heyting-algebras 0.0.1.2 → 0.0.2.0

raw patch · 14 files changed

+889/−732 lines, 14 filesdep +semiring-simpledep ~QuickCheckdep ~free-algebrasdep ~latticesPVP: major bump suggested

API removals or changes: PVP suggests a major version bump

Dependencies added: semiring-simple

Dependency ranges changed: QuickCheck, free-algebras, lattices

API changes (from Hackage documentation)

Files

• ChangeLog.md +12/−1
• LICENSE +19/−370
• README.md +26/−10
• heyting-algebras.cabal +14/−21
• src/Algebra/Boolean.hs +17/−114
• src/Algebra/Boolean/Free.hs +17/−19
• src/Algebra/Boolean/Properties.hs +36/−0
• src/Algebra/Heyting.hs +144/−92
• src/Algebra/Heyting/BoolRing.hs +48/−0
• src/Algebra/Heyting/CounterExample.hs +102/−0
• src/Algebra/Heyting/Free.hs +28/−21
• src/Algebra/Heyting/Layered.hs +10/−11
• src/Algebra/Heyting/Properties.hs +189/−0
• test/Main.hs +227/−73

@@ -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.

@@ -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. 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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. 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@@ -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).

@@ -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

@@ -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

@@ -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

@@ -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)

@@ -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)

@@ -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

@@ -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 ===

@@ -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

@@ -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

@@ -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

@@ -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)     ]