-- needs ghc -fglasgow-exts

{- HetCATS/Grothendieck.hs

$Id$

Till Mossakowski

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.

References:

R. Diaconescu:

Grothendieck institutions

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

T. Mossakowski:

Heterogeneous development graphs and heterogeneous borrowing

Fossacs 2002, LNCS 2303, p. 326-341

T. Mossakowski: Foundations of heterogeneous specification

Submitted

T. Mossakowski:

Relating CASL with Other Specification Languages:

the Institution Level

Theoretical Computer Science 286, 2002, p. 367-475

Todo:

-}

module Grothendieck where

import Logic

import LogicRepr

import PrettyPrint

import Pretty

import PPUtils (fsep_latex, comma_latex)

import Result

import Id

import Dynamic

------------------------------------------------------------------

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

------------------------------------------------------------------

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

instance Show G_basic_spec where

show (G_basic_spec _ s) = show s

instance PrettyPrint G_basic_spec where

printText0 ga (G_basic_spec _ s) = printText0 ga s

printLatex0 ga (G_basic_spec _ s) = printLatex0 ga s

instance Eq G_basic_spec where

(G_basic_spec i1 s1) == (G_basic_spec i2 s2) =

coerce i1 i2 s1 == Just s2

data G_sentence = 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_sentence lid sentence

instance Show G_sentence where

show (G_sentence _ s) = show s

data G_l_sentence_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_l_sentence lid [(String,sentence)]

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 lid sign

instance Typeable G_sign where

typeOf _ = mkAppTy (mkTyCon "G_sign") []

instance Eq G_sign where

(G_sign i1 sigma1) == (G_sign i2 sigma2) =

coerce i1 i2 sigma1 == Just sigma2

instance Show G_sign where

show (G_sign _ s) = show s

data G_sign_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_sign_list lid [sign]

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

instance Show G_symbol where

show (G_symbol _ s) = show s

instance Eq G_symbol where

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

coerce i1 i2 s1 == Just s2

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]

instance Show G_symb_items_list where

show (G_symb_items_list _ l) = show l

instance PrettyPrint G_symb_items_list where

printText0 ga (G_symb_items_list _ l) =

fsep $ punctuate comma $ map (printText0 ga) l

printLatex0 ga (G_symb_items_list _ l) =

fsep_latex $ punctuate comma_latex $ map (printLatex0 ga) l

instance Eq G_symb_items_list where

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

coerce i1 i2 s1 == Just s2

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]

instance Show G_symb_map_items_list where

show (G_symb_map_items_list _ l) = show l

instance PrettyPrint G_symb_map_items_list where

printText0 ga (G_symb_map_items_list _ l) =

fsep $ punctuate comma $ map (printText0 ga) l

printLatex0 ga (G_symb_map_items_list _ l) =

fsep_latex $ punctuate comma_latex $ map (printLatex0 ga) l

instance Eq G_symb_map_items_list where

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

coerce i1 i2 s1 == Just s2

data G_diagram = 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_diagram lid (Diagram sign morphism)

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

instance Show G_sublogics where

show (G_sublogics _ l) = show l

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 lid morphism

instance Show G_morphism where

show (G_morphism _ l) = show l

----------------------------------------------------------------

-- Existential types for the logic graph

----------------------------------------------------------------

data AnyLogic = 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 =>

Logic lid

data AnyRepresentation = forall 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 .

(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) =>

Repr(LogicRepr 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)

type LogicGraph = ([AnyLogic],[AnyRepresentation])

{- This does not work due to needed ordering:

instance Functor Set where

fmap = mapSet

instance Monad Set where

return = unitSet

m >>= k = unionManySets (setToList (fmap k m))

-}

------------------------------------------------------------------

-- The Grothendieck signature category

------------------------------------------------------------------

-- composition in diagrammatic order

comp_anyrepr :: AnyRepresentation -> AnyRepresentation -> Maybe AnyRepresentation

comp_anyrepr

(Repr (r1 :: LogicRepr 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))

(Repr (r2 :: LogicRepr lid3 sublogics3 basic_spec3 sentence3

symb_items3 symb_map_items3

sign3 morphism3 symbol3 raw_symbol3 proof_tree3

lid4 sublogics4 basic_spec4 sentence4 symb_items4 symb_map_items4

sign4 morphism4 symbol4 raw_symbol4 proof_tree4)) =

do r3 <- coerce (target r1) (source r2) r2 :: Maybe(

LogicRepr lid2 sublogics2 basic_spec2 sentence2

symb_items2 symb_map_items2

sign2 morphism2 symbol2 raw_symbol2 proof_tree2

lid4 sublogics4 basic_spec4 sentence4 symb_items4 symb_map_items4

sign4 morphism4 symbol4 raw_symbol4 proof_tree4)

r4 <- comp_repr r1 r3

return (Repr r4)

data GMorphism = forall 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 .

