{-# OPTIONS -cpp #-}

{-|

Module : $Header$

Copyright : (c) Jorina F. Gerken, Till Mossakowski, Uni Bremen 2002-2004

Licence : similar to LGPL, see HetCATS/LICENCE.txt or LIZENZ.txt

Maintainer : jfgerken@tzi.de

Stability : provisional

Portability : non-portable(Logic)

Proofs in development graphs.

Follows Sect. IV:4.4 of the CASL Reference Manual.

-}

{-

References:

T. Mossakowski, S. Autexier and D. Hutter:

Extending Development Graphs With Hiding.

H. Hussmann (ed.): Fundamental Approaches to Software Engineering 2001,

Lecture Notes in Computer Science 2029, p. 269-283,

Springer-Verlag 2001.

todo in general:

Order of rule application: try to use local links induced by %implies

for subsumption proofs (such that the %implied formulas need not be

re-proved)

Integrate stuff from Saarbr�cken

Add proof status information

what should be in proof status:

- proofs of thm links according to rules

- cons, def and mono annos, and their proofs

todo for Jorina:

todo:

- bei GlobDecomp hinzuf�gen:

zus�tzlich alle Pfade K<--theta-- M --sigma-->N in den aktuellen

Knoten N, die mit einem HidingDef anfangen, und danach nur GlobalDef

theta ist der Signaturmorphismus des HidingDef's (geht "falsch rum")

sigma ist die Komposition der Signaturmorphismen auf dem restl. Pfad

f�r jeden solchen Pfad: einen HidingThm theta einf�gen. sigma ist

der normale Morphismus (wie bei jedem anderen Link)

siehe auch Seite 302 des CASL Reference Manual

-}

module Proofs.Proofs where

import Options

import Data.Dynamic

import Logic.Logic

import Logic.Prover

import Logic.Grothendieck

import Logic.Comorphism

import Static.DevGraph

import Common.Result

import Common.Lib.Graph

import Common.Lib.Set (fromList)

import qualified Common.Lib.Map as Map

import Data.List(nub)

import Data.Maybe

import Common.Id

import Common.AS_Annotation

import Syntax.AS_Library

import Syntax.Print_AS_Library

#ifdef UNI_PACKAGE

import GUI.HTkUtils

#endif

{-

proof status = (DG0,[(R1,DG1),...,(Rn,DGn)])

DG0 is the development graph resulting from the static analysis

Ri is a list of rules that transforms DGi-1 to DGi

With the list of intermediate proof states, one can easily implement

an undo operation

-}

type ProofStatus = (GlobalContext,LibEnv,[([DGRule],[DGChange])],DGraph)

data DGRule =

TheoremHideShift

| HideTheoremShift

| Borrowing

| ConsShift

| DefShift

| MonoShift

| ConsComp

| DefComp

| MonoComp

| DefToMono

| MonoToCons

| FreeIsMono

| MonoIsFree

| Composition

| GlobDecomp (LEdge DGLinkLab) -- edge in the conclusion

| LocDecomp (LEdge DGLinkLab)

| LocSubsumption (LEdge DGLinkLab)

| GlobSubsumption (LEdge DGLinkLab)

| LocalInference

| BasicInferendge BasicProof

| BasicConsInference Edge BasicConsProof

deriving (Eq, Show)

data DGChange = InsertNode (LNode DGNodeLab)

| DeleteNode (LNode DGNodeLab)

| InsertEdge (LEdge DGLinkLab)

| DeleteEdge (LEdge DGLinkLab)

deriving (Eq,Show)

data BasicProof =

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 =>

BasicProof lid (Proof_status proof_tree)

| Guessed

| Conjectured

| Handwritten

instance Eq BasicProof where

Guessed == Guessed = True

Conjectured == Conjectured = True

Handwritten == Handwritten = True

BasicProof lid1 p1 == BasicProof lid2 p2 =

coerce lid1 lid2 p1 == Just p2

instance Show BasicProof where

show (BasicProof lid1 p1) = show p1

show Guessed = "Guessed"

show Conjectured = "Conjectured"

show Handwritten = "Handwritten"

data BasicConsProof = BasicConsProof -- more detail to be added ...

deriving (Eq, Show)

{- todo: implement apply for GlobDecomp and Subsumption

the list of DGChage must be constructed in parallel to the

new DGraph -}

applyRule :: DGRule -> DGraph -> Maybe ([DGChange],DGraph)

applyRule = error "Proofs.hs:applyRule"

automatic :: ProofStatus -> ProofStatus

automatic proofStatus =

(globalContext, libEnv, newHistory, dgraph)

where

(globalContext, libEnv, history, dgraph)

= automaticAux proofStatus

(newHistoryPart, oldHistory) = splitAt 4 history

(rules, changes) = concatHistoryElems (reverse newHistoryPart)

newHistoryElem = (rules, removeContraryChanges changes)

newHistory = newHistoryElem:oldHistory

removeContraryChanges :: [DGChange] -> [DGChange]

removeContraryChanges [] = []

removeContraryChanges (change:changes) =

if elem contraryChange changes

then removeContraryChanges (removeChange contraryChange changes)

else change:(removeContraryChanges changes)

where

contraryChange = getContraryChange change

getContraryChange :: DGChange -> DGChange

getContraryChange change =

case change of

InsertEdge edge -> DeleteEdge edge

DeleteEdge edge -> InsertEdge edge

InsertNode node -> DeleteNode node

DeleteNode node -> InsertNode node

removeChange :: DGChange -> [DGChange] -> [DGChange]

removeChange change changes =

(takeWhile (change /=) changes)

++(tail (dropWhile (change /=) changes))

automaticAux :: ProofStatus -> ProofStatus

automaticAux = locDecomp.locSubsume.globDecomp.globSubsume

concatHistoryElems :: [([DGRule],[DGChange])] -> ([DGRule],[DGChange])

concatHistoryElems [] = ([],[])

concatHistoryElems ((rules,changes):elems) =

(rules++(fst (concatHistoryElems elems)),changes++(snd (concatHistoryElems elems)))

