{- |

Module : $Header$

Description : Grothendieck logic (flattening of logic graph to a single logic)

Copyright : (c) Till Mossakowski, and Uni Bremen 2002-2006

License : similar to LGPL, see HetCATS/LICENSE.txt or LIZENZ.txt

Maintainer : till@informatik.uni-bremen.de

Stability : provisional

Portability : non-portable (overlapping instances, dynamics, existentials)

Grothendieck logic (flattening of logic graph to a single logic)

The Grothendieck logic is defined to be the

heterogeneous logic over the logic graph.

This will be the logic over which the data

structures and algorithms for specification in-the-large

are built.

This module heavily works with existential types, see

<http://haskell.org/hawiki/ExistentialTypes> and chapter 7 of /Heterogeneous

specification and the heterogeneous tool set/

(<http://www.informatik.uni-bremen.de/~till/papers/habil.ps>).

References:

R. Diaconescu:

Grothendieck institutions

J. applied categorical structures 10, 2002, p. 383-402.

T. Mossakowski:

Comorphism-based Grothendieck institutions.

In K. Diks, W. Rytter (Eds.), Mathematical foundations of computer science,

LNCS 2420, pp. 593-604

T. Mossakowski:

Heterogeneous specification and the heterogeneous tool set.

-}

module Logic.Grothendieck

( G_basic_spec (..)

, G_sign (..)

, SigId(..)

, startSigId

, isHomSubGsign

, isSubGsign

, langNameSig

, G_symbol (..)

, G_symb_items_list (..)

, G_symb_map_items_list (..)

, G_sublogics (..)

, isProperSublogic

, G_morphism (..)

, MorId(..)

, startMorId

, mkG_morphism

, AnyComorphism (..)

, idComorphism

, isIdComorphism

, isInclComorphism

, isModelTransportable

, hasModelExpansion

, isWeaklyAmalgamable

, compComorphism

, lessSublogicComor

, AnyMorphism (..)

, AnyModification (..)

, idModification

, vertCompModification

, horCompModification

, LogicGraph (..)

, emptyLogicGraph

, lookupLogic

, lookupCurrentLogic

, logicUnion

, lookupCompComorphism

, lookupComorphism

, lookupModification

, GMorphism (..)

, isHomogeneous

, Grothendieck (..)

, normalize

, gEmbed

, gEmbed2

, gEmbedComorphism

, gsigUnion

, gsigManyUnion

, homogeneousMorManyUnion

, logicInclusion

, updateMorIndex

, toG_morphism

, ginclusion

, compInclusion

, compHomInclusion

, findComorphismPaths

, findComorphism

, isTransportable

, G_prover (..)

, getProverName

, coerceProver

, G_cons_checker (..)

, coerceConsChecker

, Square (..)

, LaxTriangle (..)

, mkIdSquare

, mirrorSquare

) where

import Logic.Logic

import Logic.ExtSign

import Logic.Prover

import Logic.Comorphism

import Logic.Morphism

import Logic.Coerce

import Common.ExtSign

import Common.Doc

import Common.DocUtils

import qualified Data.Map as Map

import qualified Data.Set as Set

import Common.Result

import Common.Utils

import Data.Typeable

import Data.List (nub)

import Data.Maybe (catMaybes)

import Control.Monad (foldM)

import Logic.Modification

import Text.ParserCombinators.Parsec (Parser, try, parse, eof, string, (<|>))

-- for looking up modifications

import Common.Token

import Common.Lexer

import Common.Id

-- * \"Grothendieck\" versions of the various parts of type class Logic

-- | Grothendieck basic specifications

data G_basic_spec = forall lid sublogics

basic_spec sentence symb_items symb_map_items

sign morphism symbol raw_symbol proof_tree .

Logic lid sublogics

basic_spec sentence symb_items symb_map_items

sign morphism symbol raw_symbol proof_tree =>

G_basic_spec lid basic_spec

deriving Typeable

instance Show G_basic_spec where

show (G_basic_spec _ s) = show s

instance Pretty G_basic_spec where

pretty (G_basic_spec _ s) = pretty s

-- | index for signatures

newtype SigId = SigId Int deriving (Typeable, Show, Eq, Ord, Enum)

startSigId :: SigId

startSigId = SigId 0

{- | Grothendieck signatures with an lookup index. Zero indices

indicate unknown ones. It would be nice to have special (may be

negative) indices for empty signatures (one for every logic). -}

data G_sign = forall lid sublogics

basic_spec sentence symb_items symb_map_items

sign morphism symbol raw_symbol proof_tree .

Logic lid sublogics

basic_spec sentence symb_items symb_map_items

sign morphism symbol raw_symbol proof_tree => G_sign

{ gSignLogic :: lid

, gSign :: (ExtSign sign symbol)

, gSignSelfIdx :: SigId -- ^ index to lookup this 'G_sign' in sign map

} deriving Typeable

instance Eq G_sign where

G_sign l1 sigma1 s1 == G_sign l2 sigma2 s2 =

(s1 > startSigId && s2 > startSigId && s1 == s2) ||

coerceSign l1 l2 "Eq G_sign" sigma1 == Just sigma2

-- | prefer a faster subsignature test if possible

isHomSubGsign :: G_sign -> G_sign -> Bool

isHomSubGsign (G_sign l1 sigma1 s1) (G_sign l2 sigma2 s2) =

(s1 > startSigId && s2 > startSigId && s1 == s2) ||

maybe False (ext_is_subsig l1 sigma1)

