Utils.hs revision c3efd4f435e954846981cf46bca64e0485266634
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova{- |
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina SojakovaModule : $Header$
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina SojakovaDescription : Utilities for CspCASLProver related to the actual
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova construction of Isabelle theories.
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina SojakovaCopyright : (c) Liam O'Reilly and Markus Roggenbach,
98890889ffb2e8f6f722b00e265a211f13b5a861Corneliu-Claudiu Prodescu Swansea University 2009
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina SojakovaLicense : similar to LGPL, see HetCATS/LICENSE.txt or LIZENZ.txt
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina SojakovaMaintainer : csliam@swansea.ac.uk
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina SojakovaStability : provisional
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina SojakovaPortability : portable
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina SojakovaUtilities for CspCASLProver related to the actual construction of
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina SojakovaIsabelle theories.
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova-}
ccaa75089b23c0f043cdbd4001cba4e076ca4fd3Kristina Sojakova
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakovamodule CspCASLProver.Utils
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova ( addAlphabetType
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova , addAllBarTypes
ccaa75089b23c0f043cdbd4001cba4e076ca4fd3Kristina Sojakova , addAllChooseFunctions
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova , addAllCompareWithFun
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova , addAllIntegrationTheorems
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova , addEqFun
669b4334be8eb3313ca146137db1b83a8873632aKristina Sojakova , addInstansanceOfEquiv
669b4334be8eb3313ca146137db1b83a8873632aKristina Sojakova , addJustificationTheorems
228124cdf2560445e7f1b5312476935b51887463Kristina Sojakova , addPreAlphabet
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova , addProcMap
0737dd44f9a47bb91233ffdb7a03bc657dfc7c5eKristina Sojakova , addProcNameDatatype
0737dd44f9a47bb91233ffdb7a03bc657dfc7c5eKristina Sojakova ) where
0737dd44f9a47bb91233ffdb7a03bc657dfc7c5eKristina Sojakova
0737dd44f9a47bb91233ffdb7a03bc657dfc7c5eKristina Sojakovaimport CASL.AS_Basic_CASL (SORT, OpKind(..))
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakovaimport qualified CASL.Sign as CASLSign
669b4334be8eb3313ca146137db1b83a8873632aKristina Sojakovaimport qualified CASL.Inject as CASLInject
53d7a124a59889b9de5c6ffa856a5e697b043c90Kristina Sojakova
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakovaimport Common.AS_Annotation (makeNamed, Named, SenAttr(..))
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakovaimport qualified Common.Lib.Rel as Rel
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakovaimport Comorphisms.CFOL2IsabelleHOL (IsaTheory)
669b4334be8eb3313ca146137db1b83a8873632aKristina Sojakova
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakovaimport CspCASL.AS_CspCASL_Process (PROCESS_NAME)
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakovaimport CspCASL.SignCSP (CspSign(..), ProcProfile(..), CspCASLSen(..)
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova , isProcessEq)
ccaa75089b23c0f043cdbd4001cba4e076ca4fd3Kristina Sojakova
ccaa75089b23c0f043cdbd4001cba4e076ca4fd3Kristina Sojakovaimport CspCASLProver.Consts
ccaa75089b23c0f043cdbd4001cba4e076ca4fd3Kristina Sojakovaimport CspCASLProver.IsabelleUtils
ccaa75089b23c0f043cdbd4001cba4e076ca4fd3Kristina Sojakovaimport CspCASLProver.TransProcesses (transProcess)
ccaa75089b23c0f043cdbd4001cba4e076ca4fd3Kristina Sojakova
ccaa75089b23c0f043cdbd4001cba4e076ca4fd3Kristina Sojakovaimport qualified Data.List as List
ccaa75089b23c0f043cdbd4001cba4e076ca4fd3Kristina Sojakovaimport qualified Data.Map as Map
ccaa75089b23c0f043cdbd4001cba4e076ca4fd3Kristina Sojakovaimport qualified Data.Set as Set
ccaa75089b23c0f043cdbd4001cba4e076ca4fd3Kristina Sojakova
ccaa75089b23c0f043cdbd4001cba4e076ca4fd3Kristina Sojakovaimport Isabelle.IsaConsts
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakovaimport Isabelle.IsaSign
228124cdf2560445e7f1b5312476935b51887463Kristina Sojakova
669b4334be8eb3313ca146137db1b83a8873632aKristina Sojakova-------------------------------------------------------------------------
53d7a124a59889b9de5c6ffa856a5e697b043c90Kristina Sojakova-- Functions for adding the PreAlphabet datatype to an Isabelle theory --
228124cdf2560445e7f1b5312476935b51887463Kristina Sojakova-------------------------------------------------------------------------
e7cedce0d43b62593b8d5d552bdc36eb5ba73409Kristina Sojakova
228124cdf2560445e7f1b5312476935b51887463Kristina Sojakova-- | Add the PreAlphabet (built from a list of sorts) to an Isabelle
e7cedce0d43b62593b8d5d552bdc36eb5ba73409Kristina Sojakova-- theory.