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

-- global decomposition

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

{- apply rule GlobDecomp to all global theorem links in the current DG

current DG = DGm

add to proof status the pair ([GlobDecomp e1,...,GlobDecomp en],DGm+1)

where e1...en are the global theorem links in DGm

DGm+1 results from DGm by application of GlobDecomp e1,...,GlobDecomp en -}

{- applies global decomposition to all unproven global theorem edges

if possible -}

globDecomp :: ProofStatus -> ProofStatus

globDecomp proofStatus@(globalContext,libEnv,history,dgraph) =

(globalContext, libEnv, ((finalHistoryElem):history), finalDGraph)

where

globalThmEdges = filter isUnprovenGlobalThm (labEdges dgraph)

(newDGraph, newHistoryElem) = globDecompAux dgraph globalThmEdges ([],[])

(finalDGraph, finalHistoryElem)

= removeSuperfluousInsertions newDGraph newHistoryElem

{- removes all superfluous insertions from the list of changes as well as

from the development graph (i.e. insertions of edges that are

equivalent to edges or paths resulting from the other insertions) -}

removeSuperfluousInsertions :: DGraph -> ([DGRule],[DGChange])

-> (DGraph,([DGRule],[DGChange]))

removeSuperfluousInsertions dgraph (rules,changes)

= (newDGraph,(rules,newChanges))

where

localThms = [edge | (InsertEdge edge)

<- filter isLocalThmInsertion changes]

(newDGraph, localThmsToInsert)

= removeSuperfluousEdges dgraph localThms

newChanges = (filter (not.isLocalThmInsertion) changes)

++ [InsertEdge edge | edge <- localThmsToInsert]

{- auxiliary function for globDecomp (above)

actual implementation -}

globDecompAux :: DGraph -> [LEdge DGLinkLab] -> ([DGRule],[DGChange])

-> (DGraph,([DGRule],[DGChange]))

globDecompAux dgraph [] historyElem = (dgraph, historyElem)

globDecompAux dgraph (edge:edges) historyElem =

globDecompAux newDGraph edges newHistoryElem

where

(newDGraph, newChanges) = globDecompForOneEdge dgraph edge

newHistoryElem = (((GlobDecomp edge):(fst historyElem)),

(newChanges++(snd historyElem)))

-- applies global decomposition to a single edge

globDecompForOneEdge :: DGraph -> LEdge DGLinkLab -> (DGraph,[DGChange])

globDecompForOneEdge dgraph edge =

globDecompForOneEdgeAux dgraph edge [] paths

where

pathsToSource

= getAllLocOrHideGlobDefPathsTo dgraph (getSourceNode edge) []

paths = [(node, path++(edge:[]))| (node,path) <- pathsToSource]

{- auxiliary funktion for globDecompForOneEdge (above)

actual implementation -}

globDecompForOneEdgeAux :: DGraph -> LEdge DGLinkLab -> [DGChange] ->

[(Node, [LEdge DGLinkLab])] -> (DGraph,[DGChange])

{- if the list of paths is empty from the beginning, nothing is done

otherwise the unprovenThm edge is replaced by a proven one -}

globDecompForOneEdgeAux dgraph edge@(source,target,edgeLab) changes [] =

if null changes then (dgraph, changes)

else

if isDuplicate provenEdge dgraph changes

then (delLEdge edge dgraph,

((DeleteEdge edge):changes))

else ((insEdge provenEdge (delLEdge edge dgraph)),

((DeleteEdge edge):((InsertEdge provenEdge):changes)))

where

(GlobalThm _ conservativity conservStatus) = (dgl_type edgeLab)

proofBasis = getLabelsOfInsertedEdges changes

provenEdge = (source,

target,

DGLink {dgl_morphism = dgl_morphism edgeLab,

dgl_type =

(GlobalThm (Proven proofBasis)

conservativity conservStatus),

dgl_origin = DGProof}

)

-- for each path an unproven localThm edge is inserted

globDecompForOneEdgeAux dgraph edge@(source,target,edgeLab) changes

((node,path):list) =

if isRedundant newEdge dgraph changes list

then globDecompForOneEdgeAux dgraph edge changes list

else globDecompForOneEdgeAux newGraph edge newChanges list

where

isHiding = not (null path) && isHidingDef (head path)

morphismPath = if isHiding then tail path else path

morphism = case calculateMorphismOfPath morphismPath of

Just morph -> morph

Nothing ->

error "globDecomp: could not determine morphism of new edge"

newEdge = if isHiding then hidingEdge else localEdge

hidingEdge =

(node,

target,

DGLink {dgl_morphism = morphism,

dgl_type = (HidingThm (dgl_morphism (getLabelOfEdge (head path))) (Static.DevGraph.Open)),

dgl_origin = DGProof})

localEdge = (node,

target,

DGLink {dgl_morphism = morphism,

dgl_type = (LocalThm (Static.DevGraph.Open)

None (Static.DevGraph.Open)),

dgl_origin = DGProof}

)

newGraph = insEdge newEdge dgraph

newChanges = ((InsertEdge newEdge):changes)

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

-- global subsumption

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

-- local debug method

printEdges :: [LEdge DGLinkLab] -> String

printEdges [] = ""

printEdges ((edge@(src,tgt,label)):edges) =

"("++(show src)++", "++(show tgt)++", dgl_type=<<"++(show (dgl_type label))++">>\ndgl_origin=<<"++(show (dgl_origin label))++">>\ndgl_morphism=<<"++(show (dgl_morphism label))++">>)\n\n"++(printEdges edges)

-- applies global subsumption to all unproven global theorem edges if possible

globSubsume :: ProofStatus -> ProofStatus

globSubsume proofStatus@(globalContext,libEnv,history,dGraph) =

(globalContext, libEnv, nextHistoryElem:history, nextDGraph)

where

globalThmEdges = filter isUnprovenGlobalThm (labEdges dGraph)

-- the 'nub' is just a workaround, because some of the edges in the graph

-- do not differ from each other in this represetation - which causes

-- problems on deletion

result = globSubsumeAux libEnv dGraph ([],[]) (nub globalThmEdges)

nextDGraph = fst result

nextHistoryElem = snd result

{- auxiliary function for globSubsume (above)

the actual implementation -}

globSubsumeAux :: LibEnv -> DGraph -> ([DGRule],[DGChange]) ->

[LEdge DGLinkLab] -> (DGraph,([DGRule],[DGChange]))

globSubsumeAux _ dgraph historyElement [] = (dgraph, historyElement)

globSubsumeAux libEnv dgraph (rules,changes) ((ledge@(src,tgt,edgeLab)):list) =

if not (null proofBasis) || isIdentityEdge ledge libEnv dgraph

then

if isDuplicate newEdge dgraph changes then

globSubsumeAux libEnv (delLEdge ledge dgraph)

