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maude-npa.maude
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***(
Maude-NPA, Version: [3.1.4] [June 2nd 2020]
Copyright (c) 2020, University of Illinois
All rights reserved.
Redistribution and use in source and binary forms, with or without modification,
are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright notice,
this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright notice,
this list of conditions and the following disclaimer in the documentation
and/or other materials provided with the distribution.
* Neither the name of the University of Illinois nor the names of its contributors
may be used to endorse or promote products derived from this software without
specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO,
THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE
FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
-----------------------------------------------------------------------------------------------------------
Copyright (c) 2020. To the extent that a federal employee is an author of
a portion of the software or a derivative work thereof, no copyright is
claimed by the United States Government, as represented by the Secretary
of the Navy ("GOVERNMENT") under Title 17, U.S. Code. All Other Rights Reserved.
Permission to use, copy, and modify this software and its documentation is
hereby granted, provided that both the copyright notice and this permission
notice appear in all copies of the software, derivative works or modified
versions, and any portions thereof, and that both notices appear in
supporting documentation.
GOVERNMENT ALLOWS FREE USE OF THIS SOFTWARE IN ITS "AS IS" CONDITION AND
DISCLAIM ANY LIABILITY OF ANY KIND FOR ANY DAMAGES WHATSOEVER RESULTING
FROM THE USE OF THIS SOFTWARE.
GOVERNMENT requests users of this software to return modifications,
improvements or extensions that they make to:
-or-
Naval Research Laboratory, Code 5543
4555 Overlook Avenue, SW
Washington, DC 20375
)***
fmod BANNER-MAUDENPA is
pr QID-LIST .
op banner : -> QidList .
eq banner =
'\n '\t '\s '\s '\s '\s 'Maude-NPA 'Version: '3.1.4 ' '`( 'June '2nd '2020 '`)
'\n '\t '\s '\s '\s '\s 'with 'direct 'composition '`, '\s 'irreducibility 'constraints 'and 'time
'\n '\t '\s '\s '\s '\s '`( 'To 'be 'run 'with 'Maude '3.0 'or 'above '`)
'\n '\t '\s '\s '\s '\s 'Copyright '\s '`( 'c '`) '\s '2020 '`, '\s 'University 'of 'Illinois
'\n '\t '\s '\s '\s '\s 'All 'rights 'reserved.
'\n
'\n '\s 'Commands:
'\n '\s 'red 'unification? '. '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s 'returns 'the 'unification 'algorithm 'to 'be 'used
'\n '\s 'red 'new-strands? '. '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s 'returns 'the 'actual 'protocol 'strands
'\n '\s 'red 'displayGrammars '. '\s '\s '\s '\s '\s '\s '\s '\s 'for 'generating 'grammars
'\n '\s 'red 'run '`( 'X '`, 'Y '`) '. '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s 'for 'Y 'backwards 'analysis 'steps 'for 'attack 'pattern 'X
'\n '\s 'red 'debug '`( 'X '`, 'Y '`) '. '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s 'more 'information 'than 'run 'command
'\n '\s 'red 'digest '`( 'X '`, 'Y '`) '. '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s 'less 'information 'than 'run 'command
'\n '\s 'red 'summary '`( 'X '`, 'Y '`) '. '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s 'for 'summary 'of 'analysis 'steps
'\n '\s 'red 'ids '`( 'X '`, 'Y '`) '. '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s 'for 'set 'of 'state 'ids
'\n '\s 'red 'initials '`( 'X '`, 'Y '`) '. '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s '\s 'for 'showing 'only 'initial 'steps
.
endfm
fmod META-LEVEL-MNPA is
pr META-LEVEL * (op `{_`,_`} : Term Nat -> SmtResult to `{_`,_`}Smt ) .
endfm
fmod UNIFICATIONTRIPLE is
protecting META-LEVEL-MNPA .
protecting INT .
--- UnificationPair --------------------------------------------
---sorts UnificationPair UnificationPair? .
---op {_,_} : Substitution Nat -> UnificationPair [ctor] .
---subsort UnificationPair < UnificationPair? .
---op noUnifier : -> UnificationPair? [ctor] .
op getSubst : UnificationPair -> Substitution .
eq getSubst({S1:Substitution, N:Nat}) = S1:Substitution .
op getNextVar : UnificationPair -> Nat .
eq getNextVar({S1:Substitution, N:Nat}) = N:Nat .