e7cedce0d43b62593b8d5d552bdc36eb5ba73409Kristina SojakovaaddPreAlphabet :: [SORT] -> IsaTheory -> IsaTheory
228124cdf2560445e7f1b5312476935b51887463Kristina SojakovaaddPreAlphabet sortList isaTh =
228124cdf2560445e7f1b5312476935b51887463Kristina Sojakova let preAlphabetDomainEntry = mkPreAlphabetDE sortList
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova -- Add to the isabelle signature the new domain entry
228124cdf2560445e7f1b5312476935b51887463Kristina Sojakova in updateDomainTab preAlphabetDomainEntry isaTh
228124cdf2560445e7f1b5312476935b51887463Kristina Sojakova
228124cdf2560445e7f1b5312476935b51887463Kristina Sojakova-- | Make a Domain Entry for the PreAlphabet from a list of sorts.
228124cdf2560445e7f1b5312476935b51887463Kristina SojakovamkPreAlphabetDE :: [SORT] -> DomainEntry
228124cdf2560445e7f1b5312476935b51887463Kristina SojakovamkPreAlphabetDE sorts =
228124cdf2560445e7f1b5312476935b51887463Kristina Sojakova (Type {typeId = preAlphabetS, typeSort = [isaTerm], typeArgs = []},
228124cdf2560445e7f1b5312476935b51887463Kristina Sojakova map (\sort ->
53d7a124a59889b9de5c6ffa856a5e697b043c90Kristina Sojakova (mkVName (mkPreAlphabetConstructor sort),
669b4334be8eb3313ca146137db1b83a8873632aKristina Sojakova [Type {typeId = convertSort2String sort,
53d7a124a59889b9de5c6ffa856a5e697b043c90Kristina Sojakova typeSort = [isaTerm],
228124cdf2560445e7f1b5312476935b51887463Kristina Sojakova typeArgs = []}])
228124cdf2560445e7f1b5312476935b51887463Kristina Sojakova ) sorts
228124cdf2560445e7f1b5312476935b51887463Kristina Sojakova )
228124cdf2560445e7f1b5312476935b51887463Kristina Sojakova
53d7a124a59889b9de5c6ffa856a5e697b043c90Kristina Sojakova----------------------------------------------------------------
228124cdf2560445e7f1b5312476935b51887463Kristina Sojakova-- Functions for adding the eq functions and the compare_with --
53d7a124a59889b9de5c6ffa856a5e697b043c90Kristina Sojakova-- functions to an Isabelle theory --
53d7a124a59889b9de5c6ffa856a5e697b043c90Kristina Sojakova----------------------------------------------------------------
53d7a124a59889b9de5c6ffa856a5e697b043c90Kristina Sojakova
228124cdf2560445e7f1b5312476935b51887463Kristina Sojakova-- | Add the eq function to an Isabelle theory using a list of sorts
228124cdf2560445e7f1b5312476935b51887463Kristina SojakovaaddEqFun :: [SORT] -> IsaTheory -> IsaTheory
228124cdf2560445e7f1b5312476935b51887463Kristina SojakovaaddEqFun sortList isaTh =
53d7a124a59889b9de5c6ffa856a5e697b043c90Kristina Sojakova let eqtype = mkFunType preAlphabetType $ mkFunType preAlphabetType boolType
228124cdf2560445e7f1b5312476935b51887463Kristina Sojakova isaThWithConst = addConst eq_PreAlphabetS eqtype isaTh
228124cdf2560445e7f1b5312476935b51887463Kristina Sojakova mkEq sort =
228124cdf2560445e7f1b5312476935b51887463Kristina Sojakova let x = mkFree "x"
228124cdf2560445e7f1b5312476935b51887463Kristina Sojakova y = mkFree "y"
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova lhs = binEq_PreAlphabet lhs_a y
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova lhs_a = termAppl (conDouble (mkPreAlphabetConstructor sort)) x
228124cdf2560445e7f1b5312476935b51887463Kristina Sojakova rhs = termAppl rhs_a y
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova rhs_a = termAppl (conDouble (mkCompareWithFunName sort)) x
669b4334be8eb3313ca146137db1b83a8873632aKristina Sojakova in binEq lhs rhs
228124cdf2560445e7f1b5312476935b51887463Kristina Sojakova eqs = map mkEq sortList
669b4334be8eb3313ca146137db1b83a8873632aKristina Sojakova in addPrimRec eqs isaThWithConst
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova-- | Add all compare_with functions for a given list of sorts to an
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova-- Isabelle theory. We need to know about the casl signature (data
27bdba808fa9637ef10b739233fde57c77245f5dKristina Sojakova-- part of a CspCASL spec) so that we can pass it on to the
27bdba808fa9637ef10b739233fde57c77245f5dKristina Sojakova-- addCompareWithFun function
27bdba808fa9637ef10b739233fde57c77245f5dKristina SojakovaaddAllCompareWithFun :: CASLSign.CASLSign -> IsaTheory -> IsaTheory
27bdba808fa9637ef10b739233fde57c77245f5dKristina SojakovaaddAllCompareWithFun caslSign isaTh =
27bdba808fa9637ef10b739233fde57c77245f5dKristina Sojakova let sortList = Set.toList(CASLSign.sortSet caslSign)
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova in foldl (addCompareWithFun caslSign) isaTh sortList
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova
14650c9e129d8dc51ed55b2edc6ec27d9f0f6d00Kristina Sojakova-- | Add a single compare_with function for a given sort to an
ccaa75089b23c0f043cdbd4001cba4e076ca4fd3Kristina Sojakova-- Isabelle theory. We need to know about the casl signature (data
ccaa75089b23c0f043cdbd4001cba4e076ca4fd3Kristina Sojakova-- part of a CspCASL spec) to work out the RHS of the equations.