(newRules,(DeleteEdge ledge):changes) list

else

globSubsumeAux libEnv (insEdge newEdge (delLEdge ledge dgraph))

(newRules,(DeleteEdge ledge):((InsertEdge newEdge):changes)) list

else

globSubsumeAux libEnv dgraph (rules,changes) list

where

morphism = dgl_morphism edgeLab

allPaths = getAllGlobPathsOfMorphismBetween dgraph morphism src tgt

filteredPaths = [path| path <- allPaths, notElem ledge path]

proofBasis = selectProofBasis edgeLab filteredPaths

(GlobalThm _ conservativity conservStatus) = dgl_type edgeLab

newEdge = (src,

tgt,

DGLink {dgl_morphism = morphism,

dgl_type = (GlobalThm (Proven proofBasis)

conservativity conservStatus),

dgl_origin = DGProof}

)

newRules = (GlobSubsumption ledge):rules

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

-- local decomposition

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

{- a merge of the rules local subsumption, local decomposition I and

local decomposition II -}

-- applies this merge of rules to all unproven localThm edges if possible

locDecomp :: ProofStatus -> ProofStatus

locDecomp proofStatus@(globalContext,libEnv,history,dGraph) =

(globalContext, libEnv, nextHistoryElem:history, nextDGraph)

where

localThmEdges = filter isUnprovenLocalThm (labEdges dGraph)

result = locDecompAux libEnv dGraph ([],[]) localThmEdges

nextDGraph = fst result

nextHistoryElem = snd result

{- auxiliary function for locDecomp (above)

actual implementation -}

locDecompAux :: LibEnv -> DGraph -> ([DGRule],[DGChange]) -> [LEdge DGLinkLab]

-> (DGraph,([DGRule],[DGChange]))

locDecompAux libEnv dgraph historyElement [] = (dgraph, historyElement)

locDecompAux libEnv dgraph (rules,changes) ((ledge@(src,tgt,edgeLab)):list) =

if (null proofBasis && not (isIdentityEdge ledge libEnv dgraph))

then

locDecompAux libEnv dgraph (rules,changes) list

else

if isDuplicate newEdge dgraph changes

then locDecompAux libEnv auxGraph

(newRules,(DeleteEdge ledge):changes) list

else locDecompAux libEnv newGraph (newRules,newChanges) list

where

morphism = dgl_morphism edgeLab

allPaths = getAllLocGlobPathsBetween dgraph src tgt

sens = getSentenceList libEnv dgraph src

pathsWithoutEdgeItself = [path|path <- allPaths, notElem ledge path]

filteredPaths = filterByTranslation sens morphism pathsWithoutEdgeItself

proofBasis = selectProofBasis edgeLab filteredPaths

auxGraph = delLEdge ledge dgraph

(LocalThm _ conservativity conservStatus) = (dgl_type edgeLab)

newEdge = (src,

tgt,

DGLink {dgl_morphism = morphism,

dgl_type =

(LocalThm (Proven proofBasis)

conservativity conservStatus),

dgl_origin = DGProof}

)

newGraph = insEdge newEdge auxGraph

newRules = (LocDecomp ledge):rules

newChanges = (DeleteEdge ledge):((InsertEdge newEdge):changes)

{- returns the sentence list of the given node -}

getSentenceList :: LibEnv -> DGraph -> Node -> Maybe G_l_sentence_list

getSentenceList libEnv dgraph node =

if isDGRef nodeLab

then case Map.lookup (dgn_libname nodeLab) libEnv of

Just (_,_,refDgraph) -> getSentenceList libEnv refDgraph (dgn_node nodeLab)

Nothing -> Nothing

else Just (dgn_sens nodeLab)

where nodeLab = lab' (context node dgraph)

{- returns the sentence list of the given node -}

getSignature :: LibEnv -> DGraph -> Node -> Maybe G_sign

getSignature libEnv dgraph node =

if isDGRef nodeLab

then case Map.lookup (dgn_libname nodeLab) libEnv of

Just (_,_,refDgraph) ->

getSignature libEnv refDgraph (dgn_node nodeLab)

Nothing -> Nothing

else Just (dgn_sign nodeLab)

where nodeLab = lab' (context node dgraph)

{- removes all paths from the given list of paths whose morphism does not translate the given sentence list to the same resulting sentence list as the given morphism-}

filterByTranslation :: Maybe G_l_sentence_list -> GMorphism -> [[LEdge DGLinkLab]] -> [[LEdge DGLinkLab]]

filterByTranslation maybeSens morphism paths =

case maybeSens of

Just sens -> [path| path <- paths, isSameTranslation sens morphism path]

Nothing -> []

-- isSameTranslation sens morphism (calculateMorphismOfPath path)]

{- checks if the given morphism and the morphism of the given path translate the given sentence list to the same resulting sentence list -}

isSameTranslation :: G_l_sentence_list -> GMorphism -> [LEdge DGLinkLab] -> Bool

isSameTranslation sens morphism path =

case calculateMorphismOfPath path of

Just morphismOfPath -> isSameTranslationAux sens morphism morphismOfPath

Nothing -> False

{- checks if the two given morphisms translate the given sentence list to the same resulting sentence list

if the calculation of one of the resulting sentence lists fails, False is returned -}

isSameTranslationAux :: G_l_sentence_list -> GMorphism -> GMorphism -> Bool

isSameTranslationAux sens mor1 mor2 =

case maybeResult maybeSens1 of

Nothing -> False

Just sens1 ->

case maybeResult maybeSens2 of

Nothing -> False

Just sens2 -> eq_G_l_sentence_set sens1 sens2

where

maybeSens1 = translateG_l_sentence_list mor1 sens

maybeSens2 = translateG_l_sentence_list mor2 sens

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

-- local subsumption

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

-- applies local subsumption to all unproven local theorem edges

locSubsume :: ProofStatus -> ProofStatus

locSubsume proofStatus@(globalContext,libEnv,history,dGraph) =

(globalContext, libEnv, nextHistoryElem:history, nextDGraph)

where

localThmEdges = filter isUnprovenLocalThm (labEdges dGraph)

result = locSubsumeAux libEnv dGraph ([],[]) localThmEdges

nextDGraph = fst result

nextHistoryElem = snd result

-- auxiliary method for locSubsume

locSubsumeAux :: LibEnv -> DGraph -> ([DGRule],[DGChange]) -> [LEdge DGLinkLab]