(coerceSign l2 l1 "isHomSubGsign" sigma2)

isSubGsign :: LogicGraph -> G_sign -> G_sign -> Bool

isSubGsign lg (G_sign lid1 (ExtSign sigma1 _) _)

(G_sign lid2 (ExtSign sigma2 _) _) =

Just True ==

do Comorphism cid <- resultToMaybe $

logicInclusion lg (Logic lid1) (Logic lid2)

sigma1' <- coercePlainSign lid1 (sourceLogic cid)

"Grothendieck.isSubGsign: cannot happen" sigma1

sigma2' <- coercePlainSign lid2 (targetLogic cid)

"Grothendieck.isSubGsign: cannot happen" sigma2

sigma1t <- resultToMaybe $ map_sign cid sigma1'

return $ is_subsig (targetLogic cid) (fst sigma1t) sigma2'

instance Show G_sign where

show (G_sign _ s _) = show s

instance Pretty G_sign where

pretty (G_sign _ (ExtSign s _) _) = pretty s

langNameSig :: G_sign -> String

langNameSig (G_sign lid _ _) = language_name lid

-- | Grothendieck symbols

data G_symbol = forall lid sublogics

basic_spec sentence symb_items symb_map_items

sign morphism symbol raw_symbol proof_tree .

Logic lid sublogics

basic_spec sentence symb_items symb_map_items

sign morphism symbol raw_symbol proof_tree =>

G_symbol lid symbol

deriving Typeable

instance Show G_symbol where

show (G_symbol _ s) = show s

instance Pretty G_symbol where

pretty (G_symbol _ s) = pretty s

instance Eq G_symbol where

(G_symbol i1 s1) == (G_symbol i2 s2) =

language_name i1 == language_name i2 && coerceSymbol i1 i2 s1 == s2

-- | Grothendieck symbol lists

data G_symb_items_list = forall lid sublogics

basic_spec sentence symb_items symb_map_items

sign morphism symbol raw_symbol proof_tree .

Logic lid sublogics

basic_spec sentence symb_items symb_map_items

sign morphism symbol raw_symbol proof_tree =>

G_symb_items_list lid [symb_items]

deriving Typeable

instance Show G_symb_items_list where

show (G_symb_items_list _ l) = show l

instance Pretty G_symb_items_list where

pretty (G_symb_items_list _ l) = ppWithCommas l

instance Eq G_symb_items_list where

(G_symb_items_list i1 s1) == (G_symb_items_list i2 s2) =

coerceSymbItemsList i1 i2 "Eq G_symb_items_list" s1 == Just s2

-- | Grothendieck symbol maps

data G_symb_map_items_list = forall lid sublogics

basic_spec sentence symb_items symb_map_items

sign morphism symbol raw_symbol proof_tree .

Logic lid sublogics

basic_spec sentence symb_items symb_map_items

sign morphism symbol raw_symbol proof_tree =>

G_symb_map_items_list lid [symb_map_items]

deriving Typeable

instance Show G_symb_map_items_list where

show (G_symb_map_items_list _ l) = show l

instance Pretty G_symb_map_items_list where

pretty (G_symb_map_items_list _ l) = ppWithCommas l

instance Eq G_symb_map_items_list where

(G_symb_map_items_list i1 s1) == (G_symb_map_items_list i2 s2) =

coerceSymbMapItemsList i1 i2 "Eq G_symb_map_items_list" s1 == Just s2

-- | Grothendieck sublogics

data G_sublogics = forall lid sublogics

basic_spec sentence symb_items symb_map_items

sign morphism symbol raw_symbol proof_tree .

Logic lid sublogics

basic_spec sentence symb_items symb_map_items

sign morphism symbol raw_symbol proof_tree =>

G_sublogics lid sublogics

deriving Typeable

instance Show G_sublogics where

show (G_sublogics lid sub) = case sublogic_names sub of

[] -> error "show G_sublogics"

h : _ -> show lid ++ (if null h then "" else "." ++ h)

instance Eq G_sublogics where

g1 == g2 = compare g1 g2 == EQ

instance Ord G_sublogics where

compare (G_sublogics lid1 l1) (G_sublogics lid2 l2) =

case compare (language_name lid1) $ language_name lid2 of

EQ -> compare (forceCoerceSublogic lid1 lid2 l1) l2

r -> r

isProperSublogic :: G_sublogics -> G_sublogics -> Bool

isProperSublogic a@(G_sublogics lid1 l1) b@(G_sublogics lid2 l2) =

isSubElem (forceCoerceSublogic lid1 lid2 l1) l2 && a /= b

-- | index for morphisms

newtype MorId = MorId Int deriving (Typeable, Show, Eq, Ord, Enum)

startMorId :: MorId

startMorId = MorId 0

{- | Homogeneous Grothendieck signature morphisms with indices. For

the domain index it would be nice it showed also the emptiness. -}

data G_morphism = forall lid sublogics

basic_spec sentence symb_items symb_map_items

sign morphism symbol raw_symbol proof_tree .

Logic lid sublogics

basic_spec sentence symb_items symb_map_items

sign morphism symbol raw_symbol proof_tree => G_morphism

{ gMorphismLogic :: lid

, gMorphism :: morphism

, gMorphismSelfIdx :: MorId -- ^ lookup index in morphism map

} deriving Typeable

instance Show G_morphism where

show (G_morphism _ m _) = show m

instance Pretty G_morphism where

pretty (G_morphism _ m _) = pretty m

mkG_morphism :: forall lid sublogics

basic_spec sentence symb_items symb_map_items

sign morphism symbol raw_symbol proof_tree .