ccaa75089b23c0f043cdbd4001cba4e076ca4fd3Kristina SojakovaaddCompareWithFun :: CASLSign.CASLSign -> IsaTheory -> SORT -> IsaTheory
ccaa75089b23c0f043cdbd4001cba4e076ca4fd3Kristina SojakovaaddCompareWithFun caslSign isaTh sort =
ccaa75089b23c0f043cdbd4001cba4e076ca4fd3Kristina Sojakova let sortList = Set.toList(CASLSign.sortSet caslSign)
sortType = Type {typeId = convertSort2String sort, typeSort = [],
typeArgs =[]}
funName = mkCompareWithFunName sort
funType = mkFunType sortType $ mkFunType preAlphabetType boolType
isaThWithConst = addConst funName funType isaTh
sortSuperSet = CASLSign.supersortsOf sort caslSign
mkEq sort' =
let x = mkFree "x"
y = mkFree "y"
sort'Constructor = mkPreAlphabetConstructor sort'
lhs = termAppl lhs_a lhs_b
lhs_a = termAppl (conDouble funName) x
lhs_b = termAppl (conDouble (sort'Constructor)) y
sort'SuperSet =CASLSign.supersortsOf sort' caslSign
commonSuperList = Set.toList (Set.intersection
sortSuperSet
sort'SuperSet)
-- If there are no tests then the rhs=false else
-- combine all tests using binConj
rhs = if (null allTests)
then false
else foldr1 binConj allTests
-- The tests produce a list of equations for each test
-- Test 1 = test equality at: current sort vs current sort
-- Test 2 = test equality at: current sort vs super sorts
-- Test 3 = test equality at: super sorts vs current sort
-- Test 4 = test equality at: super sorts vs super sorts
allTests = concat [test1, test2, test3, test4]
test1 = if (sort == sort') then [binEq x y] else []
test2 = if (Set.member sort sort'SuperSet)
then [binEq x (mkInjection sort' sort y)]
else []
test3 = if (Set.member sort' sortSuperSet)
then [binEq (mkInjection sort sort' x) y]
else []
test4 = if (null commonSuperList)
then []
else map test4_atSort commonSuperList
test4_atSort s = binEq (mkInjection sort s x)
(mkInjection sort' s y)
in binEq lhs rhs
eqs = map mkEq sortList
in addPrimRec eqs isaThWithConst
--------------------------------------------------------
-- Functions for producing the Justification theorems --
--------------------------------------------------------
-- | Add all justification theorems to an Isabelle Theory. We need to
-- the CASL signature (from the datapart) and the PFOL Signature to
-- pass it on. We could recalculate the PFOL signature from the CASL
-- signature here, but we dont as it can be passed in. We need the
-- PFOL signature which is the data part CASL signature afetr
-- translation to PCFOL (i.e. without subsorting)
addJustificationTheorems :: CASLSign.CASLSign -> CASLSign.CASLSign ->
IsaTheory -> IsaTheory
addJustificationTheorems caslSign pfolSign isaTh =
let sortRel = Rel.toList(CASLSign.sortRel caslSign)
sorts = Set.toList(CASLSign.sortSet caslSign)
in addTransitivityTheorem sorts sortRel
$ addAllInjectivityTheorems pfolSign sorts sortRel
$ addAllDecompositionTheorem pfolSign sorts sortRel
$ addSymmetryTheorem sorts
$ addReflexivityTheorem
$ isaTh
-- | Add the reflexivity theorem and proof to an Isabelle Theory
addReflexivityTheorem :: IsaTheory -> IsaTheory
addReflexivityTheorem isaTh =
let name = reflexivityTheoremS
x = mkFree "x"
thmConds = []
thmConcl = binEq_PreAlphabet x x
proof' = IsaProof {
proof = [Apply (Induct x),
Apply Auto],
end = Done
}
in addTheoremWithProof name thmConds thmConcl proof' isaTh
-- | Add the symmetry theorem and proof to an Isabelle Theory. We need
-- to know the number of sorts, but instead we are given a list of
-- all sorts
addSymmetryTheorem :: [SORT] -> IsaTheory -> IsaTheory
addSymmetryTheorem sorts isaTh =
let numSorts = length(sorts)
name = symmetryTheoremS
x = mkFree "x"
y = mkFree "y"
thmConds = [binEq_PreAlphabet x y]
thmConcl = binEq_PreAlphabet y x
inductY = concat $ map (\i -> [Prefer (i*numSorts+1),
Apply (Induct y)])
[0..(numSorts-1)]
proof' = IsaProof {
