heyting-algebras (empty) → 0.0.1.0
raw patch · 10 files changed
+1275/−0 lines, 10 filesdep +QuickCheckdep +basedep +containerssetup-changed
Dependencies added: QuickCheck, base, containers, free-algebras, hashable, heyting-algebras, lattices, tagged, tasty, tasty-quickcheck, universe-base, unordered-containers
Files
- ChangeLog.md +5/−0
- LICENSE +373/−0
- Setup.hs +2/−0
- heyting-algebras.cabal +72/−0
- src/Algebra/Boolean.hs +153/−0
- src/Algebra/Boolean/Free.hs +72/−0
- src/Algebra/Heyting.hs +257/−0
- src/Algebra/Heyting/Free.hs +76/−0
- src/Algebra/Heyting/Layered.hs +68/−0
- test/Main.hs +197/−0
+ ChangeLog.md view
@@ -0,0 +1,5 @@+# Revision history for heyting-algebra++## 0.0.1.0 -- YYYY-mm-dd++* First version. Released on an unsuspecting world.
+ LICENSE view
@@ -0,0 +1,373 @@+Mozilla Public License Version 2.0+==================================++1. Definitions+--------------++1.1. "Contributor"+ means each individual or legal entity that creates, contributes to+ the creation of, or owns Covered Software.++1.2. "Contributor Version"+ means the combination of the Contributions of others (if any) used+ by a Contributor and that particular Contributor's Contribution.++1.3. "Contribution"+ means Covered Software of a particular Contributor.++1.4. "Covered Software"+ means Source Code Form to which the initial Contributor has attached+ the notice in Exhibit A, the Executable Form of such Source Code+ Form, and Modifications of such Source Code Form, in each case+ including portions thereof.++1.5. "Incompatible With Secondary Licenses"+ means++ (a) that the initial Contributor has attached the notice described+ in Exhibit B to the Covered Software; or++ (b) that the Covered Software was made available under the terms of+ version 1.1 or earlier of the License, but not also under the+ terms of a Secondary License.++1.6. "Executable Form"+ means any form of the work other than Source Code Form.++1.7. "Larger Work"+ means a work that combines Covered Software with other material, in+ a separate file or files, that is not Covered Software.++1.8. "License"+ means this document.++1.9. "Licensable"+ means having the right to grant, to the maximum extent possible,+ whether at the time of the initial grant or subsequently, any and+ all of the rights conveyed by this License.++1.10. "Modifications"+ means any of the following:++ (a) any file in Source Code Form that results from an addition to,+ deletion from, or modification of the contents of Covered+ Software; or++ (b) any new file in Source Code Form that contains any Covered+ Software.++1.11. "Patent Claims" of a Contributor+ means any patent claim(s), including without limitation, method,+ process, and apparatus claims, in any patent Licensable by such+ Contributor that would be infringed, but for the grant of the+ License, by the making, using, selling, offering for sale, having+ made, import, or transfer of either its Contributions or its+ Contributor Version.++1.12. "Secondary License"+ means either the GNU General Public License, Version 2.0, the GNU+ Lesser General Public License, Version 2.1, the GNU Affero General+ Public License, Version 3.0, or any later versions of those+ licenses.++1.13. "Source Code Form"+ means the form of the work preferred for making modifications.++1.14. "You" (or "Your")+ means an individual or a legal entity exercising rights under this+ License. For legal entities, "You" includes any entity that+ controls, is controlled by, or is under common control with You. For+ purposes of this definition, "control" means (a) the power, direct+ or indirect, to cause the direction or management of such entity,+ whether by contract or otherwise, or (b) ownership of more than+ fifty percent (50%) of the outstanding shares or beneficial+ ownership of such entity.++2. License Grants and Conditions+--------------------------------++2.1. Grants++Each Contributor hereby grants You a world-wide, royalty-free,+non-exclusive license:++(a) under intellectual property rights (other than patent or trademark)+ Licensable by such Contributor to use, reproduce, make available,+ modify, display, perform, distribute, and otherwise exploit its+ Contributions, either on an unmodified basis, with Modifications, or+ as part of a Larger Work; and++(b) under Patent Claims of such Contributor to make, use, sell, offer+ for sale, have made, import, and otherwise transfer either its+ Contributions or its Contributor Version.++2.2. Effective Date++The licenses granted in Section 2.1 with respect to any Contribution+become effective for each Contribution on the date the Contributor first+distributes such Contribution.++2.3. Limitations on Grant Scope++The licenses granted in this Section 2 are the only rights granted under+this License. No additional rights or licenses will be implied from the+distribution or licensing of Covered Software under this License.+Notwithstanding Section 2.1(b) above, no patent license is granted by a+Contributor:++(a) for any code that a Contributor has removed from Covered Software;+ or++(b) for infringements caused by: (i) Your and any other third party's+ modifications of Covered Software, or (ii) the combination of its+ Contributions with other software (except as part of its Contributor+ Version); or++(c) under Patent Claims infringed by Covered Software in the absence of+ its Contributions.++This License does not grant any rights in the trademarks, service marks,+or logos of any Contributor (except as may be necessary to comply with+the notice requirements in Section 3.4).++2.4. Subsequent Licenses++No Contributor makes additional grants as a result of Your choice to+distribute the Covered Software under a subsequent version of this+License (see Section 10.2) or under the terms of a Secondary License (if+permitted under the terms of Section 3.3).++2.5. Representation++Each Contributor represents that the Contributor believes its+Contributions are its original creation(s) or it has sufficient rights+to grant the rights to its Contributions conveyed by this License.++2.6. Fair Use++This License is not intended to limit any rights You have under+applicable copyright doctrines of fair use, fair dealing, or other+equivalents.++2.7. Conditions++Sections 3.1, 3.2, 3.3, and 3.4 are conditions of the licenses granted+in Section 2.1.++3. Responsibilities+-------------------++3.1. Distribution of Source Form++All distribution of Covered Software in Source Code Form, including any+Modifications that You create or to which You contribute, must be under+the terms of this License. You must inform recipients that the Source+Code Form of the Covered Software is governed by the terms of this+License, and how they can obtain a copy of this License. You may not+attempt to alter or restrict the recipients' rights in the Source Code+Form.++3.2. Distribution of Executable Form++If You distribute Covered Software in Executable Form then:++(a) such Covered Software must also be made available in Source Code+ Form, as described in Section 3.1, and You must inform recipients of+ the Executable Form how they can obtain a copy of such Source Code+ Form by reasonable means in a timely manner, at a charge no more+ than the cost of distribution to the recipient; and++(b) You may distribute such Executable Form under the terms of this+ License, or sublicense it under different terms, provided that the+ license for the Executable Form does not attempt to limit or alter+ the recipients' rights in the Source Code Form under this License.++3.3. Distribution of a Larger Work++You may create and distribute a Larger Work under terms of Your choice,+provided that You also comply with the requirements of this License for+the Covered Software. If the Larger Work is a combination of Covered+Software with a work governed by one or more Secondary Licenses, and the+Covered Software is not Incompatible With Secondary Licenses, this+License permits You to additionally distribute such Covered Software+under the terms of such Secondary License(s), so that the recipient of+the Larger Work may, at their option, further distribute the Covered+Software under the terms of either this License or such Secondary+License(s).