Logic lid sublogics

basic_spec sentence symb_items symb_map_items

sign morphism symbol raw_symbol proof_tree

=> lid -> morphism -> G_morphism

mkG_morphism l m = G_morphism l m startMorId

-- * Comorphisms and existential types for the logic graph

-- | Existential type for comorphisms

data AnyComorphism = forall cid lid1 sublogics1

basic_spec1 sentence1 symb_items1 symb_map_items1

sign1 morphism1 symbol1 raw_symbol1 proof_tree1

lid2 sublogics2

basic_spec2 sentence2 symb_items2 symb_map_items2

sign2 morphism2 symbol2 raw_symbol2 proof_tree2 .

Comorphism cid

lid1 sublogics1 basic_spec1 sentence1

symb_items1 symb_map_items1

sign1 morphism1 symbol1 raw_symbol1 proof_tree1

lid2 sublogics2 basic_spec2 sentence2

symb_items2 symb_map_items2

sign2 morphism2 symbol2 raw_symbol2 proof_tree2 =>

Comorphism cid

deriving Typeable

instance Eq AnyComorphism where

Comorphism cid1 == Comorphism cid2 =

constituents cid1 == constituents cid2

-- need to be refined, using comorphism translations !!!

instance Ord AnyComorphism where

Comorphism cid1 < Comorphism cid2 =

constituents cid1 < constituents cid2

instance Show AnyComorphism where

show (Comorphism cid) = language_name cid

++ " : " ++ language_name (sourceLogic cid)

++ " -> " ++ language_name (targetLogic cid)

instance Pretty AnyComorphism where

pretty = text . show

-- | compute the identity comorphism for a logic

idComorphism :: AnyLogic -> AnyComorphism

idComorphism (Logic lid) = Comorphism (mkIdComorphism lid (top_sublogic lid))

-- | Test whether a comporphism is the identity

isIdComorphism :: AnyComorphism -> Bool

isIdComorphism (Comorphism cid) =

constituents cid == []

-- | Test whether a comorphism is an ad-hoc inclusion

isInclComorphism :: AnyComorphism -> Bool

isInclComorphism (Comorphism cid) =

Logic (sourceLogic cid) == Logic (targetLogic cid) &&

(isProperSublogic (G_sublogics (sourceLogic cid) (sourceSublogic cid))

(G_sublogics (targetLogic cid) (targetSublogic cid)))

-- * Properties of comorphisms

-- | Test whether a comorphism is model-transportable

isModelTransportable :: AnyComorphism -> Bool

isModelTransportable (Comorphism cid) = is_model_transportable cid

-- | Test whether a comorphism has model expansion

hasModelExpansion :: AnyComorphism -> Bool

hasModelExpansion (Comorphism cid) = has_model_expansion cid

-- | Test whether a comorphism is weakly amalgamable

isWeaklyAmalgamable :: AnyComorphism -> Bool

isWeaklyAmalgamable (Comorphism cid) = is_weakly_amalgamable cid

-- | Compose comorphisms

compComorphism :: Monad m => AnyComorphism -> AnyComorphism -> m AnyComorphism

compComorphism (Comorphism cid1) (Comorphism cid2) =

let l1 = targetLogic cid1

l2 = sourceLogic cid2 in

if language_name l1 == language_name l2 then

if isSubElem (forceCoerceSublogic l1 l2 $ targetSublogic cid1)

$ sourceSublogic cid2

then {- case (isIdComorphism cm1,isIdComorphism cm2) of

(True,_) -> return cm2

(_,True) -> return cm1

_ -> -} return $ Comorphism (CompComorphism cid1 cid2)

else fail ("Sublogic mismatch in composition of "++language_name cid1++

" and "++language_name cid2)

else fail ("Logic mismatch in composition of "++language_name cid1++

" and "++language_name cid2)

-- | check if sublogic fits for comorphism

lessSublogicComor :: G_sublogics -> AnyComorphism -> Bool

lessSublogicComor (G_sublogics lid1 sub1) (Comorphism cid) =

let lid2 = sourceLogic cid

in language_name lid2 == language_name lid1

&& isSubElem (forceCoerceSublogic lid1 lid2 sub1) (sourceSublogic cid)

-- * Morphisms

-- | Existential type for morphisms

data AnyMorphism = forall cid lid1 sublogics1

basic_spec1 sentence1 symb_items1 symb_map_items1

sign1 morphism1 symbol1 raw_symbol1 proof_tree1

lid2 sublogics2

basic_spec2 sentence2 symb_items2 symb_map_items2

sign2 morphism2 symbol2 raw_symbol2 proof_tree2 .

Morphism cid

lid1 sublogics1 basic_spec1 sentence1

symb_items1 symb_map_items1

sign1 morphism1 symbol1 raw_symbol1 proof_tree1

lid2 sublogics2 basic_spec2 sentence2

symb_items2 symb_map_items2

sign2 morphism2 symbol2 raw_symbol2 proof_tree2 =>

Morphism cid

deriving Typeable

{-

instance Eq AnyMorphism where

Morphism cid1 == Morphism cid2 =

constituents cid1 == constituents cid2

-- need to be refined, using morphism translations !!!