proof = [Apply (Induct x)] ++ inductY ++ [Apply Auto],
end = Done
}
in addTheoremWithProof name thmConds thmConcl proof' isaTh
-- | Add all the decoposition theorems and proofs
addAllDecompositionTheorem :: CASLSign.CASLSign -> [SORT] -> [(SORT,SORT)] ->
IsaTheory -> IsaTheory
addAllDecompositionTheorem pfolSign sorts sortRel isaTh =
let tripples = [(s1,s2,s3) |
(s1,s2) <- sortRel, (s2',s3) <- sortRel, s2==s2']
in foldl (addDecompositionTheorem pfolSign sorts sortRel) isaTh tripples
-- | Add the decomposition theorem and proof which should be deduced
-- from the transitivity axioms from the translation CASL2PCFOL;
-- CASL2SubCFOL for a pair of injections represented as a tripple of
-- sorts. e.g. (S,T,U) means produce the lemma and proof for
-- inj_T_U(inj_S_T(x)) = inj_S_U(x). As a work around, we need to
-- know all sorts to pass them on.
addDecompositionTheorem :: CASLSign.CASLSign -> [SORT] -> [(SORT,SORT)] ->
IsaTheory -> (SORT,SORT,SORT) -> IsaTheory
addDecompositionTheorem pfolSign sorts sortRel isaTh (s1,s2,s3) =
let x = mkFree "x"
-- These 5 lines make use of currying
injOp_s1_s2 = mkInjection s1 s2
injOp_s2_s3 = mkInjection s2 s3
injOp_s1_s3 = mkInjection s1 s3
inj_s1_s2_s3_x = injOp_s2_s3 (injOp_s1_s2 x)
inj_s1_s3_x = injOp_s1_s3 x
definedOp_s1 = getDefinedOp sorts s1
definedOp_s3 = getDefinedOp sorts s3
collectionTransAx = getCollectionTransAx sortRel
collectionTotAx = getCollectionTotAx pfolSign
collectionNotDefBotAx = getCollectionNotDefBotAx sorts
name = getDecompositionThmName(s1, s2, s3)
conds = []
concl = binEq inj_s1_s2_s3_x inj_s1_s3_x
proof' = IsaProof [Apply(CaseTac (definedOp_s3 inj_s1_s2_s3_x)),
-- Case 1
Apply(SubgoalTac(definedOp_s1 x)),
Apply(Insert collectionTransAx),
Apply(Simp),
Apply(SimpAdd Nothing collectionTotAx),
-- Case 2
Apply(SubgoalTac(
termAppl notOp(definedOp_s3 inj_s1_s3_x))),
Apply(SimpAdd Nothing collectionNotDefBotAx),
Apply(SimpAdd Nothing collectionTotAx)]
Done
in addTheoremWithProof name conds concl proof' isaTh
-- | Add all the injectivity theorems and proofs
addAllInjectivityTheorems :: CASLSign.CASLSign -> [SORT] -> [(SORT,SORT)] ->
IsaTheory -> IsaTheory
addAllInjectivityTheorems pfolSign sorts sortRel isaTh =
foldl (addInjectivityTheorem pfolSign sorts sortRel) isaTh sortRel
-- | Add the injectivity theorem and proof which should be deduced
-- from the embedding_Injectivity axioms from the translation
-- CASL2PCFOL; CASL2SubCFOL for a single injection represented as a
-- pair of sorts. As a work around, we need to know all sorts to
-- pass them on
addInjectivityTheorem :: CASLSign.CASLSign -> [SORT] -> [(SORT,SORT)] ->
IsaTheory -> (SORT,SORT) -> IsaTheory
addInjectivityTheorem pfolSign sorts sortRel isaTh (s1,s2) =
let x = mkFree "x"
y = mkFree "y"
injX = mkInjection s1 s2 x
injY = mkInjection s1 s2 y
definedOp_s1 = getDefinedOp sorts s1
definedOp_s2 = getDefinedOp sorts s2
collectionEmbInjAx = getCollectionEmbInjAx sortRel
collectionTotAx = getCollectionTotAx pfolSign
collectionNotDefBotAx = getCollectionNotDefBotAx sorts
name = getInjectivityThmName(s1, s2)
conds = [binEq injX injY]
concl = binEq x y
proof' = IsaProof [Apply(CaseTac (definedOp_s2 injX)),
-- Case 1
Apply(SubgoalTac(definedOp_s1 x)),
Apply(SubgoalTac(definedOp_s1 y)),
Apply(Insert collectionEmbInjAx),
Apply(Simp),
Apply(SimpAdd Nothing collectionTotAx),
Apply(SimpAdd (Just No_asm_use) collectionTotAx),
-- Case 2
Apply(SubgoalTac(termAppl notOp (definedOp_s1 x))),
Apply(SubgoalTac(termAppl notOp(definedOp_s1 y))),
Apply(SimpAdd Nothing collectionNotDefBotAx),
Apply(SimpAdd Nothing collectionTotAx),
Apply(SimpAdd (Just No_asm_use) collectionTotAx)]
Done
in addTheoremWithProof name conds concl proof' isaTh
-- | Add the transitivity theorem and proof to an Isabelle Theory. We
-- need to know the number of sorts to know how much induction to
