.QuickSpec.Term.symbolType s)])) + +splitCommas s = if ',' `elem` s + then takeWhile (/= ',') s : + splitCommas (tail (dropWhile (/= ',') s)) + else [s] + +pruneEqsN :: [Equation] -> [Equation] +pruneEqsN eqs = result + where pruned = doPrune classes sig + sig = let sig' = renderN (sigFromEqs eqs) + sig'' = Test.QuickSpec.Signature.signature sig' + in sig'' `mappend` Test.QuickSpec.Main.undefinedsSig sig'' + result = showEqsOnLinesN pruned + classes = unSomeClassesN2 eqs sig + +instance Show (Test.QuickSpec.Utils.Typed.Tagged Test.QuickSpec.Term.Term) where + show t = show ("Tagged" :: String, + ("tag" :: String, + ("type" :: String, Test.QuickSpec.Utils.Typed.witnessType + (Test.QuickSpec.Utils.Typed.tag t))), + ("erase" :: String, Test.QuickSpec.Utils.Typed.erase t)) + +doPrune clss sig = pruned + where univ = concatMap (Test.QuickSpec.Utils.Typed.some2 + (map (Test.QuickSpec.Utils.Typed.tagged term))) + clss + reps = map (Test.QuickSpec.Utils.Typed.some2 + (Test.QuickSpec.Utils.Typed.tagged term . head)) + clss + eqs = Test.QuickSpec.Equation.equations clss + + ctx = Test.QuickSpec.Reasoning.NaiveEquationalReasoning.initial + (Test.QuickSpec.Signature.maxDepth sig) + (Test.QuickSpec.Signature.symbols sig) + univ + + allEqs = map (Test.QuickSpec.Utils.Typed.some + Test.QuickSpec.Equation.eraseEquation) + eqs + + pruned = Test.QuickSpec.Main.prune + ctx + (filter (not . Test.QuickSpec.Term.isUndefined) + (map Test.QuickSpec.Utils.Typed.erase reps)) + id + allEqs diff --git a/src/Algebra/Equation/Internal/Types.hs b/src/Algebra/Equation/Internal/Types.hs new file mode 100644 --- /dev/null +++ b/src/Algebra/Equation/Internal/Types.hs @@ -0,0 +1,332 @@ +{-# LANGUAGE OverloadedStrings, TypeSynonymInstances, FlexibleInstances #-} + +module Algebra.Equation.Internal.Types where + +import Control.Monad +import Data.Aeson +import Data.List +import Data.Maybe +import Data.Ord (comparing) +import Data.Stringable hiding (length) +import qualified Data.Text.Lazy as T +import qualified Data.Text.Lazy.Encoding as TE +import qualified Language.Haskell.Exts.Parser as HSE.Parser +import qualified Language.Haskell.Exts.Pretty as HSE.Pretty +import qualified Language.Haskell.Exts.Syntax as HSE.Syntax +import System.Environment +import qualified Test.QuickSpec.Utils +import Text.Read (readMaybe) -- Uses String as part of base, not Text + +-- Types to represent equations, constants, variables, etc. and functions for +-- converting between representations + +data Equation = Eq Term Term + +instance Eq Equation where + (Eq l1 r1) == (Eq l2 r2) = (l1 == l2 && r1 == r2) || + (l1 == r2 && r1 == l2) + +instance ToJSON Equation where + toJSON (Eq l r) = object [ + "relation" .= toJSON ("~=" :: String), + "lhs" .= toJSON l, + "rhs" .= toJSON r + ] + +instance FromJSON Equation where + parseJSON (Object v) = Eq <$> v .: "lhs" <*> v .: "rhs" + parseJSON _ = mzero + +instance Show Equation where + show = T.unpack . TE.decodeUtf8 . encode . toJSON + +data Term = App Term Term (Maybe Type) | C Const | V Var + +instance Eq Term where + (C c1) == (C c2) = c1 == c2 + (V v1) == (V v2) = v1 == v2 + (App l1 r1 _) == (App l2 r2 _) = l1 == l2 && r1 == r2 + _ == _ = False + +-- Duplicates the sorting ofTest.QuickSpec.Term.Term, but without the type +-- juggling of "Some" and friends +instance Ord Term where + compare = comparing stamp + where stamp t = (depth t, size 0 t, -occur t, body t) + + occur t = length (Test.QuickSpec.Utils.usort (vars t [])) + + body (V x) = Left (Left x) + body (C x) = Left (Right x) + body (App f x _) = Right (f, x) + + vars (V x) = (x:) + vars (App f x _) = vars f . vars x + vars (C _) = id + + size v (App f x _) = size v f + size v x + size v (V _) = v + size v (C _) = 1 + + depth (App f x _) = depth f `max` (1 + depth x) + depth _ = 1 + +instance Show Term where + show (C c) = "C (" ++ show c ++ ")" + show (V v) = "V (" ++ show v ++ ")" + show (App l r _) = "App (" ++ show l ++ ") (" ++ show r ++ ")" + +instance ToJSON Term where + toJSON t = case t of + C c -> toJSON c + V v -> toJSON v + App l r _ -> object ["role" .= ("application" :: String), + "lhs" .= toJSON l, + "rhs" .= toJSON r] + +instance FromJSON Term where + parseJSON (Object v) = do + role <- v .: "role" + case (role :: String) of + "variable" -> V <$> parseJSON (Object v) + "constant" -> C <$> parseJSON (Object v) + "application" -> App <$> v .: "lhs" <*> v .: "rhs" <*> return Nothing + parseJSON _ = mzero + +data Sig = Sig [Const] [Var] deriving (Show) + +instance Eq Sig where + (Sig cs1 vs1) == (Sig cs2 vs2) = all (`elem` cs1) cs2 && + all (`elem` cs2) cs1 && + all (`elem` vs1) vs2 && + all (`elem` vs2) vs1 + +data Var = Var Type Int Arity deriving (Show, Eq, Ord) + +instance ToJSON Var where + toJSON (Var t i a) = object ["role" .= ("variable" :: String), + "type" .= toJSON t, + "id" .= toJSON i, + "arity" .= toJSON a] + +instance FromJSON Var where + parseJSON (Object v) = do + t <- v .: "type" + i <- v .: "id" + return (Var (unwrapParens t) i (Arity (countArity t))) + parseJSON _ = mzero + +countArity (HSE.Syntax.TyFun _ i o) = 1 + countArity o +countArity _ = 0 + +doCount haystack needle = T.count (T.pack needle) (T.pack haystack) + +data Const = Const Arity Name Type deriving (Show, Eq, Ord) + +instance ToJSON Const where + toJSON (Const a n t) = object ["role" .= ("constant" :: String), + "symbol" .= toJSON n, + "type" .= toJSON t, + "arity" .= toJSON a] + +instance FromJSON Const where + parseJSON (Object v) = do + t <- v .: "type" + s <- v .: "symbol" + return (Const (Arity (countArity t)) s (unwrapParens t)) + parseJSON _ = mzero + +type Type = HSE.Syntax.Type () + +instance ToJSON Type where + toJSON = String . fromString . HSE.Pretty.prettyPrint + +instance FromJSON Type where + parseJSON (String s) = case HSE.Parser.parseType (toString s) of + HSE.Parser.ParseOk t -> return (stripLoc t) + _ -> mzero + parseJSON _ = mzero + +stripLoc :: HSE.Syntax.Type a -> Type +stripLoc = fmap (const ()) + +data Name = Name String deriving (Show, Eq, Ord) + +instance ToJSON Name where + toJSON (Name n) = toJSON n + +instance FromJSON Name where + parseJSON (String s) = return (Name (toString s)) + parseJSON _ = mzero + +data Arity = Arity Int deriving (Show, Eq) + +instance ToJSON Arity where + toJSON (Arity a) = toJSON a + +instance FromJSON Arity where + parseJSON (Number n) = Arity <$> parseJSON (Number n) + parseJSON _ = mzero + +instance Num Arity where + fromInteger = Arity . fromInteger + (Arity a) + (Arity b) = Arity (a + b) + (Arity a) - (Arity b) = Arity (a - b) + (Arity a) * (Arity b) = Arity (a * b) + negate (Arity a) = Arity (negate a) + abs (Arity a) = Arity (abs a) + signum (Arity a) = Arity (signum a) + +instance Ord Arity where + Arity a <= Arity b = a <= b + +-- Sig construction + +emptySig :: Sig +emptySig = Sig [] [] + +instance Monoid Sig where + mempty = emptySig + mappend (Sig cs1 vs1) (Sig cs2 vs2) = Sig (nub (cs1 ++ cs2)) (nub (vs1 ++ vs2)) + +withConsts :: [Const] -> Sig -> Sig +withConsts cs (Sig cs' vs') = Sig (cs ++ cs') vs' + +withVars :: [Var] -> Sig -> Sig +withVars vs (Sig cs' vs') = Sig cs' (vs ++ vs') + +sigFromEqs :: [Equation] -> Sig +sigFromEqs = foldr (mappend . sigFromEq) emptySig + +sigFromEq :: Equation -> Sig +sigFromEq e = withVars (eqVars e) . + withConsts (eqConsts e) $ emptySig + +-- Accessors + +eqVars :: Equation -> [Var] +eqVars (Eq lhs rhs) = nub (termVars lhs ++ termVars rhs) + +eqConsts :: Equation -> [Const] +eqConsts (Eq lhs rhs) = nub (termConsts lhs ++ termConsts rhs) + +termConsts :: Term -> [Const] +termConsts t = nub $ case t of + App lhs rhs _ -> termConsts lhs ++ termConsts rhs + C c -> [c] + V v -> [] + +termVars :: Term -> [Var] +termVars t = nub $ case t of + App lhs rhs _ -> termVars lhs ++ termVars rhs + C c -> [] + V v -> [v] + +sigConsts :: Sig -> [Const] +sigConsts (Sig cs vs) = cs + +sigVars :: Sig -> [Var] +sigVars (Sig cs vs) = vs + +varName :: Var -> Name +varName (Var t i a) = Name ("(var, " ++ typeName t ++ ", " ++ show i ++ ")") + +typeName = fixUp . HSE.Pretty.prettyPrint + where fixUp = collapseSpace . filter (/= '\n') + collapseSpace (' ':' ':s) = collapseSpace (' ':s) + collapseSpace (c:s) = c : collapseSpace s + collapseSpace "" = "" + +varArity (Var t i a) = a + +varType (Var t i a) = t + +constName :: Const -> Name +constName (Const a n t) = n + +constArity :: Const -> Arity +constArity (Const a n t) = a + +constType :: Const -> Type +constType (Const a n t) = t + +termType :: Term -> Maybe Type +termType (C c) = Just (constType c) +termType (V v) = Just (varType v) +termType (App l r t) = t + +hasType (C c) = True +hasType (V v) = True +hasType (App l r Nothing) = False +hasType (App l r _) = hasType l && hasType r + +setAllTypes :: [Equation] -> [Equation] +setAllTypes = map setForEq + +setForEq (Eq l r) = Eq (setForTerm l) (setForTerm r) + +setForTerm (C c) = C c +setForTerm (V v) = V v +setForTerm (App l r (Just t)) = App (setForTerm l) (setForTerm r) (Just t) +setForTerm (App l r Nothing) = + let l' = setForTerm l + r' = setForTerm r + in case termType' l' of + HSE.Syntax.TyFun _ _ o -> App l' r' (Just o) + x -> error ("Expected function type, got " ++ show x) + +termType' :: Term -> Type +termType' t = let Just x = termType t in x + +-- JSON conversion + +sigFrom :: [Object] -> Sig +sigFrom xs = withConsts (constsFrom xs) . withVars (varsFrom xs) $ emptySig + +constsFrom :: [Object] -> [Const] +constsFrom _ = [] + +varsFrom :: [Object] -> [Var] +varsFrom _ = [] + +termArity :: Term -> Arity +termArity (C c) = constArity c +termArity (V v) = varArity v +termArity (App l r _) = termArity l - 1 + +unName (Name n) = n + +-- Make sure we don't have one name with multiple types or arities +consistent :: [Equation] -> Bool +consistent [] = True +consistent eqs = all haveOnce names + where db = nub (symTypes [] eqs) + + symTypes acc [] = acc + symTypes acc (Eq l r:xs) = symTypes (constsOf l ++ constsOf r ++ acc) xs + + constsOf (C (Const a n t)) = [(n, (a, t))] + constsOf (V _) = [] + constsOf (App l r _) = constsOf l ++ constsOf r + + names = map fst db + + haveOnce n = Data.List.length (filter ((== n) . fst) db) == 1 + +-- Required, since 'parse (prettyPrint t)' might have TyParens which 't' doesn't +unwrapParens (HSE.Syntax.TyFun _ i o) = HSE.Syntax.TyFun () + (unwrapParens i) + (unwrapParens o) +unwrapParens (HSE.Syntax.TyApp _ i o) = HSE.Syntax.TyApp () + (unwrapParens i) + (unwrapParens o) +unwrapParens (HSE.Syntax.TyParen _ t) = unwrapParens t +unwrapParens t = t + +replaceInType db (HSE.Syntax.TyFun _ i o) = HSE.Syntax.TyFun + () + (replaceInType db i) + (replaceInType db o) +replaceInType db t = fromMaybe + (error (show t ++ " not in " ++ show db)) + (lookup t db) diff --git a/src/Algebra/Equation/Reduce.hs b/src/Algebra/Equation/Reduce.hs new file mode 100644 --- /dev/null +++ b/src/Algebra/Equation/Reduce.hs @@ -0,0 +1,73 @@ +{-# LANGUAGE OverloadedStrings #-} + +module Algebra.Equation.Reduce where + +import Data.Aeson +import qualified Data.ByteString.Lazy.Char8 as BS +import qualified Data.ByteString.Internal as BI +import Data.List +import Data.Maybe +import qualified Data.Stringable as S +import qualified Data.Text.Lazy as T +import qualified Data.Text.Lazy.Encoding as TE +import qualified Language.Haskell.Exts.Syntax as HSE.Syntax +import Algebra.Equation.Internal.Eval +import Algebra.Equation.Internal.Types + +doReduce = parseAndReduce <$> BS.getContents >>= showEqs + +showEqs = mapM_ (BS.putStrLn . encode) + +parseAndReduce s = case eitherDecode s of + Left err -> error ("Failed to parse eqs: " ++ err) + Right eqs -> reduction eqs + +reduction :: [Equation] -> [Equation] +reduction eqs = if consistent eqs' + then result + else error "Inconsistent types in parsed equations" + where eqs' = setAllTypes eqs + (db, eqs'') = replaceTypes eqs' + o = pruneEqsN eqs'' + result = restoreTypes db o + +replaceTypes :: [Equation] -> ([(Type, Type)], [Equation]) +replaceTypes eqs = let db = zip typs new + in (db, map (replaceEqTypes db) eqs) + where typs = allTypes eqs + new = iterate s z + z = tyCon "Z" + s = HSE.Syntax.TyApp () (tyCon "S") + +replaceEqTypes db (Eq l r) = Eq (replaceTermTypes db l) (replaceTermTypes db r) + +replaceTermTypes db trm = case trm of + C (Const a n t) -> C (Const a n (replace t)) + V (Var t i a) -> V (Var (replace t) i a) + App l r t -> App (replaceTermTypes db l) + (replaceTermTypes db r) + (replace <$> t) + where replace = replaceInType db . unwrapParens + +tyCon = HSE.Syntax.TyCon () . HSE.Syntax.UnQual () . HSE.Syntax.Ident () + +allTypes :: [Equation] -> [Type] +allTypes = nub . filter notFunc + . concatMap components + . catMaybes + . nub + . concatMap eqTypes + where eqTypes (Eq l r) = termTypes l ++ termTypes r + termTypes (App l r t) = [unwrapParens <$> t] ++ termTypes l ++ termTypes r + termTypes t = [unwrapParens <$> termType t] + + notFunc (HSE.Syntax.TyFun _ _ _) = False + notFunc _ = True + + components (HSE.Syntax.TyFun _ i o) = components i ++ components o + components t = [t] + +restoreTypes :: [(Type, Type)] -> [Equation] -> [Equation] +restoreTypes db = map (replaceEqTypes db') + where db' = map swap db + swap (x, y) = (y, x) diff --git a/test.sh b/test.sh new file mode 100644 --- /dev/null +++ b/test.sh @@ -0,0 +1,30 @@ +#!/usr/bin/env bash + +ERR=0 +function report { + if [[ "$1" -eq 0 ]] + then + echo "ok - $2" + else + echo "not ok - $2" + ERR=1 + fi +} + +cabal build +report "$?" "cabal build" + +cabal test --show-details=streaming +report "$?" "cabal test" + +for F in test/data/*.json +do + OUTPUT=$(NIX_EVAL_EXTRA_IMPORTS='[("runtime-arbitrary", "TestInstances")]' \ + cabal run -v0 reduce-equations < "$F") + report "$?" "Reducing $F" + + echo "$OUTPUT" | grep '^{' + report "$?" "Got equations from $F" +done + +exit "$ERR" diff --git a/test/Main.hs b/test/Main.hs new file mode 100644 --- /dev/null +++ b/test/Main.hs @@ -0,0 +1,1133 @@ +{-# LANGUAGE BangPatterns, OverloadedStrings, PartialTypeSignatures, ScopedTypeVariables, TypeSynonymInstances, FlexibleInstances, FlexibleContexts #-} + +module Main where + +import Data.Aeson +import qualified Data.ByteString as B +import qualified Data.ByteString.Lazy as LB +import Data.Char +import Data.List +import Data.Maybe +import qualified Data.Sequence as Seq +import qualified Data.Stringable as S +import qualified Data.String as DS +import Data.String.Utils +import Data.Text.Encoding +import Data.Typeable +import qualified Language.Haskell.Exts.Parser as HSE.Parser +import qualified Language.Haskell.Exts.Syntax as HSE.Syntax +import Algebra.Equation.Internal +import Algebra.Equation.Reduce +import Numeric.Natural +import System.Directory +import System.IO.Unsafe +import Test.QuickCheck +import Test.QuickCheck.Monadic +import qualified Test.QuickSpec.Equation +import qualified Test.QuickSpec.Generate +import qualified Test.QuickSpec.Main +import qualified Test.QuickSpec.Reasoning.CongruenceClosure +import qualified Test.QuickSpec.Reasoning.NaiveEquationalReasoning +import qualified Test.QuickSpec.Signature +import qualified Test.QuickSpec.TestTree +import qualified Test.QuickSpec.Utils.Typeable +import qualified Test.QuickSpec.Utils.Typed +import qualified Test.QuickSpec.Utils.TypeMap +import qualified Test.QuickSpec.Term +import Test.Tasty (defaultMain, testGroup, localOption) +import Test.Tasty.QuickCheck + +main = defaultMain $ testGroup "All tests" [ + testProperty "Can parse equations" canParseEquations + , testProperty "Can parse terms" canParseTerms + , testProperty "Can parse variables" canParseVars + , testProperty "Can parse constants" canParseConsts + , testProperty "Can parse examples" canParseExamples + , testProperty "Can evaluate examples" canEvalExamples + , testProperty "Can make example signature" canMakeSignature + , testProperty "Constants added" constantsAdded + , testProperty "Variables added" variablesAdded + , testProperty "Sigs render" sigsRender + , testProperty "Sigs have constants" sigsHaveConsts + , testProperty "Sigs have variables" sigsHaveVars + , testProperty "Constants are distinct" sigConstsUniqueIndices + , testProperty "Variables are distinct" sigVarsUniqueIndices + , testProperty "Can find closure of term" canFindClosure + , testProperty "No classes without equations" noClassesFromEmptyEqs + , testProperty "Equation induces a class" getClassFromEq + , testProperty "Classes contain given terms" classesHaveTerms + , testProperty "Equal terms in same class" eqPartsAppearInSameClass + , testProperty "Terms appear in one class" classesHaveNoDupes + , testProperty "Class elements are equal" classElementsAreEqual + , testProperty "Non-equal elements separate" nonEqualElementsSeparate + , testProperty "Classes have one arity" classHasSameArity + , testProperty "Class length more than one" classesNotSingletons + , testProperty "Can get classes from sig" canGetClassesFromEqs + , testProperty "Can get sig from equations" canGetSigFromEqs + , testProperty "Sig has equation variables" eqSigHasVars + , testProperty "Sig has equation constants" eqSigHasConsts + , testProperty "Equations have one arity" equationsHaveSameArity + , testProperty "Can get type of terms" canGetTermType + , testProperty "No trivial terms" noTrivialTerms + , testProperty "Equations are consistent" eqsAreConsistent + , testProperty "Switch function types" switchFunctionTypes + , testProperty "Can prune equations" canPruneEqs + , testProperty "Type parsing regression" regressionTypeParse + , testProperty "Nat example has eqs" natHasEqs + , testProperty "Nat example outputs eqs" natKeepsEqs + , testProperty "Nat classes are nontrivial" natClassesNontrivial + , testProperty "Commutativity is nontrivial" commClassesNontrivial + , testProperty "Nat equations are pruned" natEqsPruned + , testProperty "Reduction matches QuickSpec" natEqsMatchQS + , testProperty "Replacement types unique" extractedTypesUnique + , testProperty "QuickSpec conversions invert" convertTypesIso + , testProperty "Symmetric equations equal" eqsSymmetric + , testProperty "Spot symmetric equations" eqsSetEq + , testProperty "Can generate eq variables" canMakeVars + , testProperty "Can generate var QS sigs" canMakeQSSigs + , testProperty "Can find vars in sig" lookupVars + , testProperty "Can prune" justPrune + , testProperty "New reduce" newReduce + , testProperty "Reduce is idempotent" reduceIdem + , testProperty "Redundant transitivity" transStripped + , testProperty "Generated terms have type" termsHaveType + , testProperty "Manual Nat finds eqs" manualNatFindsEqs + , testProperty "Fresh sig + parsed eqs works" manualNatAllowsGiven + , testProperty "Fresh classes match parsed" manualNatClassesMatch + , testProperty "Fresh classes have parsed terms" topNatTermsFound + , testProperty "Parsed classes have fresh terms" qsNatTermsFound + , testProperty "Singleton classes generated" natClassesIncludeSingletons + , testProperty "Class contents match exactly" exactClassMatch + , testProperty "Pruned eqs match" parsedEqsPrune + , testProperty "Fresh Nat reduces own eqs" manualNatReducesSelf + ] + +-- Tests + +canParseEquations = all try [ + "{\"relation\":\"~=\",\"lhs\":{\"role\":\"application\",\"lhs\":{\"role\":\"application\",\"lhs\":{\"role\":\"constant\",\"type\":\"[Integer] -> [Integer] -> Ordering\",\"symbol\":\"lengthCompare\"},\"rhs\":{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":7}},\"rhs\":{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":7}},\"rhs\":{\"role\":\"application\",\"lhs\":{\"role\":\"application\",\"lhs\":{\"role\":\"constant\",\"type\":\"[Integer] -> [Integer] -> Ordering\",\"symbol\":\"lengthCompare\"},\"rhs\":{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":6}},\"rhs\":{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":6}}}", + "{\"relation\":\"~=\",\"lhs\":{\"role\":\"application\",\"lhs\":{\"role\":\"application\",\"lhs\":{\"role\":\"constant\",\"type\":\"[Integer] -> [Integer] -> Ordering\",\"symbol\":\"lengthCompare\"},\"rhs\":{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":8}},\"rhs\":{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":8}},\"rhs\":{\"role\":\"application\",\"lhs\":{\"role\":\"application\",\"lhs\":{\"role\":\"constant\",\"type\":\"[Integer] -> [Integer] -> Ordering\",\"symbol\":\"lengthCompare\"},\"rhs\":{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":6}},\"rhs\":{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":6}}}" + ] + where try x = case decode x of + Nothing -> False + Just (Eq l r) -> True + +canParseTerms = all try [ + "{\"role\":\"constant\",\"type\":\"[Integer] -> [Integer] -> Ordering\",\"symbol\":\"lengthCompare\"}", + "{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":7}", + "{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":6}", + "{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":8}", + "{\"role\":\"application\",\"lhs\":{\"role\":\"constant\",\"type\":\"[Integer] -> [Integer] -> Ordering\",\"symbol\":\"lengthCompare\"},\"rhs\":{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":7}}", + "{\"role\":\"application\",\"lhs\":{\"role\":\"constant\",\"type\":\"[Integer] -> [Integer] -> Ordering\",\"symbol\":\"lengthCompare\"},\"rhs\":{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":6}}", + "{\"role\":\"application\",\"lhs\":{\"role\":\"constant\",\"type\":\"[Integer] -> [Integer] -> Ordering\",\"symbol\":\"lengthCompare\"},\"rhs\":{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":8}}", + "{\"role\":\"application\",\"lhs\":{\"role\":\"constant\",\"type\":\"[Integer] -> [Integer] -> Ordering\",\"symbol\":\"lengthCompare\"},\"rhs\":{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":6}}", + "{\"role\":\"application\",\"lhs\":{\"role\":\"application\",\"lhs\":{\"role\":\"constant\",\"type\":\"[Integer] -> [Integer] -> Ordering\",\"symbol\":\"lengthCompare\"},\"rhs\":{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":7}},\"rhs\":{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":7}}", + "{\"role\":\"application\",\"lhs\":{\"role\":\"application\",\"lhs\":{\"role\":\"constant\",\"type\":\"[Integer] -> [Integer] -> Ordering\",\"symbol\":\"lengthCompare\"},\"rhs\":{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":6}},\"rhs\":{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":6}}", + "{\"role\":\"application\",\"lhs\":{\"role\":\"application\",\"lhs\":{\"role\":\"constant\",\"type\":\"[Integer] -> [Integer] -> Ordering\",\"symbol\":\"lengthCompare\"},\"rhs\":{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":8}},\"rhs\":{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":8}}" + ] + where try x = case decode x :: Maybe Term of + Nothing -> error ("Couldn't decode " ++ S.toString x) + Just _ -> True + +canParseVars = all try [ + "{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":7}", + "{\"role\":\"variable\",\"type\":\"[Integer]\",\"id\":6}" + ] + where try x = case decode x :: Maybe Var of + Nothing -> error ("Couldn't decode " ++ S.toString x) + Just _ -> True + +canParseConsts = all try [ + "{\"role\":\"constant\",\"type\":\"[Integer] -> [Integer] -> Ordering\",\"symbol\":\"lengthCompare\"}" + ] + where try x = case decode x :: Maybe Const of + Nothing -> error ("Couldn't decode " ++ S.toString x) + Just _ -> True + +canParseExamples = not (null exampleEqs) + +canEvalExamples = withExamples allStrict + where allStrict = foldr ((&&) . strict) True + strict (Eq !l !r) = True + +canMakeSignature = withExamples makeSig + where makeSig xs = case