-}

instance Show AnyMorphism where

show (Morphism cid) =

language_name cid

++" : "++language_name (morSourceLogic cid)

++" -> "++language_name (morTargetLogic cid)

-- * Modifications

data AnyModification = forall

lid cid1 cid2

log1 sublogics1 basic_spec1 sentence1 symb_items1 symb_map_items1

sign1 morphism1 symbol1 raw_symbol1 proof_tree1

log2 sublogics2 basic_spec2 sentence2 symb_items2 symb_map_items2

sign2 morphism2 symbol2 raw_symbol2 proof_tree2

log3 sublogics3 basic_spec3 sentence3 symb_items3 symb_map_items3

sign3 morphism3 symbol3 raw_symbol3 proof_tree3

log4 sublogics4 basic_spec4 sentence4 symb_items4 symb_map_items4

sign4 morphism4 symbol4 raw_symbol4 proof_tree4.

Modification lid cid1 cid2

log1 sublogics1 basic_spec1 sentence1 symb_items1 symb_map_items1

sign1 morphism1 symbol1 raw_symbol1 proof_tree1

log2 sublogics2 basic_spec2 sentence2 symb_items2 symb_map_items2

sign2 morphism2 symbol2 raw_symbol2 proof_tree2

log3 sublogics3 basic_spec3 sentence3 symb_items3 symb_map_items3

sign3 morphism3 symbol3 raw_symbol3 proof_tree3

log4 sublogics4 basic_spec4 sentence4 symb_items4 symb_map_items4

sign4 morphism4 symbol4 raw_symbol4 proof_tree4

=> Modification lid

deriving Typeable

{--

instance Eq AnyModification where

rules for equality

--}

instance Show AnyModification where

show (Modification lid) = language_name lid ++

":" ++ language_name (sourceComorphism lid) ++

"=>" ++ language_name (targetComorphism lid)

idModification :: AnyComorphism -> AnyModification

idModification (Comorphism cid) = Modification (IdModif cid)

-- | vertical compositions of modifications

vertCompModification :: Monad m => AnyModification -> AnyModification

-> m AnyModification

vertCompModification (Modification lid1) (Modification lid2) =

let cid1 = targetComorphism lid1

cid2 = sourceComorphism lid2

in

if language_name cid1 == language_name cid2

then return $ Modification (VerCompModif lid1 lid2)

else fail $ "Comorphism mismatch in composition of" ++ language_name lid1

++ "and" ++ language_name lid2

-- | horizontal composition of modifications

horCompModification :: Monad m => AnyModification -> AnyModification

-> m AnyModification

horCompModification (Modification lid1) (Modification lid2) =

let cid1 = sourceComorphism lid1

cid2 = sourceComorphism lid2

in if language_name (targetLogic cid1) == language_name (sourceLogic cid2)

then return $ Modification (HorCompModif lid1 lid2)

else fail $ "Logic mismatch in composition of" ++ language_name lid1

++ "and" ++ language_name lid2

-- | Logic graph

data LogicGraph = LogicGraph

{ logics :: Map.Map String AnyLogic

, currentLogic :: String

, comorphisms :: Map.Map String AnyComorphism

, inclusions :: Map.Map (String, String) AnyComorphism

, unions :: Map.Map (String, String) (AnyComorphism, AnyComorphism)

, morphisms :: Map.Map String AnyMorphism

, modifications :: Map.Map String AnyModification

, squares :: Map.Map (AnyComorphism, AnyComorphism) [Square]

} deriving Show

emptyLogicGraph :: LogicGraph

emptyLogicGraph = LogicGraph

{ logics = Map.empty

, currentLogic = "CASL"

, comorphisms = Map.empty

, inclusions = Map.empty

, unions = Map.empty

, morphisms = Map.empty

, modifications = Map.empty

, squares = Map.empty }

instance Pretty LogicGraph where

pretty lg = text ("current logic is: " ++ currentLogic lg)

$+$ text "all logics:"

$+$ sepByCommas (map text $ Map.keys $ logics lg)

$+$ text "comorphism inclusions:"

$+$ vcat (map pretty $ Map.elems $ inclusions lg)

$+$ text "all comorphisms:"

$+$ vcat (map pretty $ Map.elems $ comorphisms lg)

-- | find a logic in a logic graph

lookupLogic :: Monad m => String -> String -> LogicGraph -> m AnyLogic

lookupLogic error_prefix logname logicGraph =

case Map.lookup logname $ logics logicGraph of

Nothing -> fail $ error_prefix ++" in LogicGraph logic \""

++ logname ++ "\" unknown"

Just lid -> return lid

lookupCurrentLogic :: Monad m => String -> LogicGraph -> m AnyLogic

lookupCurrentLogic msg lg = lookupLogic msg (currentLogic lg) lg

-- | union to two logics

logicUnion :: LogicGraph -> AnyLogic -> AnyLogic

-> Result (AnyComorphism, AnyComorphism)

logicUnion lg l1@(Logic lid1) l2@(Logic lid2) =

case logicInclusion lg l1 l2 of

Result _ (Just c) -> return (c,idComorphism l2)

_ -> case logicInclusion lg l2 l1 of

Result _ (Just c) -> return (idComorphism l1,c)

_ -> case Map.lookup (ln1,ln2) (unions lg) of

Just u -> return u

Nothing -> case Map.lookup (ln2,ln1) (unions lg) of

Just (c2,c1) -> return (c1,c2)