-> (DGraph,([DGRule],[DGChange]))

locSubsumeAux libEnv dgraph historyElement [] = (dgraph, historyElement)

locSubsumeAux libEnv dgraph (rules,changes) ((ledge@(src,tgt,edgeLab)):list) =

case (getDGNode libEnv dgraph tgt, maybeAxiomsSrc) of

(Just (target,_), Just axiomsSrc) ->

case (maybeResult (computeTheory libEnv dgraph target), maybeResult (translateG_l_sentence_list morphism axiomsSrc)) of

(Just theoryTgt, Just (G_l_sentence_list lidSrc sensSrc)) ->

case maybeResult (coerceTheory lidSrc theoryTgt) of

Nothing -> locSubsumeAux libEnv dgraph (rules,changes) list

Just (_,sentencesTgt) ->

if (all (`elem` sentencesTgt) sensSrc)

then locSubsumeAux libEnv newGraph (newRules,newChanges) list

else locSubsumeAux libEnv dgraph (rules,changes) list

otherwise -> locSubsumeAux libEnv dgraph (rules,changes) list

otherwise -> -- showDiags defaultHetcatsOpts (errSrc++errTgt)

locSubsumeAux libEnv dgraph (rules,changes) list

where

morphism = dgl_morphism edgeLab

maybeAxiomsSrc = getLocalAxioms libEnv dgraph src

auxGraph = delLEdge ledge dgraph

(LocalThm _ conservativity conservStatus) = (dgl_type edgeLab)

newEdge = (src,

tgt,

DGLink {dgl_morphism = morphism,

dgl_type =

(LocalThm (Proven [])

conservativity conservStatus),

dgl_origin = DGProof}

)

newGraph = insEdge newEdge auxGraph

newRules = (LocSubsumption ledge):rules

newChanges = (DeleteEdge ledge):((InsertEdge newEdge):changes)

{- gets the local axioms of the given node. If this node is a DGRef the referenced node is looked up first.-}

getLocalAxioms :: LibEnv -> DGraph -> Node -> Maybe G_l_sentence_list

getLocalAxioms libEnv dgraph node =

case getDGNode libEnv dgraph node of

Just dgNode -> Just (dgn_sens (snd dgNode))

Nothing -> Nothing

{- if the given node is a DGRef, the referenced node is returned (as a labeled node). Otherwise the node itself is returned (as a labeled node).-}

getDGNode :: LibEnv -> DGraph -> Node -> Maybe (LNode DGNodeLab)

getDGNode libEnv dgraph node =

if isDGRef nodeLab then case Map.lookup (dgn_libname nodeLab) libEnv of

Just (_,_,refDgraph) ->

getDGNode libEnv refDgraph (dgn_node nodeLab)

Nothing -> Nothing

else Just (labNode' contextOfNode)

where contextOfNode = (context node dgraph)

nodeLab = lab' contextOfNode

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

-- methods that calculate paths of certain types

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

{- returns a list of all proven global paths of the given morphism between

the given source and target node-}

getAllGlobPathsOfMorphismBetween :: DGraph -> GMorphism -> Node -> Node

-> [[LEdge DGLinkLab]]

getAllGlobPathsOfMorphismBetween dgraph morphism src tgt =

filterPathsByMorphism morphism allPaths

where

allPaths = getAllGlobPathsBetween dgraph src tgt

{- returns all paths from the given list whose morphism is equal to the

given one-}

filterPathsByMorphism :: GMorphism -> [[LEdge DGLinkLab]]

-> [[LEdge DGLinkLab]]

filterPathsByMorphism morphism paths =

[path| path <- paths, (calculateMorphismOfPath path) == (Just morphism)]

{- returns a list of all paths to the given node

that consist of globalDef edge only

or

that consist of a localDef or hidingDef edge

followed by any number of globalDef edges -}

getAllLocOrHideGlobDefPathsTo :: DGraph -> Node -> [LEdge DGLinkLab]

-> [(Node, [LEdge DGLinkLab])]

getAllLocOrHideGlobDefPathsTo =

getAllGlobDefPathsBeginningWithTypesTo

([isLocalDef, isHidingDef]::[LEdge DGLinkLab -> Bool])

{- returns a list of all paths to the given node

that consist of globalDef edges only

or

that consist of a localDef edge followed by any number of globalDef edges -}

getAllLocGlobDefPathsTo :: DGraph -> Node -> [LEdge DGLinkLab]

-> [(Node, [LEdge DGLinkLab])]

getAllLocGlobDefPathsTo = getAllGlobDefPathsBeginningWithTypesTo

([isLocalDef]::[LEdge DGLinkLab -> Bool])

{- (node,path):(locGlobPaths ++

(concat (

[getAllLocGlobDefPathsTo dgraph (getSourceNode edge) path|

(_, path@(edge:edges)) <- globalPaths]))

)

where

inEdges = inn dgraph node

globalEdges = [edge| edge <- filter isGlobalDef inEdges, notElem edge path]

localEdges = [edge| edge <- filter isLocalDef inEdges, notElem edge path]

globalPaths = [(getSourceNode edge, (edge:path))| edge <- globalEdges]

locGlobPaths = [(getSourceNode edge, (edge:path))| edge <- localEdges]

-}

{- returns a list of all paths to the given node

that consist of globalDef edges only

or

that consist of an edge of one of the given types

followed by any number of globalDef edges -}

getAllGlobDefPathsBeginningWithTypesTo :: [LEdge DGLinkLab -> Bool] -> DGraph -> Node -> [LEdge DGLinkLab]

-> [(Node, [LEdge DGLinkLab])]

getAllGlobDefPathsBeginningWithTypesTo types dgraph node path =

(node,path):(typeGlobPaths ++

(concat ( [getAllGlobDefPathsBeginningWithTypesTo

types dgraph (getSourceNode edge) path |

(_, path@(edge:edges)) <- globalPaths])

)

)

where

inEdges = inn dgraph node

globalEdges = [edge| edge <- filter isGlobalDef inEdges, notElem edge path]

edgesOfTypes

= [edge| edge <- filterByTypes types inEdges, notElem edge path]

globalPaths = [(getSourceNode edge, (edge:path))| edge <- globalEdges]

typeGlobPaths = [(getSourceNode edge, (edge:path))| edge <- edgesOfTypes]