++3.4. Notices++You may not remove or alter the substance of any license notices+(including copyright notices, patent notices, disclaimers of warranty,+or limitations of liability) contained within the Source Code Form of+the Covered Software, except that You may alter any license notices to+the extent required to remedy known factual inaccuracies.++3.5. Application of Additional Terms++You may choose to offer, and to charge a fee for, warranty, support,+indemnity or liability obligations to one or more recipients of Covered+Software. However, You may do so only on Your own behalf, and not on+behalf of any Contributor. You must make it absolutely clear that any+such warranty, support, indemnity, or liability obligation is offered by+You alone, and You hereby agree to indemnify every Contributor for any+liability incurred by such Contributor as a result of warranty, support,+indemnity or liability terms You offer. You may include additional+disclaimers of warranty and limitations of liability specific to any+jurisdiction.++4. Inability to Comply Due to Statute or Regulation+---------------------------------------------------++If it is impossible for You to comply with any of the terms of this+License with respect to some or all of the Covered Software due to+statute, judicial order, or regulation then You must: (a) comply with+the terms of this License to the maximum extent possible; and (b)+describe the limitations and the code they affect. Such description must+be placed in a text file included with all distributions of the Covered+Software under this License. Except to the extent prohibited by statute+or regulation, such description must be sufficiently detailed for a+recipient of ordinary skill to be able to understand it.++5. Termination+--------------++5.1. The rights granted under this License will terminate automatically+if You fail to comply with any of its terms. However, if You become+compliant, then the rights granted under this License from a particular+Contributor are reinstated (a) provisionally, unless and until such+Contributor explicitly and finally terminates Your grants, and (b) on an+ongoing basis, if such Contributor fails to notify You of the+non-compliance by some reasonable means prior to 60 days after You have+come back into compliance. Moreover, Your grants from a particular+Contributor are reinstated on an ongoing basis if such Contributor+notifies You of the non-compliance by some reasonable means, this is the+first time You have received notice of non-compliance with this License+from such Contributor, and You become compliant prior to 30 days after+Your receipt of the notice.++5.2. If You initiate litigation against any entity by asserting a patent+infringement claim (excluding declaratory judgment actions,+counter-claims, and cross-claims) alleging that a Contributor Version+directly or indirectly infringes any patent, then the rights granted to+You by any and all Contributors for the Covered Software under Section+2.1 of this License shall terminate.++5.3. In the event of termination under Sections 5.1 or 5.2 above, all+end user license agreements (excluding distributors and resellers) which+have been validly granted by You or Your distributors under this License+prior to termination shall survive termination.++************************************************************************+* *+* 6. Disclaimer of Warranty *+* ------------------------- *+* *+* Covered Software is provided under this License on an "as is" *+* basis, without warranty of any kind, either expressed, implied, or *+* statutory, including, without limitation, warranties that the *+* Covered Software is free of defects, merchantable, fit for a *+* particular purpose or non-infringing. The entire risk as to the *+* quality and performance of the Covered Software is with You. *+* Should any Covered Software prove defective in any respect, You *+* (not any Contributor) assume the cost of any necessary servicing, *+* repair, or correction. This disclaimer of warranty constitutes an *+* essential part of this License. No use of any Covered Software is *+* authorized under this License except under this disclaimer. *+* *+************************************************************************++************************************************************************+* *+* 7. Limitation of Liability *+* -------------------------- *+* *+* Under no circumstances and under no legal theory, whether tort *+* (including negligence), contract, or otherwise, shall any *+* Contributor, or anyone who distributes Covered Software as *+* permitted above, be liable to You for any direct, indirect, *+* special, incidental, or consequential damages of any character *+* including, without limitation, damages for lost profits, loss of *+* goodwill, work stoppage, computer failure or malfunction, or any *+* and all other commercial damages or losses, even if such party *+* shall have been informed of the possibility of such damages. This *+* limitation of liability shall not apply to liability for death or *+* personal injury resulting from such party's negligence to the *+* extent applicable law prohibits such limitation. Some *+* jurisdictions do not allow the exclusion or limitation of *+* incidental or consequential damages, so this exclusion and *+* limitation may not apply to You. *+* *+************************************************************************++8. Litigation+-------------++Any litigation relating to this License may be brought only in the+courts of a jurisdiction where the defendant maintains its principal+place of business and such litigation shall be governed by laws of that+jurisdiction, without reference to its conflict-of-law provisions.+Nothing in this Section shall prevent a party's ability to bring+cross-claims or counter-claims.++9. Miscellaneous+----------------++This License represents the complete agreement concerning the subject+matter hereof. If any provision of this License is held to be+unenforceable, such provision shall be reformed only to the extent+necessary to make it enforceable. Any law or regulation which provides+that the language of a contract shall be construed against the drafter+shall not be used to construe this License against a Contributor.++10. Versions of the License+---------------------------++10.1. New Versions++Mozilla Foundation is the license steward. Except as provided in Section+10.3, no one other than the license steward has the right to modify or+publish new versions of this License. Each version will be given a+distinguishing version number.++10.2. Effect of New Versions++You may distribute the Covered Software under the terms of the version+of the License under which You originally received the Covered Software,+or under the terms of any subsequent version published by the license+steward.++10.3. Modified Versions++If you create software not governed by this License, and you want to+create a new license for such software, you may create and use a+modified version of this License if you rename the license and remove+any references to the name of the license steward (except to note that+such modified license differs from this License).++10.4. Distributing Source Code Form that is Incompatible With Secondary+Licenses++If You choose to distribute Source Code Form that is Incompatible With+Secondary Licenses under the terms of this version of the License, the+notice described in Exhibit B of this License must be attached.++Exhibit A - Source Code Form License Notice+-------------------------------------------++ This Source Code Form is subject to the terms of the Mozilla Public+ License, v. 2.0. If a copy of the MPL was not distributed with this+ file, You can obtain one at http://mozilla.org/MPL/2.0/.++If it is not possible or desirable to put the notice in a particular+file, then You may include the notice in a location (such as a LICENSE+file in a relevant directory) where a recipient would be likely to look+for such a notice.++You may add additional accurate notices of copyright ownership.++Exhibit B - "Incompatible With Secondary Licenses" Notice+---------------------------------------------------------++ This Source Code Form is "Incompatible With Secondary Licenses", as+ defined by the Mozilla Public License, v. 2.0.