Nothing -> fail $ "Union of logics " ++ ln1 ++

" and " ++ ln2 ++ " does not exist"

where ln1 = language_name lid1

ln2 = language_name lid2

-- | find a comorphism composition in a logic graph

lookupCompComorphism :: Monad m => [String] -> LogicGraph -> m AnyComorphism

lookupCompComorphism nameList logicGraph = do

cs <- sequence $ map lookupN nameList

case cs of

c:cs1 -> foldM compComorphism c cs1

_ -> fail "Illegal empty comorphism composition"

where

hasSublogic name (Logic lid) = elem name (sublogic_names $ top_sublogic lid)

lookupN name =

case name of

'i':'d':'_':logic -> do

let (mainLogic, subLogicD) = span (/= '.') logic

--subLogicD will begin with a . which has to be removed

l <- maybe (fail ("Cannot find Logic "++mainLogic)) return

$ Map.lookup mainLogic (logics logicGraph)

case hasSublogic (tail subLogicD) l of

True -> return $ idComorphism l

False -> fail ("Error in sublogic name"++logic)

_ -> maybe (fail ("Cannot find logic comorphism "++name)) return

$ Map.lookup name (comorphisms logicGraph)

-- | find a comorphism in a logic graph

lookupComorphism :: Monad m => String -> LogicGraph -> m AnyComorphism

lookupComorphism coname = lookupCompComorphism $ splitOn ';' coname

-- | find a modification in a logic graph

lookupModification :: (Monad m) => String -> LogicGraph -> m AnyModification

lookupModification input lG

= case parse (parseModif lG << eof) "" input of

Left err -> fail $ show err

Right x -> x

parseModif :: (Monad m) => LogicGraph -> Parser (m AnyModification)

parseModif lG = do

(xs, _) <- separatedBy (vertcomp lG) crossT

let r = do y <- sequence xs

case y of

m:ms -> return $ foldM horCompModification m ms

_ -> Nothing

case r of

Nothing -> fail "Illegal empty horizontal composition"

Just m -> return m

vertcomp :: (Monad m) => LogicGraph -> Parser (m AnyModification)

vertcomp lG = do

(xs, _) <- separatedBy (pm lG) semiT

let r = do

y <- sequence xs

case y of

m:ms -> return $ foldM vertCompModification m ms

_ -> Nothing

--r has type Maybe (m AnyModification)

case r of

Nothing -> fail "Illegal empty vertical composition"

Just m -> return m

pm :: (Monad m) => LogicGraph -> Parser (m AnyModification)

pm lG = parseName lG <|> bracks lG

bracks :: (Monad m) => LogicGraph -> Parser (m AnyModification)

bracks lG = do

oParenT

modif <- parseModif lG

cParenT

return modif

parseIdentity :: (Monad m) => LogicGraph -> Parser (m AnyModification)

parseIdentity lG = do

try (string "id_")

tok <- simpleId

let name = tokStr tok

case Map.lookup name (comorphisms lG) of

Nothing -> fail $ "Cannot find comorphism" ++ name

Just x -> return $ return $ idModification x

parseName :: (Monad m) => LogicGraph -> Parser (m AnyModification)

parseName lG = parseIdentity lG <|> do

tok <- simpleId

let name = tokStr tok

case Map.lookup name (modifications lG) of

Nothing -> fail $ "Cannot find modification" ++ name

Just x -> return $ return x

-- * The Grothendieck signature category

-- | Grothendieck signature morphisms with indices

data GMorphism = forall cid lid1 sublogics1

basic_spec1 sentence1 symb_items1 symb_map_items1

sign1 morphism1 symbol1 raw_symbol1 proof_tree1

lid2 sublogics2

basic_spec2 sentence2 symb_items2 symb_map_items2

sign2 morphism2 symbol2 raw_symbol2 proof_tree2 .

Comorphism cid

lid1 sublogics1 basic_spec1 sentence1

symb_items1 symb_map_items1

sign1 morphism1 symbol1 raw_symbol1 proof_tree1

lid2 sublogics2 basic_spec2 sentence2

symb_items2 symb_map_items2

sign2 morphism2 symbol2 raw_symbol2 proof_tree2 => GMorphism

{ gMorphismComor :: cid

, gMorphismSign :: ExtSign sign1 symbol1

, gMorphismSignIdx :: SigId -- ^ 'G_sign' index of source signature

, gMorphismMor :: morphism2

, gMorphismMorIdx :: MorId -- ^ `G_morphism index of target morphism

} deriving Typeable

instance Eq GMorphism where

GMorphism cid1 sigma1 in1 mor1 in1' == GMorphism cid2 sigma2 in2 mor2 in2'

= Comorphism cid1 == Comorphism cid2

&& G_sign (sourceLogic cid1) sigma1 in1 ==

G_sign (sourceLogic cid2) sigma2 in2

&& (in1' > startMorId && in2' > startMorId && in1' == in2'

|| coerceMorphism (targetLogic cid1) (targetLogic cid2)