{- returns all paths consisting of global edges only

or

of one local edge followed by any number of global edges-}

getAllLocGlobPathsBetween :: DGraph -> Node -> Node -> [[LEdge DGLinkLab]]

getAllLocGlobPathsBetween dgraph src tgt =

locGlobPaths ++ globPaths

where

outEdges = out dgraph src

locEdges = [(edge,target)|edge@(_,target,_) <-

(filterByTypes [isLocalEdge] outEdges)]

locGlobPaths = (concat [map ([edge]++)

(getAllPathsOfTypesBetween dgraph [isGlobalEdge] node tgt [])|

(edge,node) <- locEdges])

globPaths = getAllPathsOfTypesBetween dgraph [isGlobalEdge] src tgt []

{- returns all paths of globalDef edges or globalThm edges

between the given source and target node -}

getAllGlobPathsBetween :: DGraph -> Node -> Node -> [[LEdge DGLinkLab]]

getAllGlobPathsBetween dgraph src tgt =

getAllPathsOfTypesBetween dgraph [isGlobalDef,isGlobalThm] src tgt []

{- gets all paths consisting of local/global thm/def edges

of the given morphism -}

getAllThmDefPathsOfMorphismBetween :: DGraph -> GMorphism -> Node -> Node

-> [[LEdge DGLinkLab]]

getAllThmDefPathsOfMorphismBetween dgraph morphism src tgt =

filterPathsByMorphism morphism allPaths

where

allPaths = getAllPathsOfTypesBetween dgraph types src tgt []

types =

[isGlobalThm,

isProvenLocalThm,

isProvenLocalThm,

isUnprovenLocalThm,

isGlobalDef,

isLocalDef]

{- returns all paths of globalDef edges between the given source and

target node -}

getAllGlobDefPathsBetween :: DGraph -> Node -> Node -> [[LEdge DGLinkLab]]

getAllGlobDefPathsBetween dgraph src tgt =

getAllPathsOfTypeBetween dgraph isGlobalDef src tgt

{- returns all paths consiting of edges of the given type between the

given node -}

getAllPathsOfTypeBetween :: DGraph -> (LEdge DGLinkLab -> Bool) -> Node

-> Node -> [[LEdge DGLinkLab]]

getAllPathsOfTypeBetween dgraph isType src tgt =

getAllPathsOfTypesBetween dgraph (isType:[]) src tgt []

{- returns all paths consisting of edges of the given types between

the given nodes -}

getAllPathsOfTypesBetween :: DGraph -> [LEdge DGLinkLab -> Bool] -> Node

-> Node -> [LEdge DGLinkLab]

-> [[LEdge DGLinkLab]]

getAllPathsOfTypesBetween dgraph types src tgt path =

[edge:path| edge <- edgesFromSrc]

++ (concat

[getAllPathsOfTypesBetween dgraph types src nextTgt (edge:path)|

(edge,nextTgt) <- nextStep] )

where

inGoingEdges = inn dgraph tgt

edgesOfTypes =

[edge| edge <- filterByTypes types inGoingEdges, notElem edge path]

edgesFromSrc =

[edge| edge@(source,_,_) <- edgesOfTypes, source == src]

nextStep =

[(edge, source)| edge@(source,_,_) <- edgesOfTypes, source /= src]

-- removes all edges that are not of the given types

filterByTypes :: [LEdge DGLinkLab -> Bool] -> [LEdge DGLinkLab]

-> [LEdge DGLinkLab]

filterByTypes [] edges = []

filterByTypes (isType:typeChecks) edges =

(filter isType edges)++(filterByTypes typeChecks edges)

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

-- methods to determine or get morphisms

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

-- determines the morphism of a given path

calculateMorphismOfPath :: [LEdge DGLinkLab] -> Maybe GMorphism

calculateMorphismOfPath [] = Nothing

calculateMorphismOfPath path@((src,tgt,edgeLab):furtherPath) =

case maybeMorphismOfFurtherPath of

Nothing -> if null furtherPath then Just morphism else Nothing

Just morphismOfFurtherPath ->

resultToMaybe $ compHomInclusion morphism morphismOfFurtherPath

where

morphism = dgl_morphism edgeLab

maybeMorphismOfFurtherPath = calculateMorphismOfPath furtherPath

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

-- methods to get the nodes of an edge

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

getSourceNode :: LEdge DGLinkLab -> Node

getSourceNode (source,_,_) = source

getTargetNode :: LEdge DGLinkLab -> Node

getTargetNode (_,target,_) = target

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

-- methods to check the type of an edge

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

isProven :: LEdge DGLinkLab -> Bool

isProven edge = isGlobalDef edge || isLocalDef edge

|| isProvenGlobalThm edge || isProvenLocalThm edge

isGlobalEdge :: LEdge DGLinkLab -> Bool

isGlobalEdge edge = isGlobalDef edge || isGlobalThm edge

isLocalEdge :: LEdge DGLinkLab -> Bool

isLocalEdge edge = isLocalDef edge || isLocalThm edge

isGlobalThm :: LEdge DGLinkLab -> Bool

isGlobalThm edge = isProvenGlobalThm edge || isUnprovenGlobalThm edge

isLocalThm :: LEdge DGLinkLab -> Bool

isLocalThm edge = isProvenLocalThm edge || isUnprovenLocalThm edge

isProvenGlobalThm :: LEdge DGLinkLab -> Bool

isProvenGlobalThm (_,_,edgeLab) =

case dgl_type edgeLab of

(GlobalThm (Proven _) _ _) -> True

otherwise -> False

isUnprovenGlobalThm :: LEdge DGLinkLab -> Bool

isUnprovenGlobalThm (_,_,edgeLab) =

case dgl_type edgeLab of

(GlobalThm Static.DevGraph.Open _ _) -> True

otherwise -> False

isProvenLocalThm :: LEdge DGLinkLab -> Bool

isProvenLocalThm (_,_,edgeLab) =

case dgl_type edgeLab of

(LocalThm (Proven _) _ _) -> True

otherwise -> False

isUnprovenLocalThm :: LEdge DGLinkLab -> Bool

isUnprovenLocalThm (_,_,edgeLab) =

case dgl_type edgeLab of

(LocalThm (Static.DevGraph.Open) _ _) -> True

otherwise -> False

isGlobalDef :: LEdge DGLinkLab -> Bool

isGlobalDef (_,_,edgeLab) =

case dgl_type edgeLab of

GlobalDef -> True

otherwise -> False

isLocalDef :: LEdge DGLinkLab -> Bool

isLocalDef (_,_,edgeLab) =

case dgl_type edgeLab of

LocalDef -> True

otherwise -> False

isHidingDef :: LEdge DGLinkLab -> Bool

isHidingDef (_,_,edgeLab) =

case dgl_type edgeLab of

HidingDef -> True

otherwise -> False

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

-- other methods on edges

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

{- returns true, if an identical edge is already in the graph or marked to be inserted,

false otherwise-}

isDuplicate :: LEdge DGLinkLab -> DGraph -> [DGChange] -> Bool

isDuplicate newEdge dgraph changes =

elem (InsertEdge newEdge) changes || elem newEdge (labEdges dgraph)

