diff --git a/ChangeLog.md b/ChangeLog.md --- a/ChangeLog.md +++ b/ChangeLog.md @@ -1,5 +1,16 @@ # Revision history for heyting-algebra -## 0.0.1.0 -- YYYY-mm-dd +## 0.0.2.0 + +* Added Algebra.Heyting.CounterExample +* Added Algebra.Heyting.Free.atom +* Added `BoolRing` a Boolean ring +* Check distributivity laws +* newtype `Ordered` adds Heyting algebra instance for every type satisfying the + `Ord` constraint. +* (<=>) operator added +* Library does not depens on QuickCheck anymore + +## 0.0.1.1 -- 2018.10.5 * First version. Released on an unsuspecting world. diff --git a/LICENSE b/LICENSE --- a/LICENSE +++ b/LICENSE @@ -1,373 +1,22 @@ -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. +Copyright (c) 2018 Marcin Szamotulski +All rights reserved. -You may add additional accurate notices of copyright ownership. +Redistribution and use in source and binary forms, with or without modification, are permitted +provided that the following conditions are met: -Exhibit B - "Incompatible With Secondary Licenses" Notice ---------------------------------------------------------- + * 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 Maximilian Bolingbroke nor the names of other contributors may be used to + endorse or promote products derived from this software without specific prior written permission. - This Source Code Form is "Incompatible With Secondary Licenses", as - defined by the Mozilla Public License, v. 2.0. +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.md b/README.md --- a/README.md +++ b/README.md @@ -7,30 +7,46 @@ [lattices](https://hackage.haskell.org/package/lattices) and [free-algebras](https://hackage.haskell.org/package/free-algebras) (to provide combinators for free Heyting algebras). The package also defines a type class -for Boolean algebras and comes with a handful of instances. +for Boolean algebras and comes with many useful instances. +A note about notation: this package is based on +[lattices](https://hackage.haskell.org/package/lattices), and both are using +notation and names common in lattice theory and logic. Where `&&` becomes `∧` +and is called `meet` and `||` is denoted by `∨` and is usually called +`join`. The `lattice` package provides `\/` and `/\` operators as well as type +classes for various flavors of posets and lattices. + A very good introduction to Heyting algebras can be found at [ncatlab](https://ncatlab.org/nlab/show/Heyting%2Balgebra). Heyting algebras are the crux of [intuitionistic logic](https://en.wikipedia.org/wiki/Intuitionistic_logic), which drops the -axiom of exluded middle. From categorical point of view, Heyting algebras are +axiom of excluded middle. From categorical point of view, Heyting algebras are posets (categories with at most one arrow between any objects), which are also Cartesian closed (and finitely (co-)complete). Note that this makes any Heyting algebra a simply typed lambda calculus; hence one more incentive to -learn how to use them. +learn about them. For example currying holds in every Heyting algebra: +`a => (b ⇒ c)` is equal to `(a ∧ b) ⇒ c` The most important operation is implication `(==>) :: HeytingAlgebra a => a -> -a -> a`; since every Boolean algebra is a Heyting algebra via `a ==> -b = not a \/ b` (using the lattice notation for `or`). It is very handy in -expression conditional logic. +a -> a` (which we might also write as ⇒ in documentation). Every Boolean +algebra is a Heyting algebra via `a ==> b = not a \/ b` (using the lattice +notation for `or`). It is very handy in expression conditional logic. -Some basic examples of Heyting algebras: +Some examples of Heyting algebras: * `Bool` is a Boolean algebra * `(Ord a, Bounded a) => a`; the implication is defined as: if `a ≤ b` then `a - ⇒ b = maxBound`; otherwise `a ⇒ b = b`; e.g. integers with both `±∞` (it can - be represented by `Levitated Int`. This type is not a Boolean algebra. + ⇒ b = maxBound`, otherwise `a ⇒ b = b`; e.g. integers with both `±∞` (it can + be represented by `Levitated Int`. Note that it is not a Boolean algebra. * The power set is a Boolean algebra, in Haskell it can be represented by `Set a` (one might need to require `a` to be finite though, otherwise `not (not - empty)` might be `undefined` rather than `empty`). + empty)` might be `undefined` rather than `empty`). It is a well known fact + that every Boolean algebra is isomorphic to a power set. +* ```haskell + type CounterExample a = Lifted (Op (Set a)) + ``` + is a Heyting algebra; it is useful for gathering counter examples in + a similar way that `Property` from `QuickCheck` library does (put pure). + This library provides some useful functions for this type, see the + `Algebra.Heyting.Properties` and tests for example usage. * More generally every type `(Ord k, Finite k, HeytingAlgebra v) => Map k a` is a Heyting algebra (though in general not a Boolean one). diff --git a/heyting-algebras.cabal b/heyting-algebras.cabal --- a/heyting-algebras.cabal +++ b/heyting-algebras.cabal @@ -1,42 +1,38 @@ name: heyting-algebras -version: 0.0.1.2 +version: 0.0.2.0 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: BSD3 license-file: LICENSE author: Marcin Szamotulski maintainer: profunctor@pm.me -copyright: (c) 2018 Marcin Szamotulski +copyright: (c) 2018-2019 Marcin Szamotulski category: Math build-type: Simple extra-source-files: ChangeLog.md README.