+ Setup.hs view
@@ -0,0 +1,2 @@+import Distribution.Simple+main = defaultMain
+ heyting-algebras.cabal view
@@ -0,0 +1,72 @@+name: heyting-algebras+version: 0.0.1.0+synopsis: Heyting and Boolean algebras+description:+ This package provides Heyting and Boolean operations together+ with various constructions of Heyting algebras.+license: MPL-2.0+license-file: LICENSE+author: Marcin Szamotulski+maintainer: profunctor@pm.me+copyright: (c) 2018 Marcin Szamotulski+category: Math+build-type: Simple+extra-source-files: ChangeLog.md+cabal-version: >=1.10+tested-with: GHC==8.2.2, GHC==8.4.3++flag export-properties+ description:+ Export quickcheck properties from library; this adds QuickCheck+ as a dependency.+ manual: True+ default: False++library+ exposed-modules: Algebra.Heyting+ Algebra.Heyting.Free+ Algebra.Heyting.Layered+ Algebra.Boolean+ Algebra.Boolean.Free+ -- other-modules:+ -- other-extensions:+ build-depends: base >= 4.9 && < 4.13+ , containers >= 0.4.2 && < 0.7+ , free-algebras >= 0.0.4 && < 0.0.6 + , hashable >= 1.2.6.1 && < 1.3+ , lattices >= 1.0 && < 1.11+ , tagged >= 0.8.5 && < 0.9+ , unordered-containers+ >= 0.2.6.0 && < 0.3+ , universe-base >= 1.0 && < 1.1+ hs-source-dirs: src+ default-language: Haskell2010+ default-extensions: FlexibleInstances+ RankNTypes+ ghc-options: -Wall+ if flag(export-properties)+ build-depends:+ QuickCheck >= 2.10 && < 2.13+ cpp-options:+ -DEXPORT_PROPERTIES++test-suite heyting-algebras-test+ type: exitcode-stdio-1.0+ hs-source-dirs: test+ main-is: Main.hs+ build-depends: base+ , containers+ , lattices+ , universe-base+ , QuickCheck+ , heyting-algebras+ , tasty+ , tasty-quickcheck+ default-language: Haskell2010+ default-extensions: FlexibleInstances+ , TypeApplications++source-repository head+ type: git+ location: https://github.com/coot/heyting-algebras+
+ src/Algebra/Boolean.hs view
@@ -0,0 +1,153 @@+{-# LANGUAGE CPP #-}+{-# LANGUAGE DeriveDataTypeable #-}+{-# LANGUAGE DeriveGeneric #-}+{-# LANGUAGE GeneralizedNewtypeDeriving #-}+module Algebra.Boolean+ ( BooleanAlgebra+ , (==>)+ , not+ , iff+ , iff'+ -- * Adjunction between Boolean and Heyting algebras+ , Boolean+ , runBoolean+ , boolean++ -- * Properties+ -- $properties+ , prop_not + , prop_BooleanAlgebra+ ) where++import Prelude hiding (not)++import Control.Applicative (Const (..))+import Data.Data (Data, Typeable)+import Data.Functor.Identity (Identity (..))+import Data.Proxy (Proxy (..))+import Data.Semigroup (All (..), Any (..), Endo (..))+import Data.Tagged (Tagged (..))+import Data.Universe.Class (Finite)+import qualified Data.Set as S+import GHC.Generics (Generic)+#ifdef EXPORT_PROPERTIES+import Test.QuickCheck hiding ((==>))+#endif++import Algebra.Lattice ( Lattice+ , BoundedLattice+ , JoinSemiLattice (..)+ , BoundedJoinSemiLattice+ , MeetSemiLattice (..)+ , BoundedMeetSemiLattice+ , bottom+ , top+ )++import Algebra.Heyting ( HeytingAlgebra (..)+ , iff+ , iff'+ , not+ , toBoolean+ , prop_HeytingAlgebra+ )++-- |+-- Boolean algebra is a Heyting algebra which negation satisfies the law of+-- excluded middle, i.e. either of the following:+--+-- prop> not . not == not+--+-- or+--+-- prop> x ∨ not x == top+--+-- Another characterisation of Boolean algebras is as+-- [complemented](https://en.wikipedia.org/wiki/Complemented_lattice)+-- [distributive lattices](https://ncatlab.org/nlab/show/distributive+lattice)+-- where the complement satisfies the following three properties:+--+-- prop> (not a) ∧ a == bottom and (not a) ∨ a == top -- excluded middle law+-- prop> not (not a) == a -- involution law+-- prop> a ≤ b ⇒ not b ≤ not a -- order-reversing+class HeytingAlgebra a => BooleanAlgebra a++-- |+-- @'Boolean'@ is the left adjoint functor from the category of Heyting algebras+-- to the category of Boolean algebras; its right adjoint is the inclusion.+newtype Boolean a = Boolean+ { runBoolean :: a -- ^ extract value from @'Boolean'@+ }+ deriving+ ( JoinSemiLattice, BoundedJoinSemiLattice, MeetSemiLattice+ , BoundedMeetSemiLattice, Lattice, BoundedLattice, HeytingAlgebra+ , Eq, Ord, Read, Show, Bounded, Typeable, Data, Generic+ )++instance HeytingAlgebra a => BooleanAlgebra (Boolean a)++-- TODO: move to tests+instance (Arbitrary a, HeytingAlgebra a) => Arbitrary (Boolean a) where+ arbitrary = boolean <$> arbitrary+ shrink (Boolean a) = [ boolean a' | a' <- shrink a ]++-- |+-- Smart constructro of the @'Boolean'@ type.