{- returns true, if the given edge is duplicate or

if there already exists a parallel path,

which starts with a localThmEdge to one of the startnodes in 'otherNewEdge'

and has the same morphism as the given edge,

false otherwise -}

isRedundant :: LEdge DGLinkLab -> DGraph -> [DGChange]

->[(Node, [LEdge DGLinkLab])] -> Bool

isRedundant newEdge@(src,_,label) dgraph changes otherNewEdges =

isDuplicate newEdge dgraph changes

|| elem (Just (dgl_morphism label)) morphismsOfParallelPaths

where

localThmEdgesToOtherSources =

[outEdge|outEdge@(_,tgt,_) <- out dgraph src,

elem tgt (map fst otherNewEdges)

&& isLocalThm outEdge]

parallelPaths

= concat (map (combineSucceedingEdges otherNewEdges)

localThmEdgesToOtherSources)

morphismsOfParallelPaths = map calculateMorphismOfPath parallelPaths

{- combines the given edge with each of those given paths that start

with the target node of the edge-}

combineSucceedingEdges :: [(Node,[LEdge DGLinkLab])] -> LEdge DGLinkLab

-> [[LEdge DGLinkLab]]

combineSucceedingEdges [] _ = []

combineSucceedingEdges ((src,path):paths) edge@(_,tgt,_) =

if tgt == src

then ((edge:path)):(combineSucceedingEdges paths edge)

else combineSucceedingEdges paths edge

isIdentityEdge :: LEdge DGLinkLab -> LibEnv -> DGraph -> Bool

isIdentityEdge (src,tgt,edgeLab) libEnv dgraph =

if isDGRef nodeLab then

case Map.lookup (dgn_libname nodeLab) libEnv of

Just globContext@(_,_,refDgraph) -> isIdentityEdge (dgn_node nodeLab,tgt,edgeLab) libEnv refDgraph

Nothing -> False

else if src == tgt && (dgl_morphism edgeLab) == (ide Grothendieck (dgn_sign nodeLab)) then True else False

where nodeLab = lab' (context src dgraph)

{- returns the DGLinkLab of the given LEdge -}

getLabelOfEdge :: (LEdge b) -> b

getLabelOfEdge (_,_,label) = label

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

-- methods to determine the labels of the inserted edges in the given dgchange

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

{- filters the list of changes for insertions of edges and returns the label

of one of these; or Nothing if no insertion was found -}

getLabelsOfInsertedEdges :: [DGChange] -> [DGLinkLab]

getLabelsOfInsertedEdges changes =

[label | (_,_,label) <- getInsertedEdges changes]

{- returns all insertions of edges from the given list of changes -}

getInsertedEdges :: [DGChange] -> [LEdge DGLinkLab]

getInsertedEdges [] = []

getInsertedEdges (change:list) =

case change of

(InsertEdge edge) -> edge:(getInsertedEdges list)

otherwise -> getInsertedEdges list

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

-- methods to check and select proof basis

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

{- determines all proven paths in the given list and tries to selet a

proof basis from these (s. selectProofBasisAux);

if this fails the same is done for the rest of the given paths, i.e.

for the unproven ones -}

selectProofBasis :: DGLinkLab -> [[LEdge DGLinkLab]] -> [DGLinkLab]

selectProofBasis label paths =

if null provenProofBasis then selectProofBasisAux label unprovenPaths

else provenProofBasis

where

provenPaths = filterProvenPaths paths

provenProofBasis = selectProofBasisAux label provenPaths

unprovenPaths = [path | path <- paths, notElem path provenPaths]

{- selects the first path that does not form a proof cycle with the given

label (if such a path exits) and returns the labels of its edges -}

selectProofBasisAux :: DGLinkLab -> [[LEdge DGLinkLab]] -> [DGLinkLab]

selectProofBasisAux _ [] = []

selectProofBasisAux label (path:list) =

if notProofCycle label path then nub (calculateProofBasis path)

else selectProofBasisAux label list

{- calculates the proofBasis of the given path,

i.e. the list of all DGLinkLabs the proofs of the edges contained in the path

are based on, plus the DGLinkLabs of the edges themselves;

duplicates are not removed here, but in the calling method

selectProofBasisAux -}

calculateProofBasis :: [LEdge DGLinkLab] -> [DGLinkLab]

calculateProofBasis [] = []

calculateProofBasis ((_,_,lab):edges) =

lab:((getProofBasis lab)++(calculateProofBasis edges))

{- returns the proofBasis contained in the given DGLinkLab -}

getProofBasis :: DGLinkLab -> [DGLinkLab]

getProofBasis lab =

case dgl_type lab of

(GlobalThm (Proven proofBasis) _ _) -> proofBasis

(LocalThm (Proven proofBasis) _ _) -> proofBasis

_ -> []

{- returns all proven paths from the given list -}

filterProvenPaths :: [[LEdge DGLinkLab]] -> [[LEdge DGLinkLab]]

filterProvenPaths [] = []

filterProvenPaths (path:list) =

if (and [isProven edge| edge <- path]) then path:(filterProvenPaths list)

else filterProvenPaths list

{- opposite of isProofCycle -}

notProofCycle :: DGLinkLab -> [LEdge DGLinkLab] -> Bool

notProofCycle label = not.(isProofCycle label)

{- checks if the given label is contained in the ProofBasis of one of the

edges of the given path -}

isProofCycle :: DGLinkLab -> [LEdge DGLinkLab] -> Bool

isProofCycle _ [] = False

isProofCycle label (edge:path) =

if (elemOfProofBasis label edge) then True else (isProofCycle label path)