--- UnificationTriple --------------------------------------------
---sorts UnificationTriple UnificationTriple? .
---op {_,_,_} : Substitution Substitution Nat -> UnificationTriple [ctor] .
---subsort UnificationTriple < UnificationTriple? .
---op noUnifier : -> UnificationTriple? [ctor] .
op getLSubst : UnificationTriple -> Substitution .
eq getLSubst({S1:Substitution, S2:Substitution, N:Nat}) = S1:Substitution .
op getRSubst : UnificationTriple -> Substitution .
eq getRSubst({S1:Substitution, S2:Substitution, N:Nat}) = S2:Substitution .
op getNextVar : UnificationTriple -> Nat .
eq getNextVar({S1:Substitution, S2:Substitution, N:Nat}) = N:Nat .
endfm
fmod TERM-HANDLING is
protecting META-TERM .
protecting META-LEVEL-MNPA .
protecting EXT-BOOL . *** For and-then
var T T' T'' : Term .
var C C' : Constant .
var QIL : QidList .
var N N' : Nat .
var NL NL' : NatList .
var Q F F' : Qid .
var AtS : AttrSet .
var EqS : EquationSet .
var Eq : Equation .
var Cond : Condition .
var TP : Type .
var TPL TPL' : TypeList .
var TL TL' TL'' : TermList .
var B : Bool .
var V V' : Variable .
var Ct : Context .
var CtL : NeCTermList .
var NeTL : NeTermList .
var M : Module .
*** root ******************************
op root : Term -> Qid .
eq root(V) = V .
eq root(C) = C .
eq root(F[TL]) = F .
*** size ******************************
op size : TermList -> Nat .
eq size(empty) = 0 .
eq size((T,TL)) = s(size(TL)) .
*** elem_of_ *****************************************************
op elem_of_ : Nat TermList ~> Term .
eq elem 1 of (T,TL) = T .
eq elem s(s(N)) of (T,TL) = elem s(N) of TL .
*** subTerm_of_ *****************************************************
op subTerm_of_ : NatList Term ~> Term .
eq subTerm NL of T = subTerm* NL of T .
op subTerm*_of_ : NatList Term ~> Term .
eq subTerm* nil of T = T .
eq subTerm* N NL of (F[TL]) = subTerm* NL of (elem N of TL) .
*** ToDo: UPGRADE THIS NOTION TO MODULO AC *********************
*** is_subTermOf_ *****************************************************
op is_subTermOf_ : Term TermList -> Bool .
eq is T subTermOf (T',NeTL) = is T subTermOf T'
or-else is T subTermOf NeTL .
eq is T subTermOf T = true .
eq is T subTermOf T' = is T subTermOf* T' [owise] .
op is_subTermOf*_ : Term TermList -> Bool .
eq is T subTermOf* (F[TL]) = is T subTermOf TL .
eq is T subTermOf* T' = false [owise] .
*** noVarOfSort_In_ *****************************************************
op noVarOfSort_In_ : Type TermList -> Bool .
eq noVarOfSort T:Type In V = getType(V) =/= T:Type .
eq noVarOfSort T:Type In (F[TL]) = noVarOfSort T:Type In TL .
eq noVarOfSort T:Type In (T',NeTL)
= noVarOfSort T:Type In T' and noVarOfSort T:Type In NeTL .
eq noVarOfSort T:Type In X:TermList = true [owise] .
*** findSubTermOf_In_ ***********************************************
op findSubTermOf_In_ : NeCTermList TermList ~> Term .
eq findSubTermOf (TL, [], TL') In (TL, T, TL') = T .
eq findSubTermOf (TL, F[CtL], TL'') In (TL, F[TL'], TL'')
= findSubTermOf CtL In TL' .
*** replaceElem_of_by_ ****************************************************
op replaceElem_of_by_ : Nat TermList Term ~> TermList .
eq replaceElem 1 of (T,TL) by T' = (T',TL) .
eq replaceElem s(s(N)) of (T,TL) by T' = (T,replaceElem s(N) of TL by T') .