+boolean :: HeytingAlgebra a => a -> Boolean a+boolean = Boolean . toBoolean++--+-- Instances+--++instance BooleanAlgebra Bool++instance BooleanAlgebra All++instance BooleanAlgebra Any++instance BooleanAlgebra ()++instance BooleanAlgebra (Proxy a)++instance BooleanAlgebra a => BooleanAlgebra (Tagged t a)++instance BooleanAlgebra b => BooleanAlgebra (a -> b)++instance BooleanAlgebra a => BooleanAlgebra (Identity a)++instance BooleanAlgebra a => BooleanAlgebra (Const a b)++instance BooleanAlgebra a => BooleanAlgebra (Endo a)++instance (BooleanAlgebra a, BooleanAlgebra b) => BooleanAlgebra (a, b)++--+-- containers+--++instance (Ord a, Finite a) => BooleanAlgebra (S.Set a)++-- +-- $properties+--+-- /Properties are exported only if @export-properties@ cabal flag is defined./+#ifdef EXPORT_PROPERTIES++-- |+-- Test that @'not'@ satisfies Boolean algebra axioms.+prop_not :: (HeytingAlgebra a, Eq a, Show a) => a -> Property+prop_not a =+ counterexample "not (not a) /= a" (not (not a) === a)+ .&&. counterexample "not a ∧ a /= bottom" (not a /\ a === bottom)+ .&&. counterexample "not a ∨ a /= top" (not a \/ a === top)++-- |+-- Test that @a@ is satisfy both @'Algebra.Heyting.prop_HeytingAlgebra'@ and+-- @'prop_not'@.+prop_BooleanAlgebra :: (BooleanAlgebra a, Eq a, Show a)+ => a -> a -> a -> Property+prop_BooleanAlgebra a b c =+ prop_HeytingAlgebra a b c+ .&&. prop_not a+#endif
+ src/Algebra/Boolean/Free.hs view
@@ -0,0 +1,72 @@+{-# LANGUAGE TypeFamilies #-}+module Algebra.Boolean.Free+ ( FreeBoolean (..)+ ) where++import Control.Monad (ap)+import Algebra.Lattice+ ( BoundedJoinSemiLattice (..)+ , JoinSemiLattice (..)+ , BoundedMeetSemiLattice (..)+ , MeetSemiLattice (..)+ , BoundedLattice+ , Lattice+ )+import Data.Algebra.Free+ ( AlgebraType0+ , AlgebraType+ , FreeAlgebra (..)+ , proof+ , fmapFree+ , bindFree+ )++import Algebra.Boolean (BooleanAlgebra)+import Algebra.Heyting (HeytingAlgebra (..))++-- |+-- Free Boolean algebra. @'FreeAlgebra'@ instance provides all the usual+-- combinators for a free algebra.+newtype FreeBoolean a = FreeBoolean+ { runFreeBoolean :: forall h . BooleanAlgebra h => (a -> h) -> h }++instance JoinSemiLattice (FreeBoolean a) where+ FreeBoolean f \/ FreeBoolean g = FreeBoolean (\inj -> f inj \/ g inj)++instance BoundedJoinSemiLattice (FreeBoolean a) where+ bottom = FreeBoolean (\_ -> bottom)++instance MeetSemiLattice (FreeBoolean a) where+ FreeBoolean f /\ FreeBoolean g = FreeBoolean (\inj -> f inj /\ g inj)++instance BoundedMeetSemiLattice (FreeBoolean a) where+ top = FreeBoolean (\_ -> top)++instance Lattice (FreeBoolean a)++instance BoundedLattice (FreeBoolean a)++instance HeytingAlgebra (FreeBoolean a) where+ FreeBoolean f ==> FreeBoolean g = FreeBoolean (\inj -> f inj ==> g inj)++instance BooleanAlgebra (FreeBoolean a)++type instance AlgebraType0 FreeBoolean a = ()+type instance AlgebraType FreeBoolean a = BooleanAlgebra a+instance FreeAlgebra FreeBoolean where+ returnFree a = FreeBoolean (\inj -> inj a)+ foldMapFree f (FreeBoolean inj) = inj f++ codom = proof+ forget = proof++instance Functor FreeBoolean where+ fmap = fmapFree++instance Applicative FreeBoolean where+ pure = returnFree+ (<*>) = ap++instance Monad FreeBoolean where+ return = returnFree+ (>>=) = bindFree
+ src/Algebra/Heyting.hs view
@@ -0,0 +1,257 @@+{-# LANGUAGE CPP #-}+module Algebra.Heyting+ ( HeytingAlgebra (..)+ , iff+ , iff'+ , toBoolean++ -- * Properties+ --+ -- $properties+ , prop_BoundedMeetSemiLattice+ , prop_BoundedJoinSemiLattice+ , prop_HeytingAlgebra+ , prop_implies+ )+ where++import Prelude hiding (not)+import qualified Prelude++import Control.Applicative (Const (..))+import Data.Functor.Identity (Identity (..))+import Data.Hashable (Hashable)+import Data.Proxy (Proxy (..))+import Data.Semigroup (All (..), Any (..), Endo (..))+import Data.Tagged (Tagged (..))+import Data.Universe.Class (Finite, universe)+import qualified Data.Map as M+#if __GLASGOW_HASKELL__ >= 822+import qualified Data.Map.Merge.Lazy as Merge+#endif+import qualified Data.Set as S+import qualified Data.HashMap.Lazy as HM+import qualified Data.HashSet as HS++import Algebra.Lattice ( BoundedJoinSemiLattice (..)