(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) =>

GMorphism lid1 lid2

(LogicRepr 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)

sign1 morphism2

{-

| 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 =>

GMHomInclusion lid sign sign

| GMHetInclusion G_sign G_sign

-}

instance Eq GMorphism where

(GMorphism lid1 lid2 r1 sigma1 mor1) == (GMorphism lid3 lid4 r2 sigma2 mor2)

= maybe False id

(do r2' <- coerce lid1 lid3 r2

sigma2' <- coerce lid1 lid3 sigma2

mor2' <- coerce lid2 lid4 mor2

return (r1 == r2' && sigma1 == sigma2' && mor1==mor2'))

data Grothendieck = Grothendieck deriving Show

instance Language Grothendieck where

instance Show GMorphism where

show (GMorphism _ _ r s m) = show r ++ "(" ++ show s ++ ")" ++ show m

instance Category Grothendieck G_sign GMorphism where

ide _ (G_sign lid sigma) =

GMorphism lid lid (id_repr lid) sigma (ide lid sigma)

comp _

(GMorphism lid1 lid2 r1 sigma1 mor1)

(GMorphism lid3 lid4 r2 _sigma2 mor2) =

do r2' <- coerce lid2 lid3 r2

r3 <- comp_repr r1 r2'

mor1' <- coerce lid2 lid3 mor1

mor1'' <- map_morphism r2 mor1'

mor <- comp lid4 mor2 mor1''

return (GMorphism lid1 lid4 r3 sigma1 mor)

dom _ (GMorphism lid1 _lid2 _r sigma _mor) = G_sign lid1 sigma

cod _ (GMorphism _lid1 lid2 _r _sigma mor) = G_sign lid2 (cod lid2 mor)

legal_obj _ (G_sign lid sigma) = legal_obj lid sigma

legal_mor _ (GMorphism _lid1 lid2 r sigma mor) =

legal_mor lid2 mor && case map_sign r sigma of

Just (sigma',_) -> sigma' == cod lid2 mor

Nothing -> False

gEmbed :: G_morphism -> GMorphism

gEmbed (G_morphism lid mor) =

GMorphism lid lid (id_repr lid) (dom lid mor) mor

-- homogeneous Union of two Grothendieck signatures

homogeneousGsigUnion :: Pos -> G_sign -> G_sign -> Result G_sign

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

sigma2' <- rcoerce lid2 lid1 pos sigma2

sigma3 <- signature_union lid1 sigma1 sigma2'

return (G_sign lid1 sigma3)

-- homogeneous Union of a list of Grothendieck signatures

homogeneousGsigManyUnion :: Pos -> [G_sign] -> Result G_sign

homogeneousGsigManyUnion pos [] =

fatal_error "homogeneous union of emtpy list of signatures" pos

homogeneousGsigManyUnion pos (G_sign lid sigma : gsigmas) = do

sigmas <- let coerce_lid (G_sign lid1 sigma1) =

rcoerce lid lid1 pos sigma1

in sequence (map coerce_lid gsigmas)

bigSigma <- let sig_union s1 s2 = do

s1' <- s1

signature_union lid s1' s2

in foldl sig_union (return sigma) sigmas

return (G_sign lid bigSigma)

-- homogeneous Union of a list of morphisms

homogeneousMorManyUnion :: Pos -> [G_morphism] -> Result G_morphism

homogeneousMorManyUnion pos [] =

fatal_error "homogeneous union of emtpy list of morphisms" pos

homogeneousMorManyUnion pos (G_morphism lid mor : gmors) = do

mors <- let coerce_lid (G_morphism lid1 mor1) =

rcoerce lid lid1 pos mor1

in sequence (map coerce_lid gmors)

bigMor <- let mor_union s1 s2 = do

s1' <- s1

morphism_union lid s1' s2

in foldl mor_union (return mor) mors

return (G_morphism lid bigMor)

inclusion :: G_sign -> G_sign -> GMorphism

inclusion (G_sign _lid1 _sigma1) (G_sign _lid2 _sigma2) = error "inclusion"