"Eq GMorphism.coerceMorphism" mor1 == Just mor2)

isHomogeneous :: GMorphism -> Bool

isHomogeneous (GMorphism cid _ _ _ _) =

isIdComorphism (Comorphism cid)

data Grothendieck = Grothendieck deriving (Typeable, Show)

instance Language Grothendieck

instance Show GMorphism where

show (GMorphism cid s _ m _) =

show (normalize (Comorphism cid)) ++ "(" ++ show s ++ ")" ++ show m

instance Pretty GMorphism where

pretty (GMorphism cid (ExtSign s _) _ m _) = let c = Comorphism cid in fsep

[ text $ show $ normalize c

, if isIdComorphism c then empty else specBraces $ space <> pretty s

, pretty m ]

normalize :: AnyComorphism -> AnyComorphism

normalize = id

{- todo: somthing like the following...

normalize (Comorphism cid) =

case cid of

CompComorphism r1 r2 ->

case (normalize (Comorphism r1), normalize (Comorphism r2)) of

(Comorphism n1, Comorphism n2) ->

if isIdComorphism (Comorphism n1) then Comorphism n2

else if isIdComorphism (Comorphism n2) then Comorphism n1

else Comorphism (CompComorphism n1 n2)

_ -> Comorphism cid

-}

-- signature category of the Grothendieck institution

instance Category G_sign GMorphism where

ide (G_sign lid sigma@(ExtSign s _) ind) =

GMorphism (mkIdComorphism lid (top_sublogic lid))

sigma ind (ide s) startMorId

-- composition of Grothendieck signature morphisms

comp (GMorphism r1 sigma1 ind1 mor1 _)

(GMorphism r2 _sigma2 _ mor2 _) =

do let lid1 = sourceLogic r1

lid2 = targetLogic r1

lid3 = sourceLogic r2

lid4 = targetLogic r2

-- coercion between target of first and

-- source of second Grothendieck morphism

mor1' <- coerceMorphism lid2 lid3 "Grothendieck.comp" mor1

-- map signature morphism component of first Grothendieck morphism

-- along the comorphism component of the second one ...

mor1'' <- map_morphism r2 mor1'

-- and then compose the result with the signature morphism component

-- of first one

mor <- comp mor1'' mor2

-- if the first comorphism is the identity...

if isIdComorphism (Comorphism r1) &&

case coerceSublogic lid2 lid3 "Grothendieck.comp"

(targetSublogic r1) of

Just sl1 -> maybe (error ("Logic.Grothendieck: Category "++

"instance.comp: no mapping for " ++

show sl1))

(isSubElem (targetSublogic r2))

(mapSublogic r2 sl1)

_ -> False

-- ... then things simplify ...

then do

sigma1' <- coerceSign lid1 lid3 "Grothendieck.comp" sigma1

return (GMorphism r2 sigma1' ind1 mor startMorId)

-- ... also, if the second comorphism is the identity ...

else if isIdComorphism (Comorphism r2)

then do mor2' <- coerceMorphism lid4 lid2 "Grothendieck.comp" mor2

mor' <- comp mor1 mor2'

return (GMorphism r1 sigma1 ind1 mor' startMorId)

-- ... if not, use the composite comorphism

else return $ GMorphism (CompComorphism r1 r2)

sigma1 ind1 mor startMorId

dom (GMorphism r sigma ind _mor _) =

G_sign (sourceLogic r) sigma ind

cod (GMorphism r (ExtSign _ sys) _ mor _) =

let lid1 = sourceLogic r

lid2 = targetLogic r

sig2 = cod mor

in case coerceSymbolSet lid1 lid2 "" sys of

Nothing -> G_sign lid2 (makeExtSign lid2 sig2) startSigId

Just sys2 -> G_sign lid2 (ExtSign sig2 $ Set.map (\ sy ->

Map.findWithDefault sy sy $ symmap_of lid2 mor) sys2) startSigId

legal_mor (GMorphism r (ExtSign s _) _ mor _) =

legal_mor mor &&

case maybeResult $ map_sign r s of

Just (sigma',_) -> sigma' == cod mor

Nothing -> False

-- | Embedding of homogeneous signature morphisms as Grothendieck sig mors

gEmbed2 :: G_sign -> G_morphism -> GMorphism

gEmbed2 (G_sign lid2 sig si) (G_morphism lid mor ind) =

let cid = mkIdComorphism lid (top_sublogic lid)

Just sig1 = coerceSign lid2 (sourceLogic cid) "gEmbed2" sig

in GMorphism cid sig1 si mor ind

-- | Embedding of homogeneous signature morphisms as Grothendieck sig mors

gEmbed :: G_morphism -> GMorphism

gEmbed (G_morphism lid mor ind) = let sig = dom mor in

GMorphism (mkIdComorphism lid (top_sublogic lid))

(makeExtSign lid sig) startSigId mor ind

-- | Embedding of comorphisms as Grothendieck sig mors

gEmbedComorphism :: AnyComorphism -> G_sign -> Result GMorphism

gEmbedComorphism (Comorphism cid) (G_sign lid sig ind) = do

sig'@(ExtSign s _) <- coerceSign lid (sourceLogic cid) "gEmbedComorphism" sig

(sigTar,_) <- map_sign cid s

return (GMorphism cid sig' ind (ide sigTar) startMorId)

-- | heterogeneous union of two Grothendieck signatures

gsigUnion :: LogicGraph -> G_sign -> G_sign -> Result G_sign

gsigUnion lg gsig1@(G_sign lid1 (ExtSign sigma1 _) _)

gsig2@(G_sign lid2 (ExtSign sigma2 _) _) =

if language_name lid1 == language_name lid2

then homogeneousGsigUnion gsig1 gsig2

else do

(Comorphism cid1, Comorphism cid2) <-

logicUnion lg (Logic lid1) (Logic lid2)

let lidS1 = sourceLogic cid1

lidS2 = sourceLogic cid2

lidT1 = targetLogic cid1

lidT2 = targetLogic cid2

sigma1' <- coercePlainSign lid1 lidS1 "Union of signaturesa" sigma1

sigma2' <- coercePlainSign lid2 lidS2 "Union of signaturesb" sigma2

(sigma1'',_) <- map_sign cid1 sigma1' -- where to put axioms???