sigFromEqs xs of + Sig !cs !vs -> True + +constantsAdded cs s = case withConsts cs s of + Sig cs' _ -> all (`elem` cs') cs + +variablesAdded vs s = case withVars vs s of + Sig _ vs' -> all (`elem` vs') vs + +sigsRender = once $ do + eqs <- resize 20 genNormalisedEqs + let sig = sigFromEqs eqs + s = show (renderN sig :: Test.QuickSpec.Signature.Sig) + return (length s >= 0) + +sigsHaveConsts = once $ do + eqs <- resize 42 genNormalisedEqs + let s@(Sig cs vs) = sigFromEqs eqs + rendered = renderN s + consts = Test.QuickSpec.Signature.constantSymbols rendered + names = map constName cs + return (checkNames names consts) + +sigsHaveVars = once (forAll (resize 42 genNormalisedEqs) sigsHaveVars') +sigsHaveVars' eqs = + let s@(Sig _ vs) = sigFromEqs eqs + + rendered = renderN s + + variables = Test.QuickSpec.Signature.variableSymbols rendered + + hasVars :: Bool + hasVars = checkVars variables + + checkVars = checkNames names + names = map varName vs + foundNames = map Test.QuickSpec.Term.name variables + dbg = show (("expect names", names), + ("found names", foundNames)) + in counterexample dbg (return hasVars :: Gen Bool) + +sigConstsUniqueIndices = once . resize 20 $ do + s <- genNormalisedSig + c <- genNormalisedConst + return (sigConstsUniqueIndices' s c) + +-- Use `c` to generate a bunch of similar constants `consts`, add them to `s` to +-- get `sig`. Render `sig` to a QuickSpec signature, then print out its constant +-- symbols and compare with those of `sig`. +sigConstsUniqueIndices' s (Const a (Name n) t) = hasConsts + where syms = Test.QuickSpec.Signature.constantSymbols (renderN sig) + names = map Test.QuickSpec.Term.name syms + hasConsts = setEq (map Name names) + (map constName (sigConsts sig)) + consts = [Const a (Name (n ++ show i)) t | i <- [0..10]] + sig = withConsts consts s + +sigVarsUniqueIndices :: Property +sigVarsUniqueIndices = once $ resize 20 $ do + s <- genNormalisedSig + v <- genNormalisedVar + return (sigVarsUniqueIndices' s v) + +-- Use `v` to generate a bunch of `Var`s of the same type, `vars`, add them to +-- `s` to get `sig`. Render `sig` to a QuickSpec signature, then print out its +-- variable symbols and compare with those of `sig`. +sigVarsUniqueIndices' :: Sig -> Var -> Bool +sigVarsUniqueIndices' s (Var t _ a) = hasVars + where syms = Test.QuickSpec.Signature.variableSymbols (renderN sig) + names = map Test.QuickSpec.Term.name syms + hasVars = setEq (map Name names) + (map varName (sigVars sig)) + vars = [Var t i a | i <- [0..10]] + sig = withVars vars s + +-- Some vars get split over multiple lines +readVars s = accumulate [] (lines s) + where accumulate (v:vs) (l@(' ':_):ls) = accumulate ((v ++ l):vs) ls + accumulate vs (l :ls) = accumulate (l :vs) ls + accumulate vs [] = vs + +noClassesFromEmptyEqs = null (classesFromEqs []) + +-- Sub-terms are added, which can make more than one class +getClassFromEq eq = length (classesFromEqs [eq]) >= 1 + +classesHaveTerms eqs = found `all` terms + where terms = concatMap termsOf eqs + termsOf (Eq l r) = [l, r] + found t = (t `elem`) `any` classes + classes = classesFromEqs eqs + +eqPartsAppearInSameClass eqs = counterexample (show debug) test + where test = all eqPartsInSameClass eqs + classes = classesFromEqs eqs + matchingClass t = head $ filter (t `elem`) classes + eqPartsInSameClass (Eq l r) = r `elem` matchingClass l && + l `elem` matchingClass r + debug = (("eqs", eqs), ("classes", classes)) + +classesHaveNoDupes eqs = counterexample (show debug) test + where test = all appearOnce terms + classes = classesFromEqs eqs + terms = concat classes + appearOnce t = length (filter (t `elem`) classes) == 1 + debug = (("eqs", eqs), ("classes", classes), ("terms", terms)) + +classHasSameArity eqs = all oneArity classes + where classes = classesFromEqs eqs + oneArity ts = length (nub (map termArity ts)) == 1 + +equationsHaveSameArity (Eqs eqs) = all sameArity eqs + where sameArity (Eq l r) = termArity l == termArity r + +nonEqualElementsSeparate ty = forAll (iterable ty) nonEqualElementsSeparate' + +nonEqualElementsSeparate' (t, v) = all found expected + where (a:b:c:d:e:f:_) = map extend [0..] + + extend 0 = t + extend n = app v (extend (n-1)) + + eqs = [Eq a b, Eq b c, Eq d e, Eq e f] + + classes = classesFromEqs eqs + + expected = [[a, b, c], [d, e, f]] + + found xs = any (setEq xs) classes + + match xs ys = all (\x -> any (setEq x) ys) xs + +classElementsAreEqual = once (forAll (resize 42 arbitrary) classElementsAreEqual') +classElementsAreEqual' (Eqs eqs) = all elementsAreEqual classes + where classes :: [[Term]] + classes = classesFromEqs eqs + + terms = nub $ concatMap termsOf eqs + termsOf (Eq x y) = [x, y] + + elementsAreEqual :: [Term] -> Bool + elementsAreEqual cls = all (equalToAll cls) cls + + equalToAll :: [Term] -> Term -> Bool + equalToAll xs y = all (equal y) xs + + equal :: Term -> Term -> Bool + equal x y = y `isElem` eqClosure eqs x + +classesNotSingletons (Eqs eqs) = all nonSingle classes' + where nonSingle c = length (nub c) > 1 + classes = classesFromEqs eqs -- All classes, even subterms + classes' = filter (\c -> any (`elem` c) terms) classes -- Top-level + terms = getTerms [] eqs + getTerms acc [] = acc + getTerms acc (Eq l r:es) = getTerms (l:r:acc) es + +canFindClosure ty = forAll (iterable ty) canFindClosure' + +canFindClosure' (t, v) = all match expected + where -- Generate unique terms by wrapping in "app c" + (a:b:c:d:e:f:g:h:_) = map extend [0..] + extend 0 = t + extend n = app v (extend (n-1)) + + match (x, xs) = setEq (eqClosure eqs x) xs + + eqs = [Eq a b, Eq a c, Eq b d, Eq b b, Eq f g, Eq f h] + + abcd = [a, b, c, d] + fgh = [f, g, h] + expected = [(a, abcd), (b, abcd), (c, abcd), (d, abcd), + (e, [e]), + (f, fgh), (g, fgh), (h, fgh)] + +canGetClassesFromEqs (Eqs eqs) = True + where typeCheck = classesFromEqs eqs + +canGetSigFromEqs eqs = case sigFromEqs eqs of + Sig _ _ -> True + +eqSigHasVars eqs = counterexample debug test + where sig = sigFromEqs eqs + sigvars = sigVars sig + eqvars = concatMap eqVars eqs + test = setEq sigvars eqvars + debug = show (("eqs", eqs), + ("sigvars", sigvars), + ("eqvars", eqvars), + ("sig", sig)) + +eqSigHasConsts eqs = counterexample debug test + where sig = sigFromEqs eqs + test = setEq sigconsts eqconsts + sigconsts = sigConsts sig + eqconsts = concatMap eqConsts eqs + debug = show (("eqs", eqs), + ("sig", sig), + ("sigconsts", sigconsts), + ("eqconsts", eqconsts)) + +canPruneEqs = once (forAll (resize 20 arbitrary) canPruneEqs') +canPruneEqs' (Eqs eqs) = counterexample (show (("eqs", eqs), ("eqs'", eqs'))) + (expected eqs') + where expected [] = null eqs -- No output when no eqs + expected (x:xs) = not (null eqs) + + eqs' = reduction eqs + +canGetTermType input output = expected (termType' term) + where term = app (C (Const undefined undefined func)) + (C (Const undefined undefined input)) + func = tyFun input output + strip = filter (/= ' ') + expected t = strip (typeName t) === strip (typeName output) + +noTrivialTerms t = forAll (termOfType t) (not . trivial) + +eqsAreConsistent (Eqs eqs) = consistentEqs eqs + +-- Given 'i' and 'o', we should be able to replace 'i -> o' +switchFunctionTypes i1 i2 o1 o2 = check <$> termOfType (HSE.Syntax.TyFun () i1 o1) + where db = [(i1, i2), (o1, o2)] + check f = let [Eq lhs rhs] = restoreTypes db [replaceEqTypes db (Eq f f)] + in termType lhs == termType rhs + +regressionTypeParse = LB.length (encode result) > 0 + where ex = "[{\"relation\":\"~=\",\"lhs\":{\"role\":\"application\",\"lhs\":{\"role\":\"constant\",\"type\":\"List Integer -> List Integer\",\"symbol\":\"reverse\"},\"rhs\":{\"role\":\"application\",\"lhs\":{\"role\":\"application\",\"lhs\":{\"role\":\"constant\",\"type\":\"Integer -> List Integer -> List Integer\",\"symbol\":\"cCons\"},\"rhs\":{\"role\":\"variable\",\"type\":\"Integer\",\"id\":3}},\"rhs\":{\"role\":\"constant\",\"type\":\"List