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: False - default: True +tested-with: GHC==8.0.2, GHC==8.2.2, GHC==8.4.4, GHC==8.6.3 library exposed-modules: Algebra.Heyting Algebra.Heyting.Free Algebra.Heyting.Layered + Algebra.Heyting.BoolRing + Algebra.Heyting.CounterExample + Algebra.Heyting.Properties Algebra.Boolean Algebra.Boolean.Free - -- other-modules: - -- other-extensions: + Algebra.Boolean.Properties build-depends: base >= 4.9 && < 4.13 , containers >= 0.4.2 && < 0.7 - , free-algebras >= 0.0.4 && < 0.0.6 + , free-algebras >= 0.0.4 && < 0.0.8 , hashable >= 1.2.6.1 && < 1.3 - , lattices >= 1.0 && < 1.11 + , lattices >= 1.0 && < 1.8 + , semiring-simple >= 1.0 && < 1.2 , tagged >= 0.8.5 && < 0.9 , unordered-containers >= 0.2.6.0 && < 0.3 @@ -45,14 +41,10 @@ default-language: Haskell2010 default-extensions: FlexibleInstances RankNTypes + PackageImports ghc-options: -Wall - if flag(export-properties) - build-depends: - QuickCheck >= 2.10 && < 2.13 - cpp-options: - -DEXPORT_PROPERTIES -test-suite heyting-algebras-test +test-suite tests type: exitcode-stdio-1.0 hs-source-dirs: test main-is: Main.hs @@ -67,6 +59,7 @@ default-language: Haskell2010 default-extensions: FlexibleInstances , TypeApplications + ghc-options: -threaded -rtsopts -with-rtsopts=-N source-repository head type: git diff --git a/src/Algebra/Boolean.hs b/src/Algebra/Boolean.hs --- a/src/Algebra/Boolean.hs +++ b/src/Algebra/Boolean.hs @@ -12,65 +12,28 @@ , 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 Prelude hiding (not) -import Algebra.Lattice ( Lattice - , BoundedLattice - , JoinSemiLattice (..) - , BoundedJoinSemiLattice - , MeetSemiLattice (..) - , BoundedMeetSemiLattice - , bottom - , top - ) +import Data.Data (Data, Typeable) +import GHC.Generics (Generic) -import Algebra.Heyting ( HeytingAlgebra (..) - , iff - , iff' - , not - , toBoolean - , prop_HeytingAlgebra - ) +import Algebra.Lattice ( Lattice + , BoundedLattice + , JoinSemiLattice (..) + , BoundedJoinSemiLattice + , MeetSemiLattice (..) + , BoundedMeetSemiLattice + ) --- | --- 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 +import Algebra.Heyting ( HeytingAlgebra (..) + , BooleanAlgebra + , iff + , iff' + , not + , toBoolean + ) -- | -- @'Boolean'@ is the left adjoint functor from the category of Heyting algebras @@ -86,68 +49,8 @@ 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 diff --git a/src/Algebra/Boolean/Free.hs b/src/Algebra/Boolean/Free.hs --- a/src/Algebra/Boolean/Free.hs +++ b/src/Algebra/Boolean/Free.hs @@ -3,26 +3,24 @@ ( 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 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 (..)) +import Algebra.Boolean (BooleanAlgebra) +import Algebra.Heyting (HeytingAlgebra (..)) -- | -- Free Boolean algebra. @'FreeAlgebra'@ instance provides all the usual diff --git a/src/Algebra/Boolean/Properties.hs b/src/Algebra/Boolean/Properties.hs new file mode 100644 --- /dev/null +++ b/src/Algebra/Boolean/Properties.hs @@ -0,0 +1,36 @@ +module Algebra.Boolean.Properties where + +import Prelude hiding (not) + +import Algebra.Lattice (bottom, top, (/\), (\/)) +import Algebra.Boolean +import Algebra.Heyting +import Algebra.Heyting.CounterExample ( CounterExample + , annotate + , fmapCounterExample + , (===) + ) +import Algebra.Heyting.Properties + +-- | +-- Test that @'not'@ satisfies Boolean algebra axioms. +prop_not :: (HeytingAlgebra a, Ord a, Eq a, Ord e) => a -> CounterExample e +prop_not a = + (not (not a) === a) + /\ (not a /\ a === bottom) + /\ (not a \/ a === top) + +data BooleanAlgebraLawViolation a + = BALVHeytingAlgebraLawViolation (HeytingAlgebraLawViolation a) + | BALVNotLawViolation a + deriving (Eq, Ord, Show) + +-- | +-- Test that @a@ is satisfy both @'Algebra.Heyting.prop_HeytingAlgebra'@ and +-- @'prop_not'@. +prop_BooleanAlgebra + :: (BooleanAlgebra a, Ord a, Eq a, Show a) + => a -> a -> a -> CounterExample (BooleanAlgebraLawViolation a) +prop_BooleanAlgebra a b c = + (fmapCounterExample BALVHeytingAlgebraLawViolation $ prop_HeytingAlgebra a b c) + /\ annotate (BALVNotLawViolation a) (prop_not a) diff --git a/src/Algebra/Heyting.hs b/src/Algebra/Heyting.hs --- a/src/Algebra/Heyting.hs +++ b/src/Algebra/Heyting.hs @@ -1,80 +1,96 @@ {-# LANGUAGE CPP #-} + module Algebra.Heyting - ( HeytingAlgebra (..) + ( -- * Heyting algebras + HeytingAlgebra (..) + , implies + , (<=>) , iff , iff' , toBoolean - - -- * Properties - -- - -- $properties - , prop_BoundedMeetSemiLattice - , prop_BoundedJoinSemiLattice - , prop_HeytingAlgebra - , prop_implies + -- * Boolean algebras + , BooleanAlgebra ) where -import Prelude hiding (not) + +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 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 Data.Set (Set) +import qualified