+ , BoundedMeetSemiLattice (..)+ , BoundedLattice+ , Meet (..)+ , Join (..)+ , (/\)+ , (\/)+ )+import Algebra.Lattice.Dropped (Dropped (..))+import Algebra.Lattice.Lifted (Lifted (..))+import Algebra.Lattice.Levitated (Levitated)+import qualified Algebra.Lattice.Levitated as L+import Algebra.PartialOrd (leq)+#ifdef EXPORT_PROPERTIES+import Test.QuickCheck hiding ((==>))+import qualified Test.QuickCheck as QC+#endif++-- |+-- Heyting algebra is a bounded semi-lattice with implication which should+-- satisfy the following law:+--+-- prop> x ∧ a ≤ b ⇔ x ≤ (a ⇒ b)+--+-- This means @a ⇒ b@ is an+-- [exponential object](https://ncatlab.org/nlab/show/exponential%2Bobject),+-- which makes any Heyting algebra+-- a [cartesian+-- closed category](https://ncatlab.org/nlab/show/cartesian%2Bclosed%2Bcategory).+--+-- Some useful properties of Heyting algebras:+-- +-- prop> (a ⇒ b) ∧ a ≤ b+-- prop> b ≤ a ⇒ a ∧ b+-- prop> a ≤ b iff a ⇒ b = ⊤+-- prop> b ≤ b' then a ⇒ b ≤ a ⇒ b'+-- prop> a'≤ a then a' ⇒ b ≤ a ⇒ b+class BoundedLattice a => HeytingAlgebra a where+ -- |+ -- Default implementation: @a ==> b = not a \/ b@, it requires @not@ to+ -- satisfy Boolean axioms, which will make it into a Boolean algebra.+ (==>) :: a -> a -> a+ (==>) a b = not a \/ b++ -- |+ -- Default implementation: @not a = a '==>' 'bottom'@+ --+ -- It is useful for optimisation reasons.+ not :: a -> a+ not a = a ==> bottom++ {-# MINIMAL (==>) | not #-}++-- |+-- Less than fixity of both @'\/'@ and @'/\'@.+infixr 4 ==>++iff :: HeytingAlgebra a => a -> a -> a+iff a b = (a ==> b) /\ (b ==> a)++iff' :: (Eq a, HeytingAlgebra a) => a -> a -> Bool+iff' a b = Meet top `leq` Meet (iff a b)+ +-- |+-- Every Heyting algebra contains a Boolean algebra. @'toBoolean'@ maps onto it;+-- moreover it is a monad (Heyting algebra is a category as every poset is) which+-- preserves finite infima.+toBoolean :: HeytingAlgebra a => a -> a+toBoolean = not . not++instance HeytingAlgebra Bool where+ not = Prelude.not++instance HeytingAlgebra All where+ All a ==> All b = All (a ==> b)+ not (All a) = All (not a)++instance HeytingAlgebra Any where+ Any a ==> Any b = Any (a ==> b)+ not (Any a) = Any (not a)++instance HeytingAlgebra () where+ _ ==> _ = ()++instance HeytingAlgebra (Proxy a) where+ _ ==> _ = Proxy++instance HeytingAlgebra a => HeytingAlgebra (Tagged t a) where+ Tagged a ==> Tagged b = Tagged (a ==> b)++instance HeytingAlgebra b => HeytingAlgebra (a -> b) where+ f ==> g = \a -> f a ==> g a++instance HeytingAlgebra a => HeytingAlgebra (Identity a) where+ (Identity a) ==> (Identity b) = Identity (a ==> b)++instance HeytingAlgebra a => HeytingAlgebra (Const a b) where+ (Const a) ==> (Const b) = Const (a ==> b)++instance HeytingAlgebra a => HeytingAlgebra (Endo a) where+ (Endo f) ==> (Endo g) = Endo (f ==> g)++instance (HeytingAlgebra a, HeytingAlgebra b) => HeytingAlgebra (a, b) where+ (a0, b0) ==> (a1, b1) = (a0 ==> a1, b0 ==> b1)++--+-- Dropped, Lifted, Levitated+--++instance (Eq a, HeytingAlgebra a) => HeytingAlgebra (Dropped a) where+ (Drop a) ==> (Drop b) | Meet a `leq` Meet b = Top+ | otherwise = Drop (a ==> b)+ Top ==> a = a+ _ ==> Top = Top++instance HeytingAlgebra a => HeytingAlgebra (Lifted a) where+ (Lift a) ==> (Lift b) = Lift (a ==> b)+ Bottom ==> _ = Lift top+ _ ==> Bottom = Bottom++instance (Eq a, HeytingAlgebra a) => HeytingAlgebra (Levitated a) where+ (L.Levitate a) ==> (L.Levitate b) | Meet a `leq` Meet b = L.Top+ | otherwise = L.Levitate (a ==> b)+ L.Top ==> a = a+ _ ==> L.Top = L.Top+ L.Bottom ==> _ = L.Top+ _ ==> L.Bottom = L.Bottom++--+-- containers+--++-- |+-- Power set: the cannoical example of a Boolean algebra+instance (Ord a, Finite a) => HeytingAlgebra (S.Set a) where+ not a = S.fromList universe `S.difference` a++instance (Eq a, Finite a, Hashable a) => HeytingAlgebra (HS.HashSet a) where+ not a = HS.fromList universe `HS.difference` a++-- |+-- It is not a boolean algebra (even if values are).