Integer\",\"symbol\":\"cNil\"}}},\"rhs\":{\"role\":\"application\",\"lhs\":{\"role\":\"application\",\"lhs\":{\"role\":\"constant\",\"type\":\"Integer -> List Integer -> List Integer\",\"symbol\":\"cCons\"},\"rhs\":{\"role\":\"variable\",\"type\":\"Integer\",\"id\":3}},\"rhs\":{\"role\":\"constant\",\"type\":\"List Integer\",\"symbol\":\"cNil\"}}}]" + result = parseAndReduce ex + +natHasEqs = case eitherDecode rawNatEqs :: Either String [Equation] of + Left err -> error err + Right [] -> error "No equations found" + Right eqs -> True + +natKeepsEqs = case parseAndReduce rawNatEqs of + [] -> error "No equations found" + _ -> True + +natClassesNontrivial = + let raw = rawNatEqs + rawEqs = parsedNatEqs + (_, rawEqs') = replaceTypes rawEqs + classes = classesFromEqs rawEqs' + in counterexample (show classes) + (any (\c -> length c > 2) classes) + +commClassesNontrivial n1 n2 i o = once $ + counterexample (show (("eqs", eqs), + ("clss", clss), + ("comm", comm))) + (all ((> 1) . length . nub) clss') + where f = C $ Const (Arity 2) n1 (HSE.Syntax.TyFun () i (HSE.Syntax.TyFun () i o)) + q = C $ Const (Arity 1) n2 (HSE.Syntax.TyFun () o o) + x = V $ Var i 0 (Arity iArity) + y = V $ Var i 1 (Arity iArity) + + iArity = countArity i + + (cl, cr) = (App (App f x Nothing) y Nothing, App (App f y Nothing) x Nothing) + (ql, qr) = (App q (App (App f x Nothing) y Nothing) Nothing, + App q (App (App f y Nothing) x Nothing) Nothing) + comm = Eq cl cr + eqs = [comm, Eq ql qr] + + clss = classesFromEqs eqs + clss' = filter topLevel clss + topLevel c = any (`elem` c) [cl, cr, ql, qr] + +natEqsPruned = length pruned < length rawEqs' + where (_, rawEqs') = replaceTypes parsedNatEqs + pruned = pruneEqsN rawEqs' + +natEqsMatchQS = once $ monadicIO $ do + expect <- run $ LB.readFile "test/data/nat-simple-expect.json" + let Right expectEqs = eitherDecode expect :: Either String [Equation] + foundEqs = parseAndReduce rawNatEqs + fLen = length foundEqs + eLen = length expectEqs + monitor . counterexample $ show (("foundEqs", foundEqs), + ("expectEqs", expectEqs), + ("length foundEqs", length foundEqs), + ("length expectEqs", length expectEqs)) + assert (length foundEqs == length expectEqs) + +extractedTypesUnique (Eqs eqs) = counterexample (show types) + (nub types == types) + where types = allTypes eqs + +convertTypesIso = + (forAll (resize 20 arbitrary) + (\x -> case convertTypesIso' x of + ([], []) -> property True + (extraEqs', extraConv) -> + counterexample (show (("extraEqs'", extraEqs'), + ("extraConv", extraConv))) + (property False))) +convertTypesIso' (Eqs eqs) = + let (_, eqs') = replaceTypes eqs + qsEqs = Test.QuickSpec.Equation.equations qsClss + sig = renderN (sigFromEqs eqs') + qsClss = unSomeClassesN2 eqs' sig + conv = map (qsEqToEq . Test.QuickSpec.Utils.Typed.some Test.QuickSpec.Equation.eraseEquation) qsEqs + conv' = map (Test.QuickSpec.Utils.Typed.several + (map (qsTermToTerm . Test.QuickSpec.Term.term))) + qsClss + clss = classesFromEqs eqs' + result = setEq clss conv' + cmp (Eq l1 r1) (Eq l2 r2) = (show l1 == show l2 && show r1 == show r2) || + (show l1 == show r2 && show r1 == show l2) + in setDiffBy setEq clss conv' + +eqsSymmetric [] = return True +eqsSymmetric eqs = do + Eq l r <- elements eqs + return (Eq r l `elem` eqs) + +eqsSetEq eqs = setEq eqs (map swap eqs) + where swap (Eq l r) = Eq r l + +canMakeVars = do + v <- genNormalisedVar + return True + +canMakeQSSigs = do + v <- genNormalisedVar + let sig = renderQSVars [v] + return (length (show sig) > 0) + +lookupVars = once $ do + eqs <- resize 42 genEqsWithVars + let Sig _ vs = sigFromEqs eqs' + (_, eqs') = replaceTypes eqs + v <- elements vs + let expectName = unName (varName v) + sig = renderQSVars [v] + symbol = sigToSymN (V v) sig + foundName = Test.QuickSpec.Term.name symbol + return (expectName == foundName) + +justPrune = once $ do + eqs <- resize 20 arbitrary + let (_, eqs') = replaceTypes eqs + o = pruneEqsN eqs' + return (length o >= 0) + +newReduce = once (forAll (resize 20 arbitrary) newReduce') +newReduce' (Eqs eqs) = counterexample (show (("eqs", eqs), + ("result", result))) + (length (show result) > 0) + where result = reduction eqs + +reduceIdem = once (forAll (resize 20 arbitrary) reduceIdem') +reduceIdem' (Eqs eqs) = setEq eqs' eqs'' + where eqs' = reduction eqs + eqs'' = reduction eqs' + +transStripped = once . resize 10 $ do + Eqs eqs <- arbitrary + t <- arbitrary + a <- termOfType t + b <- termOfType t + c <- termOfType t + ds <- listOf (termOfType t) + + -- Add a bunch of redundant equations + eqs' <- renameEqs (eqs ++ [Eq x y | x <- b:c:ds, y <- b:c:ds]) + let (_, eqs'') = replaceTypes eqs' + pruned = reduction eqs'' + + -- Check if (at least) our redundant equations got stripped out + return (length pruned <= length eqs + 2 + length ds) + +termsHaveType ty = forAll (termOfType ty) checkType + where checkType trm = termType (setForTerm trm) == Just ty + +manualNatFindsEqs = once . monadicIO $ do + -- Our raw equations, for comparison + let rawEqs = parsedNatEqs + clss = rawNatClasses + eqs = Test.QuickSpec.Equation.equations clss + eqs' = dbgEqs eqs + monitor . counterexample . show $ ("eqs'", eqs') + + -- Our golden input should match these + assert (length eqs == length rawEqs) + +manualNatAllowsGiven = counterexample (show eqs') + ((length eqs' > 0) && + (length eqs' <= length parsedNatEqs)) + where clss = classesFromEqs naturalEqs + clss' = sort (map (sort . mkUnSomeClassN natSig) clss) + eqs' = unSomePruneN clss' natSig + + naturalEqs = map (replaceEqTypes db) parsedNatEqs + db = [(tyCon "Nat", tyCon "Natural")] + +manualNatClassesMatch = counterexample dbg result + where ourClss = classesFromEqs eqs' + ourClss' = sort (map (sort . mkUnSomeClassN natSig) ourClss) + ourClss'' = map (map Test.QuickSpec.Term.term) ourClss' + + eqs' = map (replaceEqTypes db) parsedNatEqs + db = [(tyCon "Nat", tyCon "Natural")] + + result = setEq natNonTrivial ourClss'' + + natNonTrivial = filter ((> 1) . length) natClasses + + dbg = show (("qs non-trivial", length natNonTrivial), + ("our classes", length ourClss'')) + +topNatTermsFound = all termInClasses (termsOf [] eqs') + where termsOf acc [] = acc + termsOf acc (Eq l r:xs) = termsOf (l:r:acc) xs + termInClasses t = let qs = any (renderTermN t natSig `elem`) natClasses + us = any (t `elem`) clss + in case (qs, us) of + (False, False) -> error $ show t ++ " not found in either" + (False, _) -> error $ show t ++ " not in QS" + (_, False) -> error $ show t ++ " not in ours" + _ -> True + clss = classesFromEqs eqs' + eqs' = map (replaceEqTypes db) parsedNatEqs + db = [(tyCon "Nat", tyCon "Natural")] + +qsNatTermsFound = case unfoundInOurs of + [] -> True + us -> error (show ("unfound", us)) + where allTerms = concat (filter ((> 1) . length) natClasses) + + clss = classesFromEqs eqs' + + inOurs t = any (t `elem`) (map (map (`renderTermN` natSig)) clss) + + unfoundInOurs = filter (not . inOurs) allTerms + + db = [(tyCon "Nat", tyCon "Natural")] + eqs' = map (replaceEqTypes db) parsedNatEqs + +natClassesIncludeSingletons = any ((== 1) . length) natClasses + +-- We compare the "show" output, to avoid irrelevant details like symbol indices +exactClassMatch = counterexample (show (("ourClasses", ourClasses), + ("qsClasses'", qsClasses'))) + (ourClasses == qsClasses') + where ourClasses = map (Test.QuickSpec.Utils.Typed.several (map term)) + (unSomeClassesN2 parsedNatEqs' natSig') + qsClasses = filter ((> 1) . length) natClasses + qsClasses' = replaceQSTypes naturalDb qsClasses + +convertPrunes = length (doPrune qsClasses natSig') == 10 + where qsClasses = filter (Test.QuickSpec.Utils.Typed.several ((> 1) . length)) + rawNatClasses' + db = [(tyCon "Natural", tyCon "Z")] + +-- We don't compare equations directly, since