Data.Set as Set 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 Algebra.Lattice ( BoundedJoinSemiLattice (..) + , BoundedMeetSemiLattice (..) + , BoundedLattice + , Meet (..) + , bottom + , top + , (/\) + , (\/) + ) +import Algebra.Lattice.Dropped (Dropped (..)) +import Algebra.Lattice.Lifted (Lifted (..)) +import Algebra.Lattice.Levitated (Levitated) import qualified Algebra.Lattice.Levitated as L -import Algebra.Lattice.Ordered (Ordered (..)) -import Algebra.PartialOrd (leq) -#ifdef EXPORT_PROPERTIES -import Test.QuickCheck hiding (Ordered, (==>)) -import qualified Test.QuickCheck as QC -#endif +import Algebra.Lattice.Ordered (Ordered (..)) +import Algebra.Lattice.Op (Op (..)) +import Algebra.PartialOrd (leq) +-- +-- Heyting algebras +-- + -- | -- Heyting algebra is a bounded semi-lattice with implication which should -- satisfy the following law: -- -- prop> x ∧ a ≤ b ⇔ x ≤ (a ⇒ b) -- +-- We also require that a Heyting algebra is a distributive lattice, which +-- means any of the two equivalent conditions holds: +-- +-- prop> a ∧ (b ∨ c) = a ∧ b ∨ a ∧ c +-- prop> a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) +-- -- 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). +-- This means that Curry isomorphism holds (which takes a form of an actual +-- equality): -- --- Some useful properties of Heyting algebras: +-- prop> a ⇒ (b ⇒ c) = (a ∧ b) ⇒ c +-- +-- Some other 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 +-- prop> not (a ∧ b) = not (a ∨ b) +-- prop> not (a ∨ b) = not a ∧ not 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. + -- + -- Fixity is less than fixity of both @'\/'@ and @'/\'@. (==>) :: a -> a -> a (==>) a b = not a \/ b @@ -87,12 +103,20 @@ {-# MINIMAL (==>) | not #-} --- | --- Less than fixity of both @'\/'@ and @'/\'@. infixr 4 ==> +-- | +-- @'implies'@ is an alias for @'==>'@ +implies :: HeytingAlgebra a => a -> a -> a +implies = (==>) + +(<=>) :: HeytingAlgebra a => a -> a -> a +a <=> b = (a ==> b) /\ (b ==> a) + +-- | +-- @'iff'@ is an alias for @'<=>'@ iff :: HeytingAlgebra a => a -> a -> a -iff a b = (a ==> b) /\ (b ==> a) +iff = (<=>) iff' :: (Eq a, HeytingAlgebra a) => a -> a -> Bool iff' a b = Meet top `leq` Meet (iff a b) @@ -105,7 +129,7 @@ toBoolean = not . not instance HeytingAlgebra Bool where - not = Prelude.not + not = Prelude.not instance HeytingAlgebra All where All a ==> All b = All (a ==> b) @@ -127,8 +151,10 @@ instance HeytingAlgebra b => HeytingAlgebra (a -> b) where f ==> g = \a -> f a ==> g a +#if MIN_VERSION_base(4,8,0) instance HeytingAlgebra a => HeytingAlgebra (Identity a) where (Identity a) ==> (Identity b) = Identity (a ==> b) +#endif instance HeytingAlgebra a => HeytingAlgebra (Const a b) where (Const a) ==> (Const b) = Const (a ==> b) @@ -143,6 +169,8 @@ -- Dropped, Lifted, Levitated, Ordered -- +-- | +-- Subdirectly irreducible Heyting algebra. instance (Eq a, HeytingAlgebra a) => HeytingAlgebra (Dropped a) where (Drop a) ==> (Drop b) | Meet a `leq` Meet b = Top | otherwise = Drop (a ==> b) @@ -171,9 +199,9 @@ -- -- | --- 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 +-- Power set: the canonical example of a Boolean algebra +instance (Ord a, Finite a) => HeytingAlgebra (Set a) where + not a = Set.fromList universe `Set.difference` a instance (Eq a, Finite a, Hashable a) => HeytingAlgebra (HS.HashSet a) where not a = HS.fromList universe `HS.difference` a @@ -184,14 +212,14 @@ -- _xx__ -- __yy_ -- T_iTT where i = x ==> y; T = top; _ missing (or removed key) -#if __GLASOW_HASKELL__ >= 822 +#if __GLASOW_HASKELL__ >= 804 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 @@ -207,56 +235,80 @@ `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)] +-- +-- Boolean algebras -- --- $properties +-- They are defined in the same module, to avoid module dependency: @Op a@ is +-- a Boolean algebra (thus Heyting algebra in the first place), whenever @a@ is +-- a Boolean algebra. -- --- /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) +-- 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 --- | --- 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)) +-- +-- Instances +-- --- | --- 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))) +instance BooleanAlgebra Bool +instance BooleanAlgebra All --- | --- 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 +instance BooleanAlgebra Any + +instance BooleanAlgebra () + +instance BooleanAlgebra (Proxy a) + +instance BooleanAlgebra a => BooleanAlgebra (Tagged t a) + +instance BooleanAlgebra b => BooleanAlgebra (a -> b) + +#if MIN_VERSION_base(4,8,0) +instance BooleanAlgebra a => BooleanAlgebra (Identity a) #endif + +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 (Set a) + + +-- +-- Op +-- + +-- | Whenever @a@ is a Boolean Algebra, @Op a@ is a Boolean algebra as well, +-- which in particular means that it is a Heyting algebra in the first place. +-- +instance BooleanAlgebra a => HeytingAlgebra (Op a) where + (Op a) ==> (Op b) = Op (not a /\ b) + +-- | Every boolean algebra is isomorphic to power set @P(X)@ of some set @X@; +-- then `not :: Op (P(X)) -> P(X)` is a lattice isomorphism, thus `Op (P(X))` is +-- a boolean algebra, since @P(X)@ is. +instance BooleanAlgebra a => BooleanAlgebra (Op a) diff --git a/src/Algebra/Heyting/BoolRing.hs b/src/Algebra/Heyting/BoolRing.hs new file mode 100644 --- /dev/null +++ b/src/Algebra/Heyting/BoolRing.hs @@ -0,0 +1,48 @@ +{-# LANGUAGE CPP #-} +module Algebra.Heyting.BoolRing + ( BoolRing (..) + , Semiring (..) + , (<+>) + ) where + +import Prelude hiding (not) + +import Data.Monoid (Monoid (..)) +#if __GLASGOW_HASKELL__ < 804 +import Data.Semigroup (Semigroup (..)) +#endif + +import Algebra.Lattice (bottom, top, (/\), (\/)) +import Data.Semiring (Semiring (..), (<+>)) + +import Algebra.Heyting + +-- | +-- Newtype wraper which captures Boolean ring structure, which holds for every +-- Heyting algebra. A Boolean ring is a ring which satisfies: +-- +-- prop> a <.> a = a +-- +-- Some other properties: +-- +-- prop> a <+> a = mempty -- thus it is a ring of characteristic 2 +-- prop> a <.> b = b <.> a -- hence it is a commutative ring +-- prop> a <+> (b <+> c) = (a <+> b) <+> c -- multiplicative associativity +newtype BoolRing a = BoolRing { getBoolRing :: a } + +-- | Sum is [symmetric differnce](https://en.wikipedia.org/wiki/Symmetric_difference). +instance HeytingAlgebra a => Semigroup (BoolRing a) where + (BoolRing a) <> (BoolRing b) = BoolRing $ (not a /\ b) \/ (a /\ not b) + +-- | In a Boolean ring @a + a = 0@, hence @negate = id@. +instance HeytingAlgebra a => Monoid (BoolRing a) where + mempty = BoolRing bottom + +#if __GLASGOW_HASKELL__ <= 804 + mappend = (<>) +#endif + +-- | Multiplication is given by @'/\'@ +instance HeytingAlgebra a => Semiring (BoolRing a) where + BoolRing a <.> BoolRing b = BoolRing (a \/ b) + one = BoolRing top diff --git a/src/Algebra/Heyting/CounterExample.hs b/src/Algebra/Heyting/CounterExample.hs new file mode 100644 --- /dev/null +++ b/src/Algebra/Heyting/CounterExample.hs @@ -0,0 +1,102 @@ +{-# LANGUAGE BangPatterns #-} +module Algebra.Heyting.CounterExample where + +import Data.Set (Set) +import qualified Data.Set as Set +import Text.Printf (printf) + +import Algebra.Lattice (top) +import Algebra.Lattice.Lifted (Lifted) +import qualified Algebra.Lattice.Lifted as Lifted +import Algebra.Lattice.Levitated (Levitated) +import qualified Algebra.Lattice.Levitated as Levitated +import Algebra.Lattice.Op (Op (..)) + + +-- | A counter example type is a Heyting algebra; it useful for tests and +-- properties. It records all failures. The truth value is represented by an +-- empty set. Since @'CounterExample' e@ is a Heyting algebra, it is useful in +-- expressing properties that require assumptions. +-- +type CounterExample a = Lifted (Op (Set a)) + +fmapCounterExample + :: (Ord a, Ord b) + => (a -> b) + -> CounterExample a + -> CounterExample b +fmapCounterExample = fmap . fmap . Set.map + +-- | A bijection from @e@ to atoms of the @'CounterExample' e@ lattice. +-- +counterExample :: e -> CounterExample e +counterExample e = Lifted.Lift (Op (Set.singleton e)) + +-- | A lattice homomorphism from @'Bool'@ to @'CounterExample'@, which lifts +-- @'False'@ to an atom of @'CounterExample'@ (uniquely determined by @e@) and +-- which preserves the top element. +-- +fromBool :: Ord e => e -> Bool -> CounterExample e +fromBool e False = counterExample e +fromBool _ True = top + +-- | +-- A homomorphism of /Heyting algebras/. +-- +toBool :: CounterExample e -> Bool +toBool (Lifted.Lift (Op s)) | Set.null s + = True +toBool _ = False + +-- | Note that this map is not a lattice homomorphism (it does not preserve +-- meet nor join). It is also not a poset map in general. Nevertheless, it preserves +-- @top@ and @bottom@. +-- +foldMapCounterExample + :: (Ord e, Monoid m) + => (e -> m) + -> CounterExample e + -> Levitated m +foldMapCounterExample f (Lifted.Lift (Op s)) | Set.null s = Levitated.Top + | otherwise = Levitated.Levitate (foldMap f s) +foldMapCounterExample _ Lifted.Bottom = Levitated.Bottom + +-- | +-- Map a CounterExample to @'Levitated' String@. Each set of counter example +-- is concatenated into a single comma separated string. This is useful for +-- printing counter examples in tests. See @'fromCounterExample''@. +-- +fromCounterExample :: Show a => CounterExample a -> Levitated String +fromCounterExample Lifted.Bottom = Levitated.Bottom +fromCounterExample (Lifted.Lift (Op s)) | Set.null s + = Levitated.Top + | otherwise + = Levitated.Levitate (Set.foldl' go "" s) + where + go "" b = show b + go !a b = printf "%s, %s" a (show b) + +-- | Map `CounterExample` to a Maybe, representing @'Levitated.Top'@ as +-- @'Nothing'@ and mapping both @'Levitated.Levitate'@ and @'Levitate.Bottom'@ to +-- a @Just String@. +-- +fromCounterExample' :: Show a => CounterExample a -> Maybe String +fromCounterExample' ce = case fromCounterExample ce of + Levitated.Top -> Nothing + Levitated.Levitate s -> Just s + Levitated.Bottom -> Just "" + +-- | Add a counter example. This is simply lifts the bottom to an atom given +-- by @e@, otherwise it preserves