+instance (Ord k, Finite k, HeytingAlgebra v) => HeytingAlgebra (M.Map k v) where+ -- _xx__+ -- __yy_+ -- T_iTT where i = x ==> y; T = top; _ missing (or removed key)+#if __GLASOW_HASKELL__ >= 822+ a ==> b =+ Merge.merge+ Merge.dropMissing -- drop if an element is missing in @b@+ (Merge.mapMissing (\_ _ -> top)) -- put @top@ if an element is missing in @a@+ (Merge.zipWithMatched (\_ -> (==>))) -- merge matching elements with @==>@+ a b+ + \/ M.fromList [(k, top) | k <- universe, not (M.member k a), not (M.member k b) ] + -- for elements which are not in a, nor in b add+ -- a @top@ key+#else+ a ==> b =+ M.intersectionWith (==>) a b+ `M.union` M.map (const top) (M.difference b a)+ `M.union` M.fromList [(k, top) | k <- universe, not (M.member k a), not (M.member k b)]+#endif++instance (Eq k, Finite k, Hashable k, HeytingAlgebra v) => HeytingAlgebra (HM.HashMap k v) where+ a ==> b = HM.intersectionWith (==>) a b+ `HM.union` HM.map (const top) (HM.difference b a)+ `HM.union` HM.fromList [(k, top) | k <- universe, not (HM.member k a), not (HM.member k b)]++--+-- $properties+--+-- /Properties are exported only if @export-properties@ cabal flag is defined./+#ifdef EXPORT_PROPERTIES++-- |+-- Verfifies bounded meet semilattice laws.+prop_BoundedMeetSemiLattice :: (BoundedMeetSemiLattice a, Eq a, Show a)+ => a -> a -> a -> Property+prop_BoundedMeetSemiLattice a b c =+ counterexample "meet associativity" ((a /\ (b /\ c)) === ((a /\ b) /\ c))+ .&&. counterexample "meet commutativity" ((a /\ b) === (b /\ a))+ .&&. counterexample "meet idempotent" ((a /\ a) === a)+ .&&. counterexample "meet identity" ((top /\ a) === a)+ .&&. counterexample "meet order" (Meet (a /\ b) `leq` Meet a)++-- |+-- Verfifies bounded join semilattice laws.+prop_BoundedJoinSemiLattice :: (BoundedJoinSemiLattice a, Eq a, Show a) => a -> a -> a -> Property+prop_BoundedJoinSemiLattice a b c =+ counterexample "join associativity" ((a \/ (b \/ c)) === ((a \/ b) \/ c))+ .&&. counterexample "join commutativity" ((a \/ b) === (b \/ a))+ .&&. counterexample "join idempotent" ((a \/ a) === a)+ .&&. counterexample "join identity" ((bottom \/ a) === a)+ .&&. counterexample "join order" (Join a `leq` Join (a \/ b))++-- |+-- Verifies the Heyting algebra law for @==>@:+-- for all @a@: @_ /\ a@ is left adjoint to @a ==>@+prop_implies :: (HeytingAlgebra a, Eq a, Show a)+ => a -> a -> a -> Property+prop_implies x a b =+ counterexample ("Failed: x ≤ (a ⇒ b) then x ∧ a ≤ b\n\ta ⇒ b = " ++ show (a ==> b))+ (Meet x `leq` Meet (a ==> b) QC.==> (Meet (x /\ a) `leq` Meet b))+ .&&.+ counterexample ("Failed: x ∧ a ≤ b then x ≤ (a ⇒ b)\n\ta ⇒ b = " ++ show (a ==> b))+ (Meet (x /\ a) `leq` Meet b QC.==> (Meet x `leq` Meet (a ==> b)))+++-- |+-- Usefull for testing valid instances of @'HeytingAlgebra'@ type class. It+-- validates:+--+-- * bounded lattice laws+-- * @'prop_implies'@+prop_HeytingAlgebra :: (HeytingAlgebra a, Eq a, Show a)+ => a -> a -> a -> Property+prop_HeytingAlgebra a b c = + prop_BoundedJoinSemiLattice a b c+ .&&. prop_BoundedMeetSemiLattice a b c+ .&&. prop_implies a b c+#endif
+ src/Algebra/Heyting/Free.hs view
@@ -0,0 +1,76 @@+{-# LANGUAGE TypeFamilies #-}+module Algebra.Heyting.Free+ ( FreeHeyting (..)+ ) where++import Prelude hiding (not)++import Control.Monad (ap)+import Algebra.Lattice+ ( BoundedJoinSemiLattice (..)+ , JoinSemiLattice (..)+ , BoundedMeetSemiLattice (..)+ , MeetSemiLattice (..)+ , BoundedLattice+ , Lattice+ )+import Data.Algebra.Free+ ( AlgebraType0+ , AlgebraType+ , FreeAlgebra (..)+ , proof+ , fmapFree+ , bindFree+ )++import Algebra.Heyting (HeytingAlgebra (..))++-- |+-- Free Heyting algebra. @'FreeAlgebra'@ instance provides all the usual+-- combinators for a free algebra.+--+-- The+-- [graph](https://en.wikipedia.org/wiki/Heyting_algebra#/media/File:Rieger-Nishimura.svg)+-- of free Heyting algebra with one generator, i.e. @'FreeHeyting' ()@.+newtype FreeHeyting a = FreeHeyting+ { runFreeHeyting :: forall h . HeytingAlgebra h => (a -> h) -> h }++instance JoinSemiLattice (FreeHeyting a) where+ FreeHeyting f \/ FreeHeyting g = FreeHeyting (\inj -> f inj \/ g inj)++instance BoundedJoinSemiLattice (FreeHeyting a) where+ bottom = FreeHeyting (\_ -> bottom)++instance MeetSemiLattice (FreeHeyting a) where+ FreeHeyting f /\ FreeHeyting g = FreeHeyting (\inj -> f inj /\ g inj)++instance BoundedMeetSemiLattice (FreeHeyting a) where+ top = FreeHeyting (\_ -> top)++instance Lattice (FreeHeyting a)++instance BoundedLattice (FreeHeyting a)++instance HeytingAlgebra (FreeHeyting a) where+ FreeHeyting f ==> FreeHeyting g = FreeHeyting (\inj -> f inj ==> g inj)+ not (FreeHeyting f) = FreeHeyting (\inj -> not (f inj))++type instance AlgebraType0 FreeHeyting a = ()+type instance AlgebraType FreeHeyting a = HeytingAlgebra a+instance FreeAlgebra FreeHeyting where+ returnFree a = FreeHeyting (\inj -> inj a)+ foldMapFree f (FreeHeyting inj) = inj f++ codom = proof+ forget = proof++instance Functor FreeHeyting where+ fmap = fmapFree++instance Applicative FreeHeyting where+ pure = returnFree+ (<*>) = ap++instance Monad FreeHeyting where+ return = returnFree+ (>>=) = bindFree