they may differ slightly e.g. due +-- to commutativity +parsedEqsPrune = nub [length ourPrune, length qsPrune] == [10] + where (db, eqs') = replaceTypes parsedNatEqs + ourPrune = doPrune classes sig + classes = unSomeClassesN2 eqs' sig + sig = let sig' = renderN (sigFromEqs eqs') + sig'' = Test.QuickSpec.Signature.signature sig' + in sig'' `mappend` Test.QuickSpec.Main.undefinedsSig sig'' + + qsPrune = doPrune rawNatClasses natSig + +manualNatReducesSelf = counterexample (show ("pruned", pruned)) + ((length pruned < 20) && + (length pruned > 1)) + where pruned = doPrune rawNatClasses natSig + +-- Helpers + +natSig = mconcat [ + Test.QuickSpec.Signature.fun0 "cZ" (0 :: Natural), + Test.QuickSpec.Signature.fun1 "cS" ((+1) :: Natural -> Natural), + Test.QuickSpec.Signature.fun2 "plus" ((+) :: Natural -> Natural -> Natural), + Test.QuickSpec.Signature.fun2 "times" ((*) :: Natural -> Natural -> Natural), + Test.QuickSpec.Signature.gvars (map (\n -> "(var, Natural, " ++ show n ++ ")") + [0, 1, 2]) + (((fromInteger . abs) <$> arbitrary) :: Gen Natural)] + +natSig' = mconcat [ + Test.QuickSpec.Signature.fun0 "cZ" (undefined :: Z), + Test.QuickSpec.Signature.fun1 "cS" (undefined :: Z -> Z), + Test.QuickSpec.Signature.fun2 "plus" (undefined :: Z -> Z -> Z), + Test.QuickSpec.Signature.fun2 "times" (undefined :: Z -> Z -> Z), + Test.QuickSpec.Signature.gvars (map (\n -> "(var, Z, " ++ show n ++ ")") + [0, 1, 2]) + (return undefined :: Gen Z)] + + +genEqsWithVars = arbitrary `suchThat` hasVars + where hasVars eqs = case sigFromEqs eqs of + Sig _ [] -> False + _ -> True + +genNormalisedVar = do + eqs' <- genNormalisedEqs + case sigFromEqs eqs' of + Sig _ [] -> scale (+1) genNormalisedVar + Sig _ (v:_) -> return v + +genNormalisedConst = do + eqs' <- genNormalisedEqs + case sigFromEqs eqs' of + Sig [] _ -> scale (+1) genNormalisedConst + Sig (c:_) _ -> return c + +genNormalisedEqs = do + eqs <- arbitrary + let (_, eqs') = replaceTypes eqs + return eqs' + +genNormalisedSig = sigFromEqs <$> genNormalisedEqs + +setEq :: (Foldable t1, Foldable t2, Eq a) => t1 a -> t2 a -> Bool +setEq = setEqBy (==) + +setEqBy :: (Foldable t1, Foldable t2) => (a -> b -> Bool) -> t1 a -> t2 b -> Bool +setEqBy f xs ys = all (\y -> any (`f` y) xs) ys && + all (\x -> any (x `f`) ys) xs + +setDiff :: Eq a => [a] -> [a] -> ([a], [a]) +setDiff = setDiffBy (==) + +setDiffBy :: (a -> b -> Bool) + -> [a] + -> [b] + -> ([a], [b]) +setDiffBy f xs ys = (xs', ys') + where xs' = filter (\x -> not (any (x `f`) ys)) xs + ys' = filter (\y -> not (any (`f` y) xs)) ys + +-- Data generators + +-- Example input from files + +exampleEqs :: [[Equation]] +exampleEqs = map (fromJust . decode) exampleJson + +exampleJson :: [LB.ByteString] +{-# NOINLINE exampleJson #-} +exampleJson = unsafePerformIO $ exampleFiles >>= mapM LB.readFile + +exampleDir = "test/data" + +exampleFiles = do + fs <- getDirectoryContents exampleDir + return (prefix (json fs)) + where prefix = map ((exampleDir ++ "/") ++) + json = filter isJson + isJson :: String -> Bool + isJson x = reverse ".json" == take 5 (reverse x) + +withExamples = forAll (elements exampleEqs) + +-- Random input generators + +newtype Equations = Eqs [Equation] deriving (Show, Eq) + +instance Arbitrary Equations where + shrink (Eqs []) = [] + shrink (Eqs eqs) = [Eqs eqs' | eqs' <- shrink eqs, consistentEqs eqs'] + + arbitrary = do + -- Make a bunch of terms and declare them equal + classEqs <- scale (\s -> min 5 (s `div` 2)) (listOf mkClass) + let eqs = concat classEqs + + -- Pad out with filler + pre <- scale (\s -> min 5 (s `div` 2)) arbitrary + post <- scale (\s -> min 5 (s `div` 2)) arbitrary + + -- Make sure it's consistent + eqs' <- renameEqs (pre ++ eqs ++ post) + return (Eqs eqs') + where mkClass = do + t <- arbitrary + ts <- scale (`div` 2) (listOf (termOfType t)) + return $ case ts of + [] -> [] + (x:xs) -> [Eq x y | y <- xs] + +-- | Keep renaming constants until each name only refers to one type +renameEqs eqs = if consistentEqs eqs + then return eqs + else mapM renameEq eqs >>= renameEqs + where renameEq (Eq l r) = Eq <$> renameTerm l <*> renameTerm r + renameTerm (C (Const a _ t)) = C <$> (Const a <$> arbitrary <*> pure t) + renameTerm (V v) = pure (V v) + renameTerm (App l r _) = app <$> renameTerm l <*> renameTerm r + +consistentEqs eqs = let nts = concatMap eqNames eqs + in consistentNames nts + +instance Arbitrary Equation where + arbitrary = do t <- arbitrary + l <- termOfType t + r <- termOfType t + if trivial l && trivial r || l == r || allVar l || allVar r + then discard + else return $ Eq l r + + shrink (Eq l r) = [Eq l' r' | (l', r') <- shrink (l, r), + not (trivial l' && trivial r'), + not (allVar l'), + not (allVar r'), + termType' l' == termType' r'] + +-- | Generate a "non-trivial" Term, i.e. containing at least one App, one Const +-- and one Var, with the given Type +termOfType :: Type -> Gen Term +termOfType t = do + -- Generate a random Term of the correct type, of the form `app l r` + term <- appOfTypeArity 0 t + -- Force one branch to contain a Var and the other to contain a Const + giveVarConst term + +-- | Force one branch of an `App` to contain a Const and the other to contain a +-- Var +giveVarConst (App l r _) = do + which <- arbitrary + l' <- if which then giveVar l else giveConst l + r' <- if not which then giveVar r else giveConst r + return (app l' r') + +-- | Randomly replace part of the given Term with a Var +giveVar :: Term -> Gen Term + +-- If the Term is already a Var, there is no work to do +giveVar (V v) = return (V v) + +-- Turn Consts into Vars. Since Consts may have higher arity (up to 5) than Vars +-- (up to 2), we rely on the App cases to prevent hitting high-arity Terms +giveVar (C c) = V <$> varOfTypeArity (constArity c) (constType c) + +-- Don't recurse any further if we reach an arity of 2, since we might reach a +-- sub-Term with an arity above 2, which cannot be turned into a Var +giveVar t | termArity t >= 2 = V <$> varOfTypeArity (termArity t) + (termType' t) + +-- If it's safe to recurse, choose a branch at random to force a Var into +giveVar (App l r _) = do + which <- arbitrary + l' <- if which then giveVar l else return l + r' <- if not which then giveVar r else return r + return (app l' r') + +-- | Randomly replace part of a Term with a Const +giveConst :: Term -> Gen Term + +-- Constants can just be returned as-is +giveConst (C c) = return (C c) + +-- Variable arity should always be lower than 3, which a Const should have no +-- problem with +giveConst (V v) = C <$> constOfTypeArity (varArity v) (varType v) + +-- Don't recurse into Terms with arity 5, since we might hit a Term with a +-- higher arity which we can't replace with a Const +giveConst t | termArity t >= 5 = C <$> constOfTypeArity (termArity t) + (termType' t) + +-- It's safe to recurse into low-arity Apps. We pick a branch randomly to put a +-- Const into +giveConst (App l r _) = do + which <- arbitrary + l' <- if which then giveConst l else return l + r' <- if not which then giveConst r else return r + return (app l' r') + +constOfTypeArity :: Arity -> Type -> Gen Const +constOfTypeArity a t = if a > 5 + then error ("Can't make Const of type " ++ show t ++ " and arity " ++ show a) + else Const a <$> arbitrary <*> pure t + +varOfTypeArity :: Arity -> Type -> Gen Var +varOfTypeArity a t = if a > 2 + then error ("Can't make Var of type " ++ show t ++ " and arity " ++ show a) + else Var t <$> choose (0, 4) <*> pure a + +appOfTypeArity :: Arity -> Type -> Gen Term +appOfTypeArity a t = do + arg <- arbitrary + r <- termOfTypeArity 0 arg + l <- termOfTypeArity (a+1) (tyFun arg t) + return $ app l r + +termOfTypeArity :: Arity -> Type -> Gen Term +termOfTypeArity a t = oneof (mkConst ++ mkVar ++ mkApp) + where -- We can only generate constants up to arity 5 + mkConst = if a > 5 + then error ("Can't gen Term of arity " ++ show a) + else [C <$> constOfTypeArity a t] + + -- We can only generate variables up to arity 2 + mkVar = if a > 2 + then [] + else [V <$> varOfTypeArity a t] + + mkApp = if a > 4 + then [] + else [appOfTypeArity a t] + +-- "Trivial" equations will be pruned, so we need to avoid generating (or +-- shrinking down to) equations where both sides only have one symbol, or which +-- don't contain any variables +trivial (C _) = True +trivial (V _) = True +trivial x | not (hasVar x) = True +trivial x | not (hasConst x) = True +trivial x = False + +hasConst (V _) = False +hasConst (C _) = True +hasConst (App l r _) = hasConst l || hasConst r + +hasVar (V _) = True +hasVar (C _) = False +hasVar (App l r _) = hasVar l || hasVar r + +allVar (V _) = True +allVar (C _) = False +allVar (App l r _) = allVar l && allVar r + +-- | Make sure no name is used for constants of two types +termNames :: Term -> [(Name, Type)] +termNames (C (Const _ n t)) = [(n, t)] +termNames (V _) = [] +termNames (App l r _) = nub (termNames l ++ termNames r) + +eqNames :: Equation -> [(Name, Type)] +eqNames (Eq l r) = termNames l ++ termNames r + +consistentNames nts = all hasOneType names + where names = map fst nts + hasOneType n = length (typesOf n) == 1 + typesOf n = nub (map snd (entriesOf n)) + entriesOf n = filter ((== n) . fst) nts + +instance Arbitrary Term where + arbitrary = do + t <- arbitrary + termOfType t + + shrink (C c) = C <$> shrink c + shrink (V v) = V <$> shrink v + shrink t@(App l r _) = C (Const (termArity t) (termName t) (termType' t)) : + [app l' r' | (l', r') <- shrink (l, r)] + +termName (C c) = constName c +termName (V v) = Name (filter isAlpha (show (varType v) ++ show (varArity v))) +termName (App l r _) = let Name l' = termName l + Name r' = termName r + in Name ("app" ++ l' ++ r') + +instance Arbitrary Var where + arbitrary = sized $ \n -> do + arity <- elements [0, 1, 2] + typ <- naryType arity n + index <- elements [0, 1, 2] + return $ Var typ index (Arity arity) + + shrink (Var t i a) = if i == 0 + then [] + else [Var t 0 a] + +instance Arbitrary Const where + arbitrary = sized $ \n -> do + arity <- elements [0..5] + name <- arbitrary + typ <- naryType arity n + return $ Const (Arity arity) name typ + + shrink (Const a n t) = do n' <- shrink n + return (Const a n' t) + +instance Arbitrary Sig where + arbitrary = Sig <$> listOf arbitrary <*> listOf arbitrary + shrink (Sig [] []) = [] + shrink (Sig cs vs) = Sig [] [] : [Sig cs' vs' | (cs', vs') <- shrink (cs, vs)] + +instance Arbitrary Type where + arbitrary = sized sizedType + +instance Arbitrary Name where + arbitrary = Name <$> listOf1 (arbitrary `suchThat` isAlpha `suchThat` isAscii) `suchThat` valid + where valid s = let t = S.fromString s + b = S.fromString s + in s == S.toString b && + s == S.toString t && + b == S.fromString s && + b == encodeUtf8 t && + t == S.fromString s && + t == decodeUtf8 b + + shrink (Name x) = let suffices = tail (tails x) + nonEmpty = filter (not . null) suffices + names = map Name nonEmpty + in reverse names -- Try shortest first + +sizedType :: Int -> Gen Type +sizedType 0 = elements [tyCon "Int", tyCon "Bool", tyCon "Float"] +sizedType n = oneof [ + sizedType 0, + do x <- sizedType (n - 1) + return $ tyCon ("[" ++ typeName x ++ "]"), + do n' <- choose (0, n - 1) + l <- sizedType n' + r <- sizedType (n - n') + return $ tyCon ("(" ++ typeName l ++ ", " ++ typeName r ++ ")") + ] + +naryType 0 n = sizedType n +naryType a n = do + arg <- sizedType n + ret <- naryType (a-1) n + return $ tyFun arg ret + +dbg :: (Show a, Monad m) => a -> PropertyM m () +dbg = monitor . counterexample . show + +doOnce :: (Show a, Arbitrary a, Testable prop) => (a -> prop) -> Property +doOnce = once . forAll (resize 42 arbitrary) + +-- | The list of all terms equal to `x`, according to `eqs` +eqClosure :: [Equation] -> Term -> Seq.Seq Term +eqClosure eqs x = indirect eqs Seq.empty (directEq eqs x) + +indirect :: [Equation] -> Seq.Seq Term -> Seq.Seq Term -> Seq.Seq Term +indirect eqs seen xs | null xs = seen +indirect eqs seen xs = indirect eqs (nub' $ seen Seq.>< unseen) unseen + where new = xs >>= directEq eqs + unseen = nub' $ Seq.filter notSeen new + notSeen a = not (a `isElem` seen) + +nub' = foldl f Seq.empty + where f acc x = if x `isElem` acc + then acc + else x Seq.<| acc + +isElem x xs = isJust (Seq.elemIndexL x xs) + +-- | The list of terms equal to `x` by definition, according to `eqs` +directEq :: [Equation] -> Term -> Seq.Seq Term +directEq eqs x = x Seq.<| Seq.filter direct terms + where terms = Seq.fromList . nub . concatMap termsOf $ eqs + termsOf (Eq a b) = [a, b] + direct a = Eq x a `elem` eqs || Eq a x `elem` eqs + +app l r = case termType l of + Just (HSE.Syntax.TyFun _ _ o) -> App l r (Just o) + _ -> x + where [Eq x _] = setAllTypes [Eq (App l r Nothing) + (App l r Nothing)] + +iterable ty = do t <- termOfType ty + v <- termOfType (tyFun ty ty) + return (t, v) + +tyFun = HSE.Syntax.TyFun () + +instance Show Test.QuickSpec.Reasoning.NaiveEquationalReasoning.Context where + show cxt = concat ["(context\n (universe ", + show (Test.QuickSpec.Reasoning.NaiveEquationalReasoning.universe cxt), + ")\n (maxDepth ", + show (Test.QuickSpec.Reasoning.NaiveEquationalReasoning.maxDepth cxt), + ")\n (rel ", + show (Test.QuickSpec.Reasoning.NaiveEquationalReasoning.rel cxt), + ")\n)"] + +instance Show Test.QuickSpec.Reasoning.CongruenceClosure.S where + show s = concat ["(S\n (funUse ", + show (Test.QuickSpec.Reasoning.CongruenceClosure.funUse s), + ")\n (argUse ", + show (Test.QuickSpec.Reasoning.CongruenceClosure.argUse s), + ")\n (lookup ", + show (Test.QuickSpec.Reasoning.CongruenceClosure.lookup s) + ] + +dbgEqs = map (Test.QuickSpec.Utils.Typed.some + (Test.QuickSpec.Equation.showTypedEquation natSig)) + +rawNatEqs = unsafePerformIO (LB.readFile "test/data/nat-simple-raw.json") + +Right parsedNatEqs = eitherDecode rawNatEqs :: Either String [Equation] + +(typeDb, parsedNatEqs') = replaceTypes parsedNatEqs + +doReps clss = map (Test.QuickSpec.Utils.Typed.some2 + (Test.QuickSpec.Utils.Typed.tagged term . head)) + clss + +natClasses = map (Test.QuickSpec.Utils.Typed.several + (map Test.QuickSpec.Term.term)) + rawNatClasses + + +rawNatClasses' = swapTypes' naturalDb rawNatClasses + +swapTypes' db = map (Test.QuickSpec.Utils.Typed.several + (\xs -> let trep = head (typeRepArgs (typeRep xs)) -- Strip off [] and Expr + typ = case HSE.Parser.parseType (show trep) of + HSE.Parser.ParseOk x -> unwrapParens (fmap (const ()) x) + typ' = replaceInType db typ + in case getVal (getRep typ') of + MkHT x -> Test.QuickSpec.Utils.Typed.Some + (Test.QuickSpec.Utils.Typed.O + (map (\e -> let trm = replaceQSType db (Test.QuickSpec.Term.term e) + in e { + Test.QuickSpec.Term.term = trm, + Test.QuickSpec.Term.eval = \env -> case trm of + Test.QuickSpec.Term.Var s -> Test.QuickSpec.Term.unValuation env (Test.QuickSpec.Term.Variable (Test.QuickSpec.Term.Atom s (Test.QuickSpec.Term.pgen (return x)))) + Test.QuickSpec.Term.Const s -> x + Test.QuickSpec.Term.App l r -> x + }) + xs)))) + +rawNatClasses :: [Test.QuickSpec.Utils.Typed.Several Test.QuickSpec.Term.Expr] +rawNatClasses = concatMap (Test.QuickSpec.Utils.Typed.some2 + (map (Test.QuickSpec.Utils.Typed.Some . + Test.QuickSpec.Utils.Typed.O) . + Test.QuickSpec.TestTree.classes)) + (Test.QuickSpec.Utils.TypeMap.toList r) + where r = unsafePerformIO $ Test.QuickSpec.Generate.generate + False + (const Test.QuickSpec.Term.partialGen) + natSig + +tType (Test.QuickSpec.Term.Var s) = Test.QuickSpec.Utils.Typeable.unTypeRep (Test.QuickSpec.Term.symbolType s) +tType (Test.QuickSpec.Term.Const s) = Test.QuickSpec.Utils.Typeable.unTypeRep (Test.QuickSpec.Term.symbolType s) +tType (Test.QuickSpec.Term.App l r) = case funResultTy (tType