-- perfom and also the sub-sort relation to build the collection of
-- injectivity theorem names
addTransitivityTheorem :: [SORT] -> [(SORT,SORT)] -> IsaTheory -> IsaTheory
addTransitivityTheorem sorts sortRel isaTh =
let colInjThmNames = getColInjectivityThmName sortRel
colDecompThmNames = getColDecompositionThmName sortRel
numSorts = length(sorts)
name = transitivityS
x = mkFree "x"
y = mkFree "y"
z = mkFree "z"
thmConds = [binEq_PreAlphabet x y, binEq_PreAlphabet y z]
thmConcl = binEq_PreAlphabet x z
inductY = concat $ map (\i -> [Prefer (i*numSorts+1),
Apply (Induct y)])
[0..(numSorts-1)]
inductZ = concat $ map (\i -> [Prefer (i*numSorts+1),
Apply (Induct z)])
[0..((numSorts^(2::Int))-1)]
proof' = IsaProof {
proof = [Apply (Induct x)] ++
inductY ++
inductZ ++
[Apply (AutoSimpAdd Nothing
(colDecompThmNames ++ colInjThmNames))],
end = Done
}
in addTheoremWithProof name thmConds thmConcl proof' isaTh
--------------------------------------------------------------
-- Functions for producing instances of equivalence classes --
--------------------------------------------------------------
-- | Function to add preAlphabet as an equivalence relation to an
-- Isabelle theory
addInstansanceOfEquiv :: IsaTheory -> IsaTheory
addInstansanceOfEquiv isaTh =
let eqvSort = [IsaClass eqvTypeClassS]
eqvProof = IsaProof [] (By (Other "intro_classes"))
equivSort = [IsaClass equivTypeClassS]
equivProof = IsaProof [Apply (Other "intro_classes"),
Apply (Other ("unfold " ++ preAlphabetSimS
++ "_def")),
Apply (Other ("rule " ++ reflexivityTheoremS)),
Apply (Other ("rule " ++ transitivityS
++", auto")),
Apply (Other ("rule " ++ symmetryTheoremS
++ ", simp"))]
Done
x = mkFree "x"
y = mkFree "y"
defLhs = binEqvSim x y
defRhs = binEq_PreAlphabet x y
in addInstanceOf preAlphabetS [] equivSort equivProof
$ addDef preAlphabetSimS defLhs defRhs
$ addInstanceOf preAlphabetS [] eqvSort eqvProof
$ isaTh
-------------------------------------------------------------
-- Functions for producing the alphabet type --
-------------------------------------------------------------
-- | Function to add the Alphabet type (type synonym) to an Isabelle theory
addAlphabetType :: IsaTheory -> IsaTheory
addAlphabetType isaTh =
let isaTh_sign = fst isaTh
isaTh_sign_tsig = tsig isaTh_sign
myabbrs = abbrs isaTh_sign_tsig
abbrsNew = Map.insert alphabetS ([], preAlphabetQuotType) myabbrs
isaTh_sign_updated = isaTh_sign {
tsig = (isaTh_sign_tsig {abbrs =abbrsNew})
}
in (isaTh_sign_updated, snd isaTh)
-------------------------------------------------------------
-- Functions for producing the bar types --
-------------------------------------------------------------
-- | Function to add all the bar types to an Isabelle theory.
addAllBarTypes :: [SORT] -> IsaTheory -> IsaTheory
addAllBarTypes sorts isaTh = foldl addBarType isaTh sorts
-- | Function to add the bar types of a sort to an Isabelle theory.
addBarType :: IsaTheory -> SORT -> IsaTheory
addBarType isaTh sort =
let sortBarString = mkSortBarString sort
barType = Type {typeId = sortBarString, typeSort = [], typeArgs =[]}
isaTh_sign = fst isaTh
isaTh_sen = snd isaTh
x = mkFree "x"
y = mkFree "y"
rhs = termAppl (conDouble (mkPreAlphabetConstructor sort)) y
bin_eq = binEq x $ termAppl (conDouble classS ) rhs
exist_eq =termAppl (conDouble exS) (Abs y bin_eq NotCont)
subset = SubSet x alphabetType exist_eq
sen = TypeDef barType subset (IsaProof [] (By Auto))
namedSen = (makeNamed sortBarString sen)
in (isaTh_sign, isaTh_sen ++ [namedSen])
-------------------------------------------------------------
-- Functions for producing the choose functions --
-------------------------------------------------------------
-- | Add all choose functions for a given list of sorts to an Isabelle
-- theory.