{- checks if the given label is contained in the ProofBasis of the given

edge -}

elemOfProofBasis :: DGLinkLab -> (LEdge DGLinkLab) -> Bool

elemOfProofBasis label (_,_,dglink) =

case dgl_type dglink of

(GlobalThm (Proven proofBasis) _ _) -> elem label proofBasis

(LocalThm (Proven proofBasis) _ _) -> elem label proofBasis

_ -> False

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

-- methods for the extension of globDecomp (avoid insertion ofredundant edges)

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

{- returns all paths consisting of local theorem links whose src and tgt nodes

are contained in the given list of nodes -}

localThmPathsBetweenNodes :: DGraph -> [Node] -> [[LEdge DGLinkLab]]

localThmPathsBetweenNodes _ [] = []

localThmPathsBetweenNodes dgraph nodes@(x:xs) =

localThmPathsBetweenNodesAux dgraph nodes nodes

{- auxiliary method for localThmPathsBetweenNodes -}

localThmPathsBetweenNodesAux :: DGraph -> [Node] -> [Node] -> [[LEdge DGLinkLab]]

localThmPathsBetweenNodesAux _ [] _ = []

localThmPathsBetweenNodesAux dgraph (node:srcNodes) tgtNodes =

(concat (map (getAllPathsOfTypeBetween dgraph isUnprovenLocalThm node) tgtNodes))

++ (localThmPathsBetweenNodesAux dgraph srcNodes tgtNodes)

{- combines each of the given paths with matching edges from the given list

(i.e. every edge that has as its source node the tgt node of the path)-}

combinePathsWithEdges :: [[LEdge DGLinkLab]] -> [LEdge DGLinkLab]

-> [[LEdge DGLinkLab]]

combinePathsWithEdges paths edges =

concat (map (combinePathsWithEdge paths) edges)

{- combines the given path with each matching edge from the given list

(i.e. every edge that has as its source node the tgt node of the path)-}

combinePathsWithEdge :: [[LEdge DGLinkLab]] -> LEdge DGLinkLab

-> [[LEdge DGLinkLab]]

combinePathsWithEdge [] _ = []

combinePathsWithEdge (path:paths) edge@(src,_,_) =

case path of

[] -> combinePathsWithEdge paths edge

(x:xs) -> if (getTargetNode (last path)) == src

then (path++[edge]):(combinePathsWithEdge paths edge)

else combinePathsWithEdge paths edge

{- todo: choose a better name for this method...

returns for each of the given paths a pair consisting of the last edge

contained in the path and - as a triple - the src, tgt and morphism of the

complete path

if there is an empty path in the given list or the morphsim cannot be

calculated, it is simply ignored -}

calculateResultingEdges :: [[LEdge DGLinkLab]] -> [(LEdge DGLinkLab,(Node,Node,GMorphism))]

calculateResultingEdges [] = []

calculateResultingEdges (path:paths) =

case path of

[] -> calculateResultingEdges paths

(x:xs) ->

case calculateMorphismOfPath path of

Nothing -> calculateResultingEdges paths

Just morphism -> (last path, (src,tgt,morphism)):(calculateResultingEdges paths)

where src = getSourceNode (head path)

tgt = getTargetNode (last path)

{- removes from the given list every edge for which there is already an

equivalent edge or path (i.e. an edge or path with the same src, tgt and

morphsim) -}

removeSuperfluousEdges :: DGraph -> [LEdge DGLinkLab]

-> (DGraph,[LEdge DGLinkLab])

removeSuperfluousEdges dgraph [] = (dgraph,[])

removeSuperfluousEdges dgraph edges

= removeSuperfluousEdgesAux dgraph edges

(calculateResultingEdges combinedPaths) []

where

localThmPaths

= localThmPathsBetweenNodes dgraph (map (getSourceNode) edges)

combinedPaths = combinePathsWithEdges localThmPaths edges

{- auxiliary method for removeSuperfluousEdges -}

removeSuperfluousEdgesAux :: DGraph -> [LEdge DGLinkLab]

-> [(LEdge DGLinkLab,(Node,Node,GMorphism))]

-> [LEdge DGLinkLab] -> (DGraph,[LEdge DGLinkLab])

removeSuperfluousEdgesAux dgraph [] _ edgesToInsert= (dgraph,edgesToInsert)

removeSuperfluousEdgesAux dgraph ((src,tgt,lab):edges)

resultingEdges edgesToInsert =

if not (null equivalentEdges)

then removeSuperfluousEdgesAux

newDGraph edges newResultingEdges edgesToInsert

else removeSuperfluousEdgesAux

dgraph edges resultingEdges ((src,tgt,lab):edgesToInsert)

where

equivalentEdges

= [e | e <- resultingEdges,(snd e) == (src,tgt,dgl_morphism lab)]

newResultingEdges = [e | e <- resultingEdges,(fst e) /= (src,tgt,lab)]

newDGraph = delLEdge (src,tgt,lab) dgraph

{- returns true, if the given change is an insertion of an local theorem edge,

false otherwise -}

isLocalThmInsertion :: DGChange -> Bool

isLocalThmInsertion change

= case change of

InsertEdge edge -> isLocalThm edge

otherwise -> False

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

-- further methods

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

-- | Calculate the morphism of a path with given start node

calculateMorphismOfPathWithStart :: DGraph -> LibEnv

-> (Node,[LEdge DGLinkLab])

-> Maybe GMorphism

calculateMorphismOfPathWithStart dg libEnv (n,[]) = do

ctx <- fst $ match n dg

case getDGNode libEnv dg (fst (labNode' ctx)) of

Just dgNode_ctx -> return $ ide Grothendieck (dgn_sign (snd (dgNode_ctx))) -- ??? to simplistic

Nothing -> Nothing

calculateMorphismOfPathWithStart _ _ (_,p) = calculateMorphismOfPath p

-- | Compute the theory of a node (CASL Reference Manual, p. 294, Def. 4.9)

computeTheory :: LibEnv -> DGraph -> Node -> Result G_theory

computeTheory libEnv dg n = do

let paths = reverse $ getAllLocGlobDefPathsTo dg n []

-- reverse needed to have a "bottom up" ordering

mors <- maybeToResult nullPos "Could not calculate morphism of path"

$ mapM (calculateMorphismOfPathWithStart dg libEnv) paths

sens <- maybeToResult nullPos "Could not calculate sentence list of node"