l) (tType r) of + Nothing -> error ("Incompatible types (" ++ show (tType l) ++ ") (" ++ show (tType r) ++ ")") + Just t -> t + +replaceQSTypes :: [(Type, Type)] -> [[Test.QuickSpec.Term.Term]] -> [[Test.QuickSpec.Term.Term]] +replaceQSTypes db = map rep + where rep = map (replaceQSType db) + +replaceQSType db t = + case t of + Test.QuickSpec.Term.Var s -> Test.QuickSpec.Term.Var ( + let ty = prnt (Test.QuickSpec.Utils.Typeable.unTypeRep + (Test.QuickSpec.Term.symbolType s)) + tr = repToQSRep (getRep ty) + n = let suf = reverse + (takeWhile (/= ',') + (reverse (Test.QuickSpec.Term.name s))) + idx = read (init suf) :: Int + in unName (varName (Var + ty + idx + (Arity (Test.QuickSpec.Term.symbolArity s)))) + in s { + Test.QuickSpec.Term.symbolType = tr, + Test.QuickSpec.Term.name = n + }) + Test.QuickSpec.Term.Const s -> Test.QuickSpec.Term.Const (s { + Test.QuickSpec.Term.symbolType = + repToQSRep + (getRep + (prnt (Test.QuickSpec.Utils.Typeable.unTypeRep + (Test.QuickSpec.Term.symbolType s)))) + }) + Test.QuickSpec.Term.App l r -> Test.QuickSpec.Term.App + (replaceQSType db l) + (replaceQSType db r) + where prs x = case HSE.Parser.parseType (show x) of + HSE.Parser.ParseOk typ -> unwrapParens (fmap (const ()) typ) + HSE.Parser.ParseFailed _ e -> error (concat [ + "Failed to replace QuickSpec type '", + show t, + "'. Error is: ", + e]) + prnt = replaceInType db . prs + +naturalDb = [(tyCon "Natural", tyCon "Z")] diff --git a/test/data/list-extras.json b/test/data/list-extras.json new file mode 100644 --- /dev/null +++ b/test/data/list-extras.json @@ -0,0 +1,92 @@ +[ + { + "relation": "~=", + "lhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "[Integer] -> [Integer] -> Ordering", + "symbol": "lengthCompare" + }, + "rhs": { + "role": "variable", + "type": "[Integer]", + "id": 7 + } + }, + "rhs": { + "role": "variable", + "type": "[Integer]", + "id": 7 + } + }, + "rhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "[Integer] -> [Integer] -> Ordering", + "symbol": "lengthCompare" + }, + "rhs": { + "role": "variable", + "type": "[Integer]", + "id": 6 + } + }, + "rhs": { + "role": "variable", + "type": "[Integer]", + "id": 6 + } + } + }, + { + "relation": "~=", + "lhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "[Integer] -> [Integer] -> Ordering", + "symbol": "lengthCompare" + }, + "rhs": { + "role": "variable", + "type": "[Integer]", + "id": 8 + } + }, + "rhs": { + "role": "variable", + "type": "[Integer]", + "id": 8 + } + }, + "rhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "[Integer] -> [Integer] -> Ordering", + "symbol": "lengthCompare" + }, + "rhs": { + "role": "variable", + "type": "[Integer]", + "id": 6 + } + }, + "rhs": { + "role": "variable", + "type": "[Integer]", + "id": 6 + } + } + } +] diff --git a/test/data/nat-simple-expect.json b/test/data/nat-simple-expect.json new file mode 100644 --- /dev/null +++ b/test/data/nat-simple-expect.json @@ -0,0 +1,604 @@ +[ + { + "relation": "~=", + "lhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "Nat -> Nat -> Nat", + "symbol": "plus" + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 1 + } + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 0 + } + }, + "rhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "Nat -> Nat -> Nat", + "symbol": "plus" + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 0 + } + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 1 + } + } + }, + { + "relation": "~=", + "lhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "Nat -> Nat -> Nat", + "symbol": "times" + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 1 + } + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 0 + } + }, + "rhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "Nat -> Nat -> Nat", + "symbol": "times" + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 0 + } + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 1 + } + } + }, + { + "relation": "~=", + "lhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "Nat -> Nat -> Nat", + "symbol": "plus" + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 0 + } + }, + "rhs": { + "role": "constant", + "type": "Nat", + "symbol": "cZ" + } + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 0 + } + }, + { + "relation": "~=", + "lhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "Nat -> Nat -> Nat", + "symbol": "times" + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 0 + } + }, + "rhs": { + "role": "constant", + "type": "Nat", + "symbol": "cZ" + } + }, + "rhs": { + "role": "constant", + "type": "Nat", + "symbol": "cZ" + } + }, + { + "relation": "~=", + "lhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "Nat -> Nat -> Nat", + "symbol": "plus" + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 1 + } + }, + "rhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "Nat -> Nat -> Nat", + "symbol": "plus" + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 0 + } + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 2 + } + } + }, + "rhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "Nat -> Nat -> Nat", + "symbol": "plus" + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 0 + } + }, + "rhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "Nat -> Nat -> Nat", + "symbol": "plus" + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 1 + } + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 2 + } + } + } + }, + { + "relation": "~=", + "lhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "Nat -> Nat -> Nat", + "symbol": "times" + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 1 + } + }, + "rhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "Nat -> Nat -> Nat", + "symbol": "times" + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 0 + } + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 2 + } + } + }, + "rhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "Nat -> Nat -> Nat", + "symbol": "times" + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 0 + } + }, + "rhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "Nat -> Nat -> Nat", + "symbol": "times" + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 1 + } + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 2 + } + } + } + }, + { + "relation": "~=", + "lhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "Nat -> Nat -> Nat", + "symbol": "plus" + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 0 + } + }, + "rhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "Nat -> Nat", + "symbol": "cS" + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 1 + } + } + }, + "rhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "Nat -> Nat", + "symbol": "cS" + }, + "rhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "Nat -> Nat -> Nat", + "symbol": "plus" + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 0 + } + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 1 + } + } + } + }, + { + "relation": "~=", + "lhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "Nat -> Nat -> Nat", + "symbol": "times" + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 0 + } + }, + "rhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "Nat -> Nat", + "symbol": "cS" + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 1 + } + } + }, + "rhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "Nat -> Nat -> Nat", + "symbol": "plus" + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 0 + } + }, + "rhs": { + "role": "application", + "lhs": { + "role": "application", + "lhs": { + "role": "constant", + "type": "Nat -> Nat -> Nat", + "symbol": "times" + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 0 + } + }, + "rhs": { + "role": "variable", + "type": "Nat", + "id": 1 + } + } + } + }, + { + "relation": "~=", + "lhs": { + "role": "application", + "lhs": { + "role": 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