(sigma2'',_) <- map_sign cid2 sigma2' -- where to put axioms???

sigma2''' <- coercePlainSign lidT2 lidT1 "Union of signaturesc" sigma2''

sigma3 <- signature_union lidT1 sigma1'' sigma2'''

return (G_sign lidT1 (makeExtSign lidT1 sigma3) startSigId)

-- | homogeneous Union of two Grothendieck signatures

homogeneousGsigUnion :: G_sign -> G_sign -> Result G_sign

homogeneousGsigUnion (G_sign lid1 sigma1 _) (G_sign lid2 sigma2 _) = do

sigma2' <- coerceSign lid2 lid1 "Union of signaturesd" sigma2

sigma3 <- ext_signature_union lid1 sigma1 sigma2'

return (G_sign lid1 sigma3 startSigId)

-- | union of a list of Grothendieck signatures

gsigManyUnion :: LogicGraph -> [G_sign] -> Result G_sign

gsigManyUnion _ [] =

fail "union of emtpy list of signatures"

gsigManyUnion lg (gsigma : gsigmas) =

foldM (gsigUnion lg) gsigma gsigmas

-- | homogeneous Union of a list of morphisms

homogeneousMorManyUnion :: [G_morphism] -> Result G_morphism

homogeneousMorManyUnion [] =

fail "homogeneous union of emtpy list of morphisms"

homogeneousMorManyUnion (gmor : gmors) =

foldM ( \ (G_morphism lid2 mor2 _) (G_morphism lid1 mor1 _) -> do

mor1' <- coerceMorphism lid1 lid2 "homogeneousMorManyUnion" mor1

mor <- morphism_union lid2 mor1' mor2

return (G_morphism lid2 mor startMorId)) gmor gmors

-- | inclusion between two logics

logicInclusion :: LogicGraph -> AnyLogic -> AnyLogic -> Result AnyComorphism

logicInclusion logicGraph l1@(Logic lid1) (Logic lid2) =

let ln1 = language_name lid1

ln2 = language_name lid2 in

if ln1==ln2 then

return (idComorphism l1)

else case Map.lookup (ln1,ln2) (inclusions logicGraph) of

Just (Comorphism i) ->

return (Comorphism i)

Nothing ->

fail ("No inclusion from "++ln1++" to "++ln2++" found")

updateMorIndex :: MorId -> GMorphism -> GMorphism

updateMorIndex i (GMorphism cid sign si mor _) = GMorphism cid sign si mor i

toG_morphism :: GMorphism -> G_morphism

toG_morphism (GMorphism cid _ _ mor i) = G_morphism (targetLogic cid) mor i

-- | inclusion morphism between two Grothendieck signatures

ginclusion :: LogicGraph -> G_sign -> G_sign -> Result GMorphism

ginclusion logicGraph (G_sign lid1 sigma1 ind) (G_sign lid2 sigma2 _) = do

Comorphism i <- logicInclusion logicGraph (Logic lid1) (Logic lid2)

ext1@(ExtSign sigma1' _) <-

coerceSign lid1 (sourceLogic i) "Inclusion of signatures" sigma1

(sigma1'',_) <- map_sign i sigma1'

ExtSign sigma2' _ <-

coerceSign lid2 (targetLogic i) "Inclusion of signatures" sigma2

mor <- inclusion (targetLogic i) sigma1'' sigma2'

return (GMorphism i ext1 ind mor startMorId)

genCompInclusion :: (G_sign -> G_sign -> Result GMorphism)

-> GMorphism -> GMorphism -> Result GMorphism

genCompInclusion f mor1 mor2 = do

let sigma1 = cod mor1

sigma2 = dom mor2

incl <- f sigma1 sigma2

mor <- comp mor1 incl

comp mor mor2

-- | Composition of two Grothendieck signature morphisms

-- | with itermediate inclusion

compInclusion :: LogicGraph -> GMorphism -> GMorphism -> Result GMorphism

compInclusion lg = genCompInclusion (ginclusion lg)

-- | Composition of two Grothendieck signature morphisms

-- | with intermediate homogeneous inclusion

compHomInclusion :: GMorphism -> GMorphism -> Result GMorphism

compHomInclusion = genCompInclusion homInclusion

-- | compose homogeneous inclusions avoiding mapping of signatures

homInclusion :: G_sign -> G_sign -> Result GMorphism

homInclusion (G_sign lid1 sigma1 ind) (G_sign lid2 sigma2 _) =

case idComorphism $ Logic lid2 of

Comorphism i -> do

let tlid = targetLogic i

ext1 <- coerceSign lid1 (sourceLogic i) "homInclusion1" sigma1

ExtSign sigma1'' _ <- coerceSign lid1 tlid "homInclusion2" sigma1

ExtSign sigma2' _ <- coerceSign lid2 tlid "homInclusion3" sigma2

mor <- inclusion tlid sigma1'' sigma2'

return (GMorphism i ext1 ind mor startMorId)