addAllChooseFunctions :: [SORT] -> IsaTheory -> IsaTheory
addAllChooseFunctions sorts isaTh =
let isaTh' = foldl addChooseFunction isaTh sorts --add function and def
in foldl addChooseFunctionLemma isaTh' sorts --add theorem and proof
-- | Add a single choose function for a given sort to an Isabelle
-- theory. The following Isabelle code is produced by this function:
-- consts choose_Nat :: "Alphabet => Nat"
-- defs choose_Nat_def: "choose_Nat x == contents{y . class(C_Nat y) = x}"
addChooseFunction :: IsaTheory -> SORT -> IsaTheory
addChooseFunction isaTh sort =
let --constant
sortType = Type {typeId = convertSort2String sort,
typeSort = [],
typeArgs =[]}
chooseFunName = mkChooseFunName sort
chooseFunType = mkFunType alphabetType sortType
-- definition
x = mkFree "x"
y = mkFree "y"
contentsOp = termAppl (conDouble "contents")
chooseOp = termAppl (conDouble chooseFunName)
sortConsOp = termAppl (conDouble (mkPreAlphabetConstructor sort))
bin_eq = binEq (classOp $ sortConsOp y) x
subset = SubSet y sortType bin_eq
lhs = chooseOp x
rhs = contentsOp (Set subset)
in addDef chooseFunName lhs rhs -- Add defintion to theory
$ addConst chooseFunName chooseFunType isaTh -- Add constant to theory
-- | Add a single choose function lemma for a given sort to an
-- Isabelle theory. The following Isabelle code is produced by this
-- function: lemma "choose_Nat (class (C_Nat x)) = x". The proof is
-- also produced.
addChooseFunctionLemma :: IsaTheory -> SORT -> IsaTheory
addChooseFunctionLemma isaTh sort =
let sortType = Type {typeId = convertSort2String sort,
typeSort = [],
typeArgs =[]}
chooseFunName = mkChooseFunName sort
x = mkFree "x"
y = mkFree "y"
chooseOp = termAppl (conDouble chooseFunName)
sortConsOp = termAppl (conDouble (mkPreAlphabetConstructor sort))
-- theorm
subgoalTacTermLhsBinEq = binEq (classOp $ sortConsOp y)
(classOp $ sortConsOp x)
subgoalTacTermLhs = Set $ SubSet y sortType subgoalTacTermLhsBinEq
subgoalTacTermRhs = Set $ FixedSet [x]
subgoalTacTerm = binEq subgoalTacTermLhs subgoalTacTermRhs
thmConds = []
thmConcl = binEq (chooseOp $ classOp $ sortConsOp x) x
proof' = IsaProof [Apply (Other ("unfold " ++ chooseFunName ++ "_def")),
Apply (SubgoalTac subgoalTacTerm),
Apply(Simp),
Apply(SimpAdd Nothing [quotEqualityS]),
Apply(Other ("unfold "++ preAlphabetSimS
++ "_def")),
Apply(Auto)]
Done
in addTheoremWithProof chooseFunName thmConds thmConcl proof' isaTh
------------------------------------------------------
-- Functions for producing the integration theorems --
------------------------------------------------------
-- | Add all the integration theorems. We need to know all the sorts
-- to produce all the theorems. We need to know the CASL signature
-- of the data part to pass it on as an argument.
addAllIntegrationTheorems :: [SORT] -> CASLSign.CASLSign -> IsaTheory ->
IsaTheory
addAllIntegrationTheorems sorts caslSign isaTh =
let pairs = [(s1,s2) | s1 <- sorts, s2 <- sorts]
in foldl (addIntegrationTheorem_A caslSign) isaTh pairs
-- | Add Integration theorem A -- Compare to elements of the Alphabet.