$ mapM (getSentenceList libEnv dg . fst) paths

sens' <- mapM (uncurry translateG_l_sentence_list)

$ zip mors sens

G_l_sentence_list lid1 sens'' <-

maybeToResult nullPos "Logic mismatch for sentences"

$ flatG_l_sentence_list sens'

G_sign lid2 sig <-

maybeToResult nullPos "Could not calculate signature of node"

$ getSignature libEnv dg n

sens''' <- rcoerce lid1 lid2 nullPos sens''

return $ G_theory lid2 sig sens'''

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

-- basic inference

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

getGoals :: LibEnv -> DGraph -> LEdge DGLinkLab -> Result G_l_sentence_list

getGoals libEnv dg (n,_,edge) = do

sens <- maybeToResult nullPos ("Could node find node "++show n)

$ getSentenceList libEnv dg n

let mor = dgl_morphism edge

translateG_l_sentence_list mor sens

getProvers :: Bool -> LogicGraph -> G_sublogics -> [(G_prover,AnyComorphism)]

getProvers consCheck lg gsub =

if consCheck then

[(G_cons_checker (targetLogic cid) p,Comorphism cid) |

Comorphism cid <- cms,

p <- cons_checkers (targetLogic cid)]

else

[(G_prover (targetLogic cid) p,Comorphism cid) |

Comorphism cid <- cms,

p <- provers (targetLogic cid)]

where cms = findComorphismPaths lg gsub

selectProver :: [(G_prover,AnyComorphism)] -> IOResult (G_prover,AnyComorphism)

selectProver [p] = return p

selectProver [] = resToIORes $ fatal_error "No pover available" nullPos

#ifdef UNI_PACKAGE

selectProver provers = do

sel <- ioToIORes $ listBox

"Choose a translation to a prover-supported logic"

(map (show.snd) provers)

i <- case sel of

Just j -> return j

_ -> resToIORes $ fatal_error "Proofs.Proofs: selection" nullPos

return (provers!!i)

#endif

-- applies basic inference to a given node

basicInferenceNode :: Bool -> LogicGraph -> (LIB_NAME,Node) -> ProofStatus

-> IO (Result ProofStatus)

basicInferenceNode checkCons lg (ln,node)

proofStatus@(globalContext,libEnv,history,dGraph) =

ioresToIO (do

-- compute the theory of the node, and its name

G_theory lid1 sign axs <-

resToIORes $ computeTheory libEnv dGraph node

ctx <- resToIORes

$ maybeToResult nullPos ("Could node find node "++show node)

$ fst $ match node dGraph

let nlab = lab' ctx

nodeName = case nlab of

DGNode _ _ _ _-> dgn_name nlab

DGRef _ _ _ -> dgn_renamed nlab

thName = showPretty (getLIB_ID ln) "_"

++ maybe (show node) (flip showPretty "") nodeName

-- compute the list of proof goals

let inEdges = inn dGraph node

localEdges = filter isUnprovenLocalThm inEdges

goalslist <- if checkCons then return []

else resToIORes $ mapM (getGoals libEnv dGraph) localEdges

G_l_sentence_list lid3 goals <-

if null goalslist then return $ G_l_sentence_list lid1 []

else resToIORes (maybeToResult nullPos

"Logic mismatch for proof goals"

(flatG_l_sentence_list goalslist))

goals1 <- resToIORes $ rcoerce lid1 lid3 nullPos goals

-- select a suitable translation and prover

let provers = getProvers checkCons lg $ sublogicOfTh $

(G_theory lid1 sign (axs++goals1))

(prover,Comorphism cid) <- selectProver provers

-- Borrowing: translate theory

let lidS = sourceLogic cid

lidT = targetLogic cid

sign' <- resToIORes $ rcoerce lidS lid1 nullPos sign

axs' <- resToIORes $ rcoerce lidS lid1 nullPos axs

(sign'',sens'') <- resToIORes

$ maybeToResult nullPos "Could not map signature"

$ map_theory cid (sign',axs')

case prover of

G_prover lid4 p -> do

-- Borrowing: translate goal

goals' <- resToIORes $ rcoerce lidS lid3 nullPos goals

let goals'' = mapMaybe (mapNamedM (map_sentence cid sign')) goals'

-- call the prover

p' <- resToIORes $ rcoerce lidT lid4 nullPos p

ps <- ioToIORes $ prove p' thName (sign'',sens'') goals''

-- update the development graph

let (nextDGraph, nextHistoryElem) = proveLocalEdges dGraph localEdges

return (globalContext, libEnv, nextHistoryElem:history, nextDGraph)

G_cons_checker lid4 p -> do

incl <- resToIORes $ inclusion lidT (empty_signature lidT) sign''

let mor = TheoryMorphism { t_source = empty_theory lidT,

t_target = (sign'',sens''),

t_morphism = incl }

p' <- resToIORes $ rcoerce lidT lid4 nullPos p

(res,_) <- ioToIORes $ cons_check p' mor

let nextHistoryElem = error "Proofs.Proofs: basic inference"

-- ??? to be implemented

return (globalContext, libEnv, nextHistoryElem:history, dGraph)

)

proveLocalEdges :: DGraph -> [LEdge DGLinkLab] -> (DGraph,([DGRule],[DGChange]))

proveLocalEdges dGraph edges = (nextDGraph,([LocalInference],changes))

where (nextDGraph,(_,changes)) = proveLocalEdgesAux ([],[]) dGraph edges

proveLocalEdgesAux :: ([DGRule],[DGChange]) -> DGraph -> [LEdge DGLinkLab]

-> (DGraph,([DGRule],[DGChange]))

proveLocalEdgesAux historyElem dGraph [] = (dGraph, historyElem)

proveLocalEdgesAux (rules,changes) dGraph ((edge@(src, tgt, edgelab)):edges) =

if True then proveLocalEdgesAux (rules,(DeleteEdge edge):((InsertEdge provenEdge):changes)) (insEdge provenEdge(delLEdge edge dGraph)) edges

else proveLocalEdgesAux (rules,changes) dGraph edges

where

morphism = dgl_morphism edgelab

(LocalThm _ conservativity conservStatus) = (dgl_type edgelab)

proofBasis = [] -- to be implemented

provenEdge = (src,

tgt,

DGLink {dgl_morphism = morphism,

dgl_type =

(LocalThm (Proven proofBasis)

conservativity conservStatus),

dgl_origin = DGProof}

)