-- | Find all (composites of) comorphisms starting from a given logic

findComorphismPaths :: LogicGraph -> G_sublogics -> [AnyComorphism]

findComorphismPaths lg (G_sublogics lid sub) =

nub $ map fst $ iterateComp (0::Int) [(idc,[idc])]

where

idc = Comorphism (mkIdComorphism lid sub)

coMors = Map.elems $ comorphisms lg

-- compute possible compositions, but only up to depth 3

iterateComp n l = -- (l::[(AnyComorphism,[AnyComorphism])]) =

if n>1 || l==newL then newL else iterateComp (n+1) newL

where

newL = nub (l ++ (concat (map extend l)))

-- extend comorphism list in all directions, but no cylces

extend (coMor, cmps) =

let addCoMor c =

case compComorphism coMor c of

Nothing -> Nothing

Just c1 -> Just (c1, c : cmps)

in catMaybes $ map addCoMor $ filter (not . (`elem` cmps)) $ coMors

-- | finds first comorphism with a matching sublogic

findComorphism ::Monad m => G_sublogics -> [AnyComorphism] -> m AnyComorphism

findComorphism _ [] = fail "No matching comorphism found"

findComorphism gsl@(G_sublogics lid sub) ((Comorphism cid):rest) =

let l2 = sourceLogic cid

rec = findComorphism gsl rest in

if language_name lid == language_name l2

then if isSubElem (forceCoerceSublogic lid l2 sub) $ sourceSublogic cid

then return $ Comorphism cid

else rec

else rec

-- | check transportability of Grothendieck signature morphisms

-- | (currently returns false for heterogeneous morphisms)

isTransportable :: GMorphism -> Bool

isTransportable (GMorphism cid _ ind1 mor ind2) =

ind1 > startSigId && ind2 > startMorId

&& isModelTransportable(Comorphism cid)

&& is_transportable (targetLogic cid) mor

-- * Provers

-- | provers and consistency checkers for specific logics

data G_prover = forall lid sublogics

basic_spec sentence symb_items symb_map_items

sign morphism symbol raw_symbol proof_tree .

Logic lid sublogics

basic_spec sentence symb_items symb_map_items

sign morphism symbol raw_symbol proof_tree =>

G_prover lid (Prover sign sentence sublogics proof_tree)

deriving Typeable

getProverName :: G_prover -> String

getProverName (G_prover _ p) = prover_name p

coerceProver ::

(Logic lid1 sublogics1 basic_spec1 sentence1 symb_items1 symb_map_items1

sign1 morphism1 symbol1 raw_symbol1 proof_tree1,

Logic lid2 sublogics2 basic_spec2 sentence2 symb_items2 symb_map_items2

sign2 morphism2 symbol2 raw_symbol2 proof_tree2,

Monad m) => lid1 -> lid2 -> String

-> Prover sign1 sentence1 sublogics1 proof_tree1

-> m (Prover sign2 sentence2 sublogics2 proof_tree2)

coerceProver l1 l2 msg m1 = primCoerce l1 l2 msg m1

data G_cons_checker = forall lid sublogics

basic_spec sentence symb_items symb_map_items

sign morphism symbol raw_symbol proof_tree .

Logic lid sublogics

basic_spec sentence symb_items symb_map_items

sign morphism symbol raw_symbol proof_tree =>

G_cons_checker lid

(ConsChecker sign sentence sublogics morphism proof_tree)

deriving Typeable

coerceConsChecker ::

(Logic lid1 sublogics1 basic_spec1 sentence1 symb_items1 symb_map_items1

sign1 morphism1 symbol1 raw_symbol1 proof_tree1,

Logic lid2 sublogics2 basic_spec2 sentence2 symb_items2 symb_map_items2

sign2 morphism2 symbol2 raw_symbol2 proof_tree2,

Monad m) => lid1 -> lid2 -> String

-> ConsChecker sign1 sentence1 sublogics1 morphism1 proof_tree1

-> m (ConsChecker sign2 sentence2 sublogics2 morphism2 proof_tree2)

coerceConsChecker l1 l2 msg m1 = primCoerce l1 l2 msg m1

-- * Lax triangles and weakly amalgamable squares of lax triangles

-- a lax triangle looks like:

-- laxTarget

-- i -------------------------------------> k

-- ^ laxModif

-- | |

-- i ------------- > j -------------------> k

-- laxFst laxSnd

--

-- and I_k is quasi-semi-exact

data LaxTriangle = LaxTriangle {

laxModif :: AnyModification,

laxFst, laxSnd, laxTarget :: AnyComorphism

} deriving Show

-- a weakly amalgamable square of lax triangles

-- consists of two lax triangles with the same laxTarget

data Square = Square {

leftTriangle, rightTriangle :: LaxTriangle

} deriving Show

-- for deriving Eq, first equality for modifications is needed

mkIdSquare :: AnyLogic -> Square

mkIdSquare (Logic lid) = let

idCom = Comorphism (mkIdComorphism lid (top_sublogic lid))

idMod = idModification idCom

idTriangle = LaxTriangle{

laxModif = idMod,

laxFst = idCom,

laxSnd = idCom,

laxTarget = idCom}

in Square{leftTriangle = idTriangle, rightTriangle = idTriangle}

mirrorSquare :: Square -> Square

mirrorSquare s = Square{

leftTriangle = rightTriangle s,

rightTriangle = leftTriangle s}