-- We add the integration theorem based on the sorts of both
-- elements of the alphabet. We need to know the subsort relation to
-- find the highest common sort, but we pass in the CASL signature
-- for the data part.
addIntegrationTheorem_A :: CASLSign.CASLSign -> IsaTheory -> (SORT,SORT) ->
IsaTheory
addIntegrationTheorem_A caslSign isaTh (s1,s2) =
let sortRel = Rel.toList(CASLSign.sortRel caslSign)
s1SuperSet = CASLSign.supersortsOf s1 caslSign
s2SuperSet = CASLSign.supersortsOf s2 caslSign
commonSuperList = Set.toList (Set.intersection
(Set.insert s1 s1SuperSet)
(Set.insert s2 s2SuperSet))
x = mkFree "x"
y = mkFree "y"
s1ConsOp = termAppl (conDouble (mkPreAlphabetConstructor s1))
s2ConsOp = termAppl (conDouble (mkPreAlphabetConstructor s2))
rhs = binEq (classOp $ s1ConsOp x) (classOp $ s2ConsOp y)
lhs = if (null commonSuperList)
then false
else
-- BUG pick any common sort for now (this does hold
-- and is valid) we should pick the top most one.
let s' = head commonSuperList
in binEq (mkInjection s1 s' x) (mkInjection s2 s' y)
thmConds = []
thmConcl = binEq rhs lhs
-- theorem names for proof
colInjThmNames = getColInjectivityThmName sortRel
colDecompThmNames = getColDecompositionThmName sortRel
proof' = IsaProof [Apply (SimpAdd Nothing [quotEqualityS]),
Apply (Other ("unfold " ++ preAlphabetSimS
++ "_def")),
Apply (AutoSimpAdd Nothing
(colDecompThmNames ++ colInjThmNames))]
Done
in addTheoremWithProof "IntegrationTheorem_A"
thmConds thmConcl proof' isaTh
--------------------------------------------------------------------------
-- Functions for adding the process name datatype to an Isabelle theory --
--------------------------------------------------------------------------
-- | Add process name datatype which has a constructor for each
-- process name (along with the arguments for the process) in the
-- CspCASL Signature to an Isabelle theory
addProcNameDatatype :: CspSign -> IsaTheory -> IsaTheory
addProcNameDatatype cspSign isaTh =
let -- Create a list of pairs of process names and thier profiles
procSetList = Map.toList (procSet cspSign)
procNameDomainEntry = mkProcNameDE procSetList
in updateDomainTab procNameDomainEntry isaTh
-- | Make a proccess name Domain Entry from a list of a Process name and profile
-- pair. This creates a data type for the process names.
mkProcNameDE :: [(PROCESS_NAME, ProcProfile)] -> DomainEntry
mkProcNameDE processes =
let -- The a list of pairs of constructors and their arguments
constructors = map mk_cons processes
-- Take a proccess name and its argument sorts (also its
-- commAlpha - thrown away) and make a pair representing the
-- constructor and the argument types
mk_cons (procName, (ProcProfile sorts _)) =
(mkVName (mkProcNameConstructor procName), map sortToTyp sorts)
-- Turn a sort in to a Typ suitable for the domain entry The
-- processes need to have arguments of the bar variants of
-- the sorts not the original sorts
sortToTyp sort = Type {typeId = mkSortBarString sort,
typeSort = [isaTerm],
typeArgs = []}
in
(procNameType, constructors)
-------------------------------------------------------------------------
-- Functions adding the process map function to an Isabelle theory --
-------------------------------------------------------------------------
-- | Add the function procMap to an Isabelle theory. This function
-- maps process names to real processes build using the same names
-- and the alphabet i.e., ProcName => (ProcName, Alphabet) proc. We
-- need to know the CspCASL sentences and the casl signature (data
-- part)
addProcMap :: [Named CspCASLSen] -> CASLSign.Sign () () -> IsaTheory ->
IsaTheory
addProcMap namedSens caslSign isaTh =
let
-- Extend Isabelle theory with additional constant
isaThWithConst = addConst procMapS procMapType isaTh
-- Get the plain sentences from the named senetences
sens = map (\namedsen -> sentence namedsen) namedSens
-- Filter so we only have proccess equations and no CASL senetences
processEqs = filter isProcessEq sens
-- the term representing the procMap that tajes a term as a
-- parameter
procMapTerm = termAppl (conDouble procMapS)
-- Make a single equation for the primrec from a process equation
-- BUG HERE - this next part is not right - underscore is bad
mkEq (ProcessEq procName _ _ proc) =
let -- Make the name (string) for this process
procNameString = convertProcessName2String procName
-- Change the name to a term
procNameTerm = conDouble procNameString
-- Turn the list of variables into a list of Isabelle
-- free variables
varTerms = [] -- BUG - should be somehting like map (\x -> mkFree (show x)) vars
-- Make a lhs term built of termAppl applied to the
-- proccess name and the variables
lhs = procMapTerm (foldl termAppl (procNameTerm) varTerms)
rhs = transProcess caslSign proc
in binEq lhs rhs
-- Build equations for primrec using process equations
eqs = map mkEq processEqs
in addPrimRec eqs isaThWithConst
----------------------------------------------
-- Function to help keep strings consistent --
----------------------------------------------
-- | Return the list of strings of all gn_totality axiom names. This
-- function is not implemented in a satisfactory way.
getCollectionTotAx :: CASLSign.CASLSign -> [String]
getCollectionTotAx pfolSign =
let opList = Map.toList $ CASLSign.opMap pfolSign
-- This filter is not quite right
totFilter (_,setOpType) = let listOpType = Set.toList setOpType
in CASLSign.opKind (head listOpType) == Total
totList = filter totFilter opList
mkTotAxName = (\i -> "ga_totality"
++ (if (i==0)
then ""
else ("_" ++ show i))
)
in map mkTotAxName [0 .. (length(totList) - 1)]
-- | Return the name of the definedness function for a sort. We need
-- to know all sorts to perform this workaround
-- This function is not implemented in a satisfactory way
getDefinedName :: [SORT] -> SORT -> String
getDefinedName sorts s =
let index = List.elemIndex s sorts
str = case index of
Nothing -> ""
Just i -> show (i + 1)
in "gn_definedX" ++ str
-- | Return the name of the injection as it is used in the alternative
-- syntax of the injection from one sort to another.