the @'CounterExample' e@. +-- +-- Note that take join of @\e es = counterExample e \\/ es@ will return @top@ if +-- @e@ if e is not in the set @es@; thus this map is not defined with using @\\/@. +-- +annotate :: Ord e => e -> CounterExample e -> CounterExample e +annotate e Lifted.Bottom = counterExample e +annotate _ es = es + +(===) :: (Ord e, Eq a) => a -> a -> CounterExample e +a === b = if a == b then Lifted.Lift (Op Set.empty) else Lifted.Bottom + +infixr 4 === diff --git a/src/Algebra/Heyting/Free.hs b/src/Algebra/Heyting/Free.hs --- a/src/Algebra/Heyting/Free.hs +++ b/src/Algebra/Heyting/Free.hs @@ -1,29 +1,28 @@ {-# LANGUAGE TypeFamilies #-} module Algebra.Heyting.Free ( FreeHeyting (..) + , atom ) where -import Prelude hiding (not) +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 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 (..)) +import Algebra.Heyting (HeytingAlgebra (..)) -- | -- Free Heyting algebra. @'FreeAlgebra'@ instance provides all the usual @@ -31,7 +30,8 @@ -- -- The -- [graph](https://en.wikipedia.org/wiki/Heyting_algebra#/media/File:Rieger-Nishimura.svg) --- of free Heyting algebra with one generator, i.e. @'FreeHeyting' ()@. +-- of free Heyting algebra with one generator\/atom, i.e. @'FreeHeyting' ()@. +-- newtype FreeHeyting a = FreeHeyting { runFreeHeyting :: forall h . HeytingAlgebra h => (a -> h) -> h } @@ -55,10 +55,17 @@ FreeHeyting f ==> FreeHeyting g = FreeHeyting (\inj -> f inj ==> g inj) not (FreeHeyting f) = FreeHeyting (\inj -> not (f inj)) +-- | +-- Construct an atom of the @'FreeHeyting'@ lattice (in the laguage of free +-- algebra, it is called a generator, e.g. @atom = returnFree@). +-- +atom :: a -> FreeHeyting a +atom = \a -> FreeHeyting $ \inj -> inj a + type instance AlgebraType0 FreeHeyting a = () type instance AlgebraType FreeHeyting a = HeytingAlgebra a instance FreeAlgebra FreeHeyting where - returnFree a = FreeHeyting (\inj -> inj a) + returnFree = atom foldMapFree f (FreeHeyting inj) = inj f codom = proof diff --git a/src/Algebra/Heyting/Layered.hs b/src/Algebra/Heyting/Layered.hs --- a/src/Algebra/Heyting/Layered.hs +++ b/src/Algebra/Heyting/Layered.hs @@ -3,22 +3,21 @@ , layer ) where -import Prelude +import Prelude -import Algebra.Lattice - ( BoundedJoinSemiLattice (..) - , JoinSemiLattice (..) - , BoundedMeetSemiLattice (..) - , MeetSemiLattice (..) - , BoundedLattice - , Lattice - ) +import Algebra.Lattice ( BoundedJoinSemiLattice (..) + , JoinSemiLattice (..) + , BoundedMeetSemiLattice (..) + , MeetSemiLattice (..) + , BoundedLattice + , Lattice + ) -import Algebra.Heyting (HeytingAlgebra (..)) +import Algebra.Heyting (HeytingAlgebra (..)) -- | -- Layer one Heyting algebra on top of the other. Note: this is not --- a categorial sum. +-- a categorical sum. data Layered a b = Lower a | Upper b diff --git a/src/Algebra/Heyting/Properties.hs b/src/Algebra/Heyting/Properties.hs new file mode 100644 --- /dev/null +++ b/src/Algebra/Heyting/Properties.hs @@ -0,0 +1,189 @@ +{-# LANGUAGE TupleSections #-} +-- | +-- Properties of Heyting algebras; useful for testing lawfulness of instances. +-- +module Algebra.Heyting.Properties where + +import Prelude hiding (not) + +import Data.List (intersperse) +import Data.Semigroup ((<>)) + +import Algebra.Lattice ( Lattice + , BoundedJoinSemiLattice + , BoundedMeetSemiLattice + , Meet (..) + , Join (..) + , top + , bottom + , (/\) + , (\/) + ) +import Algebra.PartialOrd (leq) +import Algebra.Heyting +import Algebra.Heyting.CounterExample + +data BoundedMeetSemiLatticeLawViolation a + = BMSLVNonAssociative a a a + | BMSLVNonCommutative a a + | BMSLVNonIdempotent a + | BMSLVNonUnital a + | BMSLVMeetOrderViolation a a + deriving (Eq, Ord) + +instance Show a => Show (BoundedMeetSemiLatticeLawViolation a) where + show (BMSLVNonAssociative a b c) = withArgs "a ∧ (b ∧ c) ≠ (a ∧ b) ∧ c" [a, b, c] + show (BMSLVNonCommutative a b) = withArgs "a ∧ b ٍ≠ b ∧ a" [a, b] + show (BMSLVNonIdempotent a) = withArgs "a ∧ a ≠ a" [a] + show (BMSLVNonUnital a) = withArgs "a ∧ top ≠ a" [a] + show (BMSLVMeetOrderViolation a b) = withArgs "a ∧ b > a" [a, b] + +withArgs :: Show a => String -> [a] -> String +withArgs s bs = s ++ "\n\t" ++ foldr (<>) mempty (intersperse "\n\t" (map show bs)) + +-- | +-- Verifies bounded meet semilattice laws. +prop_BoundedMeetSemiLattice + :: (BoundedMeetSemiLattice a, Ord a, Eq a, Show a) + => a -> a -> a -> CounterExample (BoundedMeetSemiLatticeLawViolation a) +prop_BoundedMeetSemiLattice a b c = + annotate (BMSLVNonAssociative a b c) ((a /\ (b /\ c)) === ((a /\ b) /\ c)) + /\ annotate (BMSLVNonCommutative a b) ((a /\ b) === (b /\ a)) + /\ annotate (BMSLVNonIdempotent a) ((a /\ a) === a) + /\ annotate (BMSLVNonUnital a) ((top /\ a) === a) + /\ annotate (BMSLVMeetOrderViolation a b) ((Meet (a /\ b) `leq` Meet a) === True) + +data BoundedJoinSemiLatticeLawViolation a + = BJSLVNonAssociative a a a + | BJSLVNonCommutative a a + | BJSLVNonIdempotent a + | BJSLVNonUnital a + | BJSLVJoinOrderViolation a a + deriving (Eq, Ord) + +instance Show a => Show (BoundedJoinSemiLatticeLawViolation a) where + show (BJSLVNonAssociative a b c) = withArgs "a ∨ (b ∨ c) ≠ (a ∨ b) ∨ c" [a, b, c] + show (BJSLVNonCommutative a b) = withArgs "a ∨ b ٍ≠ b ∨ a" [a, b] + show (BJSLVNonIdempotent a) = withArgs "a ∨ a ≠ a" [a] + show (BJSLVNonUnital a) = withArgs "a ∨ top ≠ a" [a] + show (BJSLVJoinOrderViolation a b) = withArgs "a ∨ b > a" [a, b] + +-- | +-- Verifies bounded join semilattice laws. +prop_BoundedJoinSemiLattice + :: (BoundedJoinSemiLattice a, Ord a, Eq a, Show a) + => a -> a -> a -> CounterExample (BoundedJoinSemiLatticeLawViolation a) +prop_BoundedJoinSemiLattice a b c = + annotate (BJSLVNonAssociative a b c) + ((a \/ (b \/ c)) === ((a \/ b) \/ c)) + /\ annotate (BJSLVNonCommutative a b) + ((a \/ b) === (b \/ a)) + /\ annotate (BJSLVNonIdempotent a) + ((a \/ a) === a) + /\ annotate (BJSLVNonUnital a) + ((bottom \/ a) === a) + /\ fromBool (BJSLVJoinOrderViolation a b) + (Join a `leq` Join (a \/ b)) + +data DistributiveLatticeLawViolation a + = DLLVJoinOverMeetViolation a a a + | DLLVMeetOverJoinViolation a a a + deriving (Eq, Ord) + +instance Show a => Show (DistributiveLatticeLawViolation a) where + show (DLLVJoinOverMeetViolation a b c) = withArgs "a ∧ (b ∨ c) ≠ a ∧ b ∨ a ∧ c" [a, b, c] + show (DLLVMeetOverJoinViolation a b c) = withArgs "a ∨ (b ∧ c) ≠ (a ∨ b) ∧ (a ∨ c)" [a, b, c] + +-- | +-- Distributivity laws for a lattice. +prop_DistributiveLattice + :: (Lattice a, Ord a, Eq a, Show a) + => a -> a -> a -> CounterExample (DistributiveLatticeLawViolation a) +prop_DistributiveLattice a b c = + annotate (DLLVJoinOverMeetViolation a b c) + ((a /\ (b \/ c)) === ((a /\ b) \/ (a /\ c))) + /\ annotate (DLLVMeetOverJoinViolation a b c) + ((a \/ (b /\ c)) === ((a \/ b) /\ (a \/ c))) + +data HeytingAlgebraLawViolation a + = HAVImplication1 a a a + | HAVImplication2 a a a + | HAVNot a a + | HAVNotAndMeet a a + | HAVNotAndJoin a a + | HAVImplicationAndOrd a a + | HAVDistributiveLatticeLawViolation (DistributiveLatticeLawViolation a) + | HAVBoundedJoinSemilatticeLawViolation (BoundedJoinSemiLatticeLawViolation a) + | HAVBoundedMeetSemilatticeLawViolation (BoundedMeetSemiLatticeLawViolation a) + deriving (Eq, Ord) + +instance Show a => Show (HeytingAlgebraLawViolation a) where + show (HAVImplication1 x a b)= + withArgs "x ≤ (a ⇒ b) then x ∧ a ≤ b" [x, a, b] + + show (HAVImplication2 x a b) = + withArgs "x ∧ a ≤ b then x ≤ (a ⇒ b)" [x, a, b] + + show (HAVNot a b) = + withArgs "a ≤ b ⇏ not a" [a, b] + + show (HAVNotAndMeet a b) = + withArgs "not (a ∧ b) ≠ not a ∨ not b" [a, b] + + show (HAVNotAndJoin a b) = + withArgs "not (a ∨ b) ≠ not a ∧ not b" [a, b] + + show (HAVImplicationAndOrd a b) = + withArgs "(a ⇒ b) ∧ a ≰ b" [a, b] + + show (HAVDistributiveLatticeLawViolation e) = show e + show (HAVBoundedJoinSemilatticeLawViolation e) = show e + show (HAVBoundedMeetSemilatticeLawViolation e) = show e + +-- | +-- Verifies the Heyting algebra law for @==>@: +-- for all @a@: @_ /\ a@ is left adjoint to @a ==>@ +-- and some other properties that are a consequence of that. +prop_implies :: (HeytingAlgebra a, Ord a, Eq a, Show a) + => a -> a -> a -> CounterExample (HeytingAlgebraLawViolation a) +prop_implies x a b = + fromBool + (HAVImplication1 x a b) + (Meet x `leq` Meet (a ==> b) ==> (Meet (x /\ a) `leq` Meet b)) + /\ fromBool + (HAVImplication2 x a b) + (Meet (x /\ a) `leq` Meet b ==> (Meet x `leq` Meet (a ==> b))) + /\ fromBool + (HAVNot a b) + (Meet a `leq` Meet b ==> (Meet (not b) `leq` Meet (not a))) + /\ annotate + (HAVNotAndMeet a b) + (not (a /\ b) === (not a \/ not b)) + /\ annotate + (HAVNotAndJoin a b) + (not (a \/ b) === (not a /\ not b)) + /\ fromBool + (HAVImplicationAndOrd a a) + (Meet ((a ==> b) /\ a) `leq` Meet b) + +-- | +-- Useful for testing valid instances of @'HeytingAlgebra'@ type class. It +-- validates: +-- +-- * bounded lattice laws +-- * @'prop_implies'@ +prop_HeytingAlgebra + :: (HeytingAlgebra a, Ord a, Eq a, Show a) + => a -> a -> a -> CounterExample (HeytingAlgebraLawViolation a) +prop_HeytingAlgebra a b c = + + fmapCounterExample HAVBoundedJoinSemilatticeLawViolation + (prop_BoundedJoinSemiLattice a b c) + + /\ fmapCounterExample HAVBoundedMeetSemilatticeLawViolation + (prop_BoundedMeetSemiLattice a b c) + + /\ fmapCounterExample HAVDistributiveLatticeLawViolation + (prop_DistributiveLattice a b c) + + /\ prop_implies a b c diff --git a/test/Main.hs b/test/Main.hs --- a/test/Main.hs +++ b/test/Main.hs @@ -1,31 +1,208 @@ {-# LANGUAGE GeneralizedNewtypeDeriving #-} -{-# LANGUAGE ScopedTypeVariables #-} +{-# 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 Data.Universe.Class (Universe (..), Finite) +import Data.Set (Set) +import qualified Data.Set as Set +import Data.Map (Map) +import qualified Data.Map as Map + +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 Algebra.Lattice.Ordered (Ordered (..)) -import Data.Universe.Class (Universe (..), Finite) -import qualified Data.Set as S -import qualified Data.Map as M +import Algebra.Lattice.Op (Op (..)) +import Algebra.Lattice.Ordered (Ordered (..)) -import Algebra.Boolean -import Algebra.Heyting -import Algebra.Heyting.Layered +import Algebra.Boolean +import Algebra.Boolean.Properties +import Algebra.Heyting +import Algebra.Heyting.CounterExample +import Algebra.Heyting.Properties +import Algebra.Heyting.Layered -import Test.Tasty -import Test.Tasty.QuickCheck hiding (Ordered) +import Test.Tasty +import Test.Tasty.QuickCheck hiding (Ordered) --- | Arbitrary wrapper +main :: IO () +main = defaultMain tests + +counterExampleProperty + :: Show e + => CounterExample e + -> Property +counterExampleProperty = maybe (property True) (flip counterexample False) . fromCounterExample' + +-- +-- List of tast cases +-- + +tests :: TestTree +tests = + testGroup "heyting-algebras tests" + [ testGroup "Boolean algebras" + [ testProperty "Bool" prop_boolean_Bool + , testProperty "(Bool, Bool)" prop_boolean_BoolBool + , testProperty "Boolean (Lifted Bool)" prop_boolean_LiftedBool + , testProperty "Boolean (Dropped Bool)" prop_boolean_DroppedBool + , testProperty "(Set S5)" prop_boolean_Set + ] + , testGroup "Non Boolean algebras" + [ testProperty "Not a BooleanAlgebra (Lifted Bool)" prop_non_boolean_LiftedBool + , testProperty "Not a BooleanAlgebra (Dropped Bool)" prop_non_boolean_DroppedBool + , testProperty "Not a BooleanAlgebra Levitated (Ordered Int)" prop_non_boolean_LevitatedOrderedInt + ] + , testGroup "Heyting algebras" + [ testProperty "Lifted Bool" prop_heyting_LiftedBool + , testProperty "Dropped Bool" prop_heyting_DroppedBool + , testProperty "Layered Bool Bool" prop_heyting_LayeredBoolBool + , testProperty "Levitated Bool" prop_heyting_LevitatedBool + , testProperty "Sum (Lifted Bool) (Dropped Bool)" prop_heyting_LayeredLiftedDropped + , testProperty "Levitated (Ordered Int)" prop_heyting_LevitatedOrderedInt + , testProperty "Map S5 Bool" prop_heyting_MapS5Bool + , testProperty "Dropped (Lifted Bool)" prop_heyting_DroppedLiftedBool + , testProperty "Lifted (Dropped Bool)" prop_heyting_LiftedDroppedBool + , testProperty "Lifted (Lifted Bool)" prop_heyting_LiftedLiftedBool + , testProperty "Dropped (Dropped Bool)" prop_heyting_DroppedDroppedBool + , testProperty "CounterExample" prop_heyting_CounterExample + ] + ] + +-- +-- Boolean algebra tests +-- + +type BooleanProp a = a -> a -> a -> Property + +prop_boolean_Bool + :: BooleanProp Bool +prop_boolean_Bool = + (fmap . fmap) counterExampleProperty . prop_BooleanAlgebra + +prop_boolean_BoolBool + :: BooleanProp (Bool, Bool) +prop_boolean_BoolBool = + (fmap . fmap) counterExampleProperty . prop_BooleanAlgebra + +prop_boolean_LiftedBool + :: BooleanProp (Arb (Boolean (Arb (Lifted Bool)))) +prop_boolean_LiftedBool = + (fmap . fmap) counterExampleProperty . prop_BooleanAlgebra + +prop_boolean_DroppedBool + :: BooleanProp (Arb (Boolean (Arb (Dropped Bool)))) +prop_boolean_DroppedBool = + (fmap . fmap) counterExampleProperty . prop_BooleanAlgebra + +prop_boolean_Set + :: BooleanProp (Arb (Set S5)) +prop_boolean_Set = + (fmap . fmap) counterExampleProperty . prop_BooleanAlgebra + +-- +-- Non Boolean algebra tests +-- + +type NonBooleanProp a = a -> Property + +prop_non_boolean_LiftedBool + :: NonBooleanProp (Arb (Lifted Bool)) +prop_non_boolean_LiftedBool = + expectFailure + . counterExampleProperty @String + . prop_not + +prop_non_boolean_DroppedBool + :: NonBooleanProp (Arb (Dropped Bool)) +prop_non_boolean_DroppedBool = + expectFailure + . counterExampleProperty @String + . prop_not + +prop_non_boolean_LevitatedOrderedInt + :: NonBooleanProp (Arb (Levitated (Arb (Ordered Int)))) +prop_non_boolean_LevitatedOrderedInt = + expectFailure + . counterExampleProperty @String + . prop_not + +-- +-- Heyting algebra tests +-- + +type HeytingProp a = a -> a -> a -> Property + +prop_heyting_LiftedBool + :: HeytingProp (Arb (Lifted Bool)) +prop_heyting_LiftedBool = + (fmap . fmap) counterExampleProperty . prop_HeytingAlgebra + +prop_heyting_DroppedBool + :: HeytingProp (Arb (Dropped Bool)) +prop_heyting_DroppedBool = + (fmap . fmap) counterExampleProperty . prop_HeytingAlgebra + +prop_heyting_LayeredBoolBool + :: HeytingProp (Arb (Layered Bool Bool)) +prop_heyting_LayeredBoolBool = + (fmap . fmap) counterExampleProperty . prop_HeytingAlgebra + +prop_heyting_LevitatedBool + :: HeytingProp (Arb (Levitated Bool)) +prop_heyting_LevitatedBool = + (fmap . fmap) counterExampleProperty . prop_HeytingAlgebra + +prop_heyting_LayeredLiftedDropped + :: HeytingProp (Arb (Layered (Arb (Lifted Bool)) (Arb (Dropped Bool)))) +prop_heyting_LayeredLiftedDropped = + (fmap . fmap) counterExampleProperty . prop_HeytingAlgebra + +prop_heyting_LevitatedOrderedInt + :: HeytingProp (Arb (Levitated (Arb (Ordered Int)))) +prop_heyting_LevitatedOrderedInt = + (fmap . fmap) counterExampleProperty . prop_HeytingAlgebra + +prop_heyting_MapS5Bool + :: HeytingProp (Arb (Map S5 Bool)) +prop_heyting_MapS5Bool = + (fmap . fmap) counterExampleProperty . prop_HeytingAlgebra + +prop_heyting_DroppedLiftedBool + :: HeytingProp (Composed Dropped Lifted Bool) +prop_heyting_DroppedLiftedBool = + (fmap . fmap) counterExampleProperty . prop_HeytingAlgebra + +prop_heyting_LiftedDroppedBool + :: HeytingProp (Composed Lifted Dropped Bool) +prop_heyting_LiftedDroppedBool = + (fmap . fmap) counterExampleProperty . prop_HeytingAlgebra + +prop_heyting_LiftedLiftedBool + :: HeytingProp (Composed Lifted Lifted Bool) +prop_heyting_LiftedLiftedBool = + (fmap . fmap) counterExampleProperty . prop_HeytingAlgebra + +prop_heyting_DroppedDroppedBool + :: HeytingProp (Composed Dropped Dropped Bool) +prop_heyting_DroppedDroppedBool = + (fmap . fmap) counterExampleProperty . prop_HeytingAlgebra + +prop_heyting_CounterExample + :: HeytingProp (Composed Lifted Op (Set S5)) +prop_heyting_CounterExample = + (fmap . fmap) counterExampleProperty . prop_HeytingAlgebra + +-- | Arbitrary wrapper for varous lattices. +-- newtype Arb a = Arb a deriving ( JoinSemiLattice , BoundedJoinSemiLattice @@ -42,24 +219,24 @@ 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 +instance (Finite k, Arbitrary k, Arbitrary v, Ord k) => Arbitrary (Arb (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) + [ (1, return $ Arb Map.empty) + , (1, Arb . Map.fromList . zip universe <$> vectorOf (length (universe @k)) arbitrary) + , (8, Arb . Map.fromList <$> arbitrary) ] instance Arbitrary a => Arbitrary (Arb (Lifted a)) where arbitrary = Arb . maybe Bottom Lift <$> arbitrary shrink (Arb Bottom) = [] shrink (Arb (Lift a)) = - Arb Bottom : [ Arb (Lift a') | a' <- shrink a ] + Arb Bottom : (Arb . Lift <$> shrink a) instance Arbitrary a => Arbitrary (Arb (Dropped a)) where arbitrary = Arb . maybe Top Drop <$> arbitrary shrink (Arb Top) = [] shrink (Arb (Drop a)) = - Arb Top : [ Arb (Drop a') | a' <- shrink a ] + Arb Top : (Arb . Drop <$> shrink a) instance Arbitrary a => Arbitrary (Arb (Levitated a)) where arbitrary = frequency @@ -75,9 +252,14 @@ : [ Arb (L.Levitate a') | a' <- shrink a ] shrink (Arb L.Top) = [] +instance (Arbitrary a, HeytingAlgebra a, Eq a) => Arbitrary (Arb (Boolean a)) where + arbitrary = Arb . boolean <$> arbitrary + shrink (Arb a) = filter (/= Arb a) (Arb . boolean <$> shrink (runBoolean a)) + + instance Arbitrary a => Arbitrary (Arb (Ordered a)) where arbitrary = Arb . Ordered <$> arbitrary - shrink (Arb (Ordered a)) = [ Arb (Ordered a') | a' <- shrink a ] + shrink (Arb (Ordered a)) = Arb . Ordered <$> shrink a data S5 = S1 | S2 | S3 | S4 | S5 deriving (Ord, Eq, Show) @@ -90,20 +272,20 @@ instance Arbitrary S5 where arbitrary = elements universe -instance (Arbitrary a, Ord a) => Arbitrary (Arb (S.Set a)) where - arbitrary = Arb . S.fromList <$> arbitrary - shrink (Arb as) = [ Arb (S.fromList as') | as' <- shrink (S.toList as) ] +instance (Arbitrary a, Ord a) => Arbitrary (Arb (Set a)) where + arbitrary = Arb . Set.fromList <$> arbitrary + shrink (Arb as) = [ Arb (Set.fromList as') | as' <- shrink (Set.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 ] + shrink (Arb (Lower a)) = Arb . Lower <$> shrink a + shrink (Arb (Upper b)) = Arb . Upper <$> shrink b --- Another arbitrary newtype wrapper; using tagged type let us avoid --- overlapping instances. +-- | Arbitrary newtype wrapper for compositions of heigher kinded types. +-- newtype Composed f g a = Composed (f (g a)) deriving ( JoinSemiLattice , BoundedJoinSemiLattice @@ -124,40 +306,40 @@ arbitrary = frequency [ (1, return $ Composed Top) , (1, return $ Composed (Drop Bottom)) - , (2, Composed . Drop . Lift <$> arbitrary) + , (8, Composed . Drop . Lift <$> arbitrary) ] shrink (Composed Top) = [] shrink (Composed (Drop Bottom)) = [Composed Top] shrink (Composed (Drop (Lift a))) = - [ Composed Top, Composed (Drop Bottom) ] + [ Composed (Drop 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 <$> arbitrary) + , (8, Composed . Lift . Drop <$> 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 <$> arbitrary) + , (8, Composed . Lift . Lift <$> arbitrary) ] shrink (Composed Bottom) = [] shrink (Composed (Lift Bottom)) = [Composed Bottom] shrink (Composed (Lift (Lift a))) = - [ Composed Bottom, Composed (Lift Bottom) ] + [ Composed (Lift 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 <$> arbitrary) + , (8, Composed . Drop . Drop <$> arbitrary) ] shrink (Composed Top) = [] @@ -166,36 +348,8 @@ [ 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 "Boolean algebras" - [ 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 "Non Boolean algebras" - [ testProperty "Not a BooleanAlgebra (Lifted Bool)" $ expectFailure $ prop_not @(Arb (Lifted Bool)) - , testProperty "Not a BooleanAlgebra (Dropped Bool)" $ expectFailure $ prop_not @(Arb (Dropped Bool)) - , testProperty "Not a BooleanAlgebra Levitated (Ordered Int)" $ expectFailure $ prop_BooleanAlgebra @(Arb (Levitated (Arb (Ordered Int)))) - ] - , testGroup "Heyting algebras" - [ 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 "Levitated (Ordered Int)" $ prop_HeytingAlgebra @(Arb (Levitated (Arb (Ordered Int)))) - , 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) - ] +instance Arbitrary a => Arbitrary (Composed Lifted Op a) where + arbitrary = frequency + [ (1, return (Composed Bottom)) + , (9, Composed . Lift . Op <$> arbitrary) ]