+ src/Algebra/Heyting/Layered.hs view
@@ -0,0 +1,68 @@+module Algebra.Heyting.Layered+ ( Layered (..)+ , layer+ ) where++import Prelude++import Algebra.Lattice+ ( BoundedJoinSemiLattice (..)+ , JoinSemiLattice (..)+ , BoundedMeetSemiLattice (..)+ , MeetSemiLattice (..)+ , BoundedLattice+ , Lattice+ )++import Algebra.Heyting (HeytingAlgebra (..))++-- |+-- Layer one Heyting algebra on top of the other. Note: this is not+-- a categorial sum.+data Layered a b+ = Lower a+ | Upper b+ deriving (Show, Eq, Ord)++instance (JoinSemiLattice a, JoinSemiLattice b) => JoinSemiLattice (Layered a b) where+ Lower _ \/ u@Upper{} = u+ u@Upper{} \/ Lower _ = u+ Lower l \/ Lower l' = Lower $ l \/ l'+ Upper u \/ Upper u' = Upper $ u \/ u'++instance (BoundedJoinSemiLattice a, JoinSemiLattice b) => BoundedJoinSemiLattice (Layered a b) where+ bottom = Lower bottom++instance (MeetSemiLattice a, MeetSemiLattice b) => MeetSemiLattice (Layered a b) where+ l@Lower{} /\ Upper _ = l+ Upper _ /\ l@Lower{} = l+ Lower l /\ Lower l' = Lower $ l /\ l'+ Upper u /\ Upper u' = Upper $ u /\ u'++instance (MeetSemiLattice a, BoundedMeetSemiLattice b) => BoundedMeetSemiLattice (Layered a b) where+ top = Upper top++instance ( Lattice a+ , Lattice b+ ) => Lattice (Layered a b)++instance ( Lattice a+ , Lattice b+ , BoundedJoinSemiLattice a+ , BoundedMeetSemiLattice b+ ) => BoundedLattice (Layered a b)++instance ( HeytingAlgebra a+ , HeytingAlgebra b+ , Eq a+ ) => HeytingAlgebra (Layered a b) where+ Lower _ ==> Upper _ = Upper top+ Upper _ ==> l@Lower{} = l+ Upper u ==> Upper u' = Upper $ u ==> u'+ Lower l ==> Lower l' = case l ==> l' of+ ll' | ll' == top -> Upper top+ | otherwise -> Lower ll'++layer :: (a -> c) -> (b -> c) -> Layered a b -> c+layer f _ (Lower a) = f a+layer _ g (Upper b) = g b
+ test/Main.hs view
@@ -0,0 +1,197 @@+{-# LANGUAGE GeneralizedNewtypeDeriving #-}+{-# LANGUAGE ScopedTypeVariables #-}+module Main where++import Algebra.Lattice ( JoinSemiLattice+ , BoundedJoinSemiLattice+ , MeetSemiLattice+ , BoundedMeetSemiLattice+ , Lattice+ , BoundedLattice+ )+import Algebra.Lattice.Dropped (Dropped (..))+import Algebra.Lattice.Lifted (Lifted (..))+import Algebra.Lattice.Levitated (Levitated)+import qualified Algebra.Lattice.Levitated as L+import Data.Universe.Class (Universe (..), Finite)+import qualified Data.Set as S+import qualified Data.Map as M++import Test.QuickCheck++import Algebra.Boolean+import Algebra.Heyting+import Algebra.Heyting.Layered++import Test.Tasty+import Test.Tasty.QuickCheck++-- | Arbitrary wrapper+newtype Arb a = Arb a+ deriving ( JoinSemiLattice+ , BoundedJoinSemiLattice+ , MeetSemiLattice+ , BoundedMeetSemiLattice+ , Lattice+ , BoundedLattice+ , BooleanAlgebra+ , HeytingAlgebra+ , Eq+ , Ord+ )++instance Show a => Show (Arb a) where+ show (Arb a) = show a++instance (Finite k, Arbitrary k, Arbitrary v, Ord k) => Arbitrary (Arb (M.Map k v)) where+ arbitrary = frequency + [ (1, Arb . M.fromList <$> arbitrary)+ , (4, Arb . M.fromList . zip universe <$> vectorOf (length (universe @k)) arbitrary)+ , (6, return $ Arb M.empty)+ ]++instance Arbitrary a => Arbitrary (Arb (Lifted a)) where+ arbitrary = Arb . maybe Bottom Lift <$> arbitrary+ shrink (Arb Bottom) = []+ shrink (Arb (Lift a)) =+ Arb Bottom : [ Arb (Lift a') | a' <- shrink a ]++instance Arbitrary a => Arbitrary (Arb (Dropped a)) where+ arbitrary = Arb . maybe Top Drop <$> arbitrary+ shrink (Arb Top) = []+ shrink (Arb (Drop a)) =+ Arb Top : [ Arb (Drop a') | a' <- shrink a ]++instance Arbitrary a => Arbitrary (Arb (Levitated a)) where+ arbitrary = frequency+ [ (1, return $ Arb L.Top)+ , (1, return $ Arb L.Bottom)+ , (2, Arb . L.Levitate <$> arbitrary)+ ]++ shrink (Arb L.Bottom) = []+ shrink (Arb (L.Levitate a))+ = Arb L.Bottom+ : Arb L.Top+ : [ Arb (L.Levitate a') | a' <- shrink a ]+ shrink (Arb L.Top) = []++data S5 = S1 | S2 | S3 | S4 | S5+ deriving (Ord, Eq, Show)++instance Universe S5 where+ universe = [S1, S2, S3, S4, S5]++instance Finite S5++instance Arbitrary S5 where+ arbitrary = elements universe++instance (Arbitrary a, Ord a) => Arbitrary (Arb (S.Set a)) where+ arbitrary = Arb . S.fromList <$> arbitrary+ shrink (Arb as) = [ Arb (S.fromList as') | as' <- shrink (S.toList as) ]++instance (Arbitrary a, Arbitrary b) => Arbitrary (Arb (Layered a b)) where+ arbitrary = oneof+ [ Arb . Lower <$> arbitrary+ , Arb . Upper <$> arbitrary+ ]+ shrink (Arb (Lower a)) = [ Arb (Lower a') | a' <- shrink a ]+ shrink (Arb (Upper b)) = [ Arb (Upper b') | b' <- shrink b ]++-- Another arbitrary newtype wrapper; using tagged type let us avoid+-- overlapping instances.+newtype Composed f g a = Composed (f (g a))+ deriving ( JoinSemiLattice+ , BoundedJoinSemiLattice+ , MeetSemiLattice+ , BoundedMeetSemiLattice+ , Lattice+ , BoundedLattice+ , BooleanAlgebra+ , HeytingAlgebra+ , Eq+ , Ord+ )++instance Show (f (g a)) => Show (Composed f g a) where+ show (Composed fga) = show fga++instance Arbitrary a => Arbitrary (Composed Dropped Lifted a) where+ arbitrary = frequency+ [ (1, return $ Composed Top)+ , (1, return $ Composed (Drop Bottom))+ , (2, Composed . Drop . Lift <$> arbitrary)+ ]++ shrink (Composed Top) = []+ shrink (Composed (Drop Bottom)) = [Composed Top]+ shrink (Composed (Drop (Lift a))) =+ [ Composed Top, Composed (Drop Bottom) ]+ ++ [ Composed (Drop (Lift a')) | a' <- shrink a ]++instance Arbitrary a => Arbitrary (Composed Lifted Dropped a) where+ arbitrary = frequency+ [ (1, return $ Composed Bottom)+ , (1, return $ Composed (Lift Top))+ , (2, Composed . Lift . Drop <$> arbitrary)+ ]++instance Arbitrary a => Arbitrary (Composed Lifted Lifted a) where+ arbitrary = frequency+ [ (1, return $ Composed Bottom)+ , (1, return $ Composed (Lift Bottom))+ , (2, Composed . Lift . Lift <$> arbitrary)+ ]++ shrink (Composed Bottom) = []+ shrink (Composed (Lift Bottom)) = [Composed Bottom]+ shrink (Composed (Lift (Lift a))) =+ [ Composed Bottom, Composed (Lift Bottom) ]+ ++ [ Composed (Lift (Lift a')) | a' <- shrink a ]++instance Arbitrary a => Arbitrary (Composed Dropped Dropped a) where+ arbitrary = frequency+ [ (1, return $ Composed Top)+ , (1, return $ Composed (Drop Top))+ , (2, Composed . Drop . Drop <$> arbitrary)+ ]++ shrink (Composed Top) = []+ shrink (Composed (Drop Top)) = [Composed Top]+ shrink (Composed (Drop (Drop a))) =+ [ Composed Top, Composed (Drop Top) ]+ ++ [ Composed (Drop (Drop a')) | a' <- shrink a ]+++main :: IO ()+main = defaultMain tests++tests :: TestTree+tests =+ testGroup "heyting-algebras tests"+ [ testGroup "BooleanAlgebra tests"+ [ testProperty "Bool" $ prop_BooleanAlgebra @Bool+ , testProperty "(Bool, Bool)" $ prop_BooleanAlgebra @(Bool, Bool)+ , testProperty "Boolean (Lifted Bool)" $ prop_BooleanAlgebra @(Boolean (Arb (Lifted Bool)))+ , testProperty "(Set S5)" $ prop_BooleanAlgebra @(Arb (S.Set S5))+ ]+ , testGroup "Not BooleanAlgebra tests"+ [ testProperty "Not BooleanAlgebra (Lifted Bool)" $ expectFailure $ prop_not @(Arb (Lifted Bool))+ , testProperty "Not BooleanAlgebra (Dropped Bool)" $ expectFailure $ prop_not @(Arb (Dropped Bool))+ , testProperty "Not BooleanAlgebra (Levitated Bool)" $ expectFailure $ prop_not @(Arb (Levitated Bool))+ ]+ , testGroup "HeytingAlgebra tests"+ [ testProperty "Lifted Bool" $ prop_HeytingAlgebra @(Arb (Lifted Bool))+ , testProperty "Dropped Bool" $ prop_HeytingAlgebra @(Arb (Dropped Bool))+ , testProperty "Layered Bool Bool" $ prop_HeytingAlgebra @(Arb (Layered Bool Bool))+ , testProperty "Levitated Bool" $ prop_HeytingAlgebra @(Arb (Levitated Bool))+ , testProperty "Sum (Lifted Bool) (Dropped Bool)"+ $ prop_HeytingAlgebra @(Arb (Layered (Arb (Lifted Bool)) (Arb (Dropped Bool))))+ , testProperty "Map S5 Bool" $ prop_HeytingAlgebra @(Arb (M.Map S5 Bool))+ , testProperty "Dropped (Lifted Bool)" $ prop_HeytingAlgebra @(Composed Dropped Lifted Bool)+ , testProperty "Lifted (Dropped Bool)" $ prop_HeytingAlgebra @(Composed Lifted Dropped Bool)+ , testProperty "Lifted (Lifted Bool)" $ prop_HeytingAlgebra @(Composed Lifted Lifted Bool)+ , testProperty "Dropped (Dropped Bool)" $ prop_HeytingAlgebra @(Composed Dropped Dropped Bool)+ ]+ ]