-- This function is not implemented in a satisfactory way
getInjectionName :: SORT -> SORT -> String
getInjectionName s s' =
let t = CASLSign.toOP_TYPE(CASLSign.OpType{CASLSign.opKind = Total,
CASLSign.opArgs = [s],
CASLSign.opRes = s'})
injName = show $ CASLInject.uniqueInjName t
in injName
-- | Return the injection name of the injection from one sort to another
-- This function is not implemented in a satisfactory way
getDefinedOp :: [SORT] -> SORT -> Term -> Term
getDefinedOp sorts s t =
termAppl (con $ VName (getDefinedName sorts s) $ Nothing) t
-- | Return the term representing the injection of a term from one sort to another
-- note: the term is returned if both sorts are the same
-- This function is not implemented in a satisfactory way
mkInjection :: SORT -> SORT -> Term -> Term
mkInjection s s' t =
let injName = getInjectionName s s'
replace string c s1 = concat (map (\x -> if x==c
then s1
else [x]) string)
injOp = Const {
termName= VName {
new = ("X_" ++ injName),
altSyn = Just (AltSyntax ((replace injName '_' "'_")
++ "/'(_')") [3] 999)
},
termType = Hide {
typ = Type {
typeId ="dummy",
typeSort = [IsaClass "type"],
typeArgs = []
},
kon = NA,
arit= Nothing
}
}
in if s == s'
then t
else termAppl injOp t
-- | Return the list of string of all embedding_injectivity axioms
-- produced by the translation CASL2PCFOL; CASL2SubCFOL. This
-- function is not implemented in a satisfactory way.
getCollectionEmbInjAx :: [(SORT,SORT)] -> [String]
getCollectionEmbInjAx sortRel =
let mkEmbInjAxName = (\i -> "ga_embedding_injectivity"
++ (if (i==0)
then ""
else ("_" ++ show i))
)
in map mkEmbInjAxName [0 .. (length(sortRel) - 1)]
-- This function is not implemented in a satisfactory way
-- Return the list of strings of all ga_notDefBottom axioms
getCollectionNotDefBotAx :: [SORT] -> [String]
getCollectionNotDefBotAx sorts =
let mkNotDefBotAxName = (\i -> "ga_notDefBottom"
++ (if (i==0)
then ""
else ("_" ++ show i)))
in map mkNotDefBotAxName [0 .. (length(sorts) - 1)]
-- This function is not implemented in a satisfactory way
-- Return the list of string of all decomposition theorem names that we generate
getColDecompositionThmName :: [(SORT,SORT)] -> [String]
getColDecompositionThmName sortRel =
let tripples = [(s1,s2,s3) |
(s1,s2) <- sortRel, (s2',s3) <- sortRel, s2==s2']
in map getDecompositionThmName tripples
-- Produce the theorem name of the decomposition theorem that we produce
getDecompositionThmName :: (SORT,SORT,SORT) -> String
getDecompositionThmName (s, s', s'') =
"decomposition_" ++ (convertSort2String s) ++ "_" ++ (convertSort2String s')
++ "_" ++ (convertSort2String s'')
-- This function is not implemented in a satisfactory way
-- Return the list of strings of the injectivity theorem names
-- that we generate
getColInjectivityThmName :: [(SORT,SORT)] -> [String]
getColInjectivityThmName sortRel = map getInjectivityThmName sortRel
-- Produce the theorem name of the injectivity theorem that we produce
getInjectivityThmName :: (SORT,SORT) -> String
getInjectivityThmName (s, s') =
"injectivity_" ++ (convertSort2String s) ++ "_" ++ (convertSort2String s')
-- This function is not implemented in a satisfactory way. Return the
-- list of strings of all transitivity axioms produced by the
-- translation CASL2PCFOL; CASL2SubCFOL
getCollectionTransAx :: [(SORT,SORT)] -> [String]
getCollectionTransAx sortRel =
let tripples = [(s1,s2,s3) |
(s1,s2) <- sortRel, (s2',s3) <- sortRel, s2==s2']
mkEmbInjAxName = (\i -> "ga_transitivity"
++ (if (i==0)
then ""
else ("_" ++ show i))
)
in map mkEmbInjAxName [0 .. (length(tripples) - 1)]