*** replaceSubTerm_of_by_ *************************************************
op replaceSubTerm_of_by_ : NatList TermList Term ~> TermList .
eq replaceSubTerm nil of T by T' = T' .
eq replaceSubTerm N NL of (F[TL]) by T'
= F[replaceSubTermL N NL of TL by T'] .
op replaceSubTermL_of_by_ : NatList TermList Term ~> TermList .
eq replaceSubTermL 1 NL of (T,TL) by T'
= (replaceSubTerm NL of T by T', TL) .
eq replaceSubTermL s(s(N)) NL of (T,TL) by T'
= (T,replaceSubTermL s(N) NL of TL by T') .
op replaceTerm_by_in_ : Term Term TermList ~> TermList .
eq replaceTerm T by T' in T = T' .
eq replaceTerm T by T' in (F[TL]) = F[replaceTerm T by T' in TL] .
eq replaceTerm T by T' in T'' = T'' [owise] .
eq replaceTerm T by T' in (T'',NeTL)
= (replaceTerm T by T' in T'',replaceTerm T by T' in NeTL) .
*** context replacement **************************************************
op _[_] : Context Context -> Context .
op _[_] : NeCTermList Context -> NeCTermList .
eq [] [ Ct ] = Ct .
eq (F[CtL])[ Ct ] = F[ CtL [ Ct ] ] .
eq (CtL,NeTL) [Ct] = (CtL [Ct] ), NeTL .
eq (NeTL,CtL) [Ct] = NeTL, (CtL [Ct] ) .
op _[_] : Context Term -> Term .
op _[_] : NeCTermList Term -> TermList .
eq [] [ T ] = T .
eq (F[CtL])[ T ] = F[ CtL [ T ] ] .
eq (CtL,NeTL) [T] = (CtL [T] ), NeTL .
eq (NeTL,CtL) [T] = NeTL, (CtL [T] ) .
*** is_substring_ *****************************************
op is_substring_ : Qid Qid -> Bool [memo] .
eq is F:Qid substring F':Qid
= rfind(string(F':Qid), string(F:Qid), length(string(F':Qid))) =/= notFound .
*** addprefix_To_ addsufix_To_ *****************************************
op addprefix_To_ : Qid Variable -> Variable [memo] .
eq addprefix Q To V
= qid(string(Q) + string(getName(V)) + ":" + string(getType(V))) .
op addprefix_To_ : Qid Constant -> Constant [ditto] .
eq addprefix Q To F
= if noUnderBar(F) and getName(F) :: Qid then
if getType(F) :: Type then
qid(string(Q) + string(getName(F)) + "." + string(getType(F)))
else
qid(string(Q) + string(getName(F)))
fi
else
qid(string(Q) + string(F))
fi .
op addsufix_To_ : Qid Variable -> Variable [memo] .
eq addsufix Q To V
= qid(string(getName(V)) + string(Q) + ":" + string(getType(V))) .
op addsufix_To_ : Qid Constant -> Constant [ditto] .
eq addsufix Q To F
= if noUnderBar(F) and getName(F) :: Qid then
if getType(F) :: Type then
qid(string(getName(F)) + string(Q) + "." + string(getType(F)))
else
qid(string(getName(F)) + string(Q))
fi
else
qid(string(F) + string(Q))
fi .
op addType_ToVar_ : Type Qid -> Variable [memo] .
eq addType TP:Qid ToVar V:Qid
= qid(string(V:Qid) + ":" + string(TP:Qid)) .
*** noUnderBar (auxiliary) ****************************
op noUnderBar : Qid -> Bool .
eq noUnderBar(F)
= rfind(string(F), "_", length(string(F))) == notFound .
*** addType ******************************
op addType : Qid Type -> Qid .
eq addType(F,TP)
= if noUnderBar(F) and getName(F) :: Qid then
qid( string(getName(F)) + "." + string(TP) )
else
qid( string(F) + "." + string(TP) )
fi .
*** addTypeVar ******************************
op addTypeVar : Qid Type -> Qid .
eq addTypeVar(F,TP)
= qid( string(F) + ":" + string(TP) ) .
endfm
fmod SUBSTITUTION-HANDLING is
protecting META-TERM .
protecting META-LEVEL-MNPA .
protecting TERM-HANDLING .
var S S' Subst Subst' : Substitution .
var V V' : Variable .
var C C' : Constant .
var Ct : Context .
var T T' T1 T2 T1' T2' T1'' T2'' : Term .
var F F' : Qid .
var TL TL' TL1 TL2 TL1' TL2' : TermList .
var Att : AttrSet .
var RLS : RuleSet .
var Rl : Rule .
var TP : Type .
var N : Nat .
var NeTL : NeTermList .
var CtL : NeCTermList .
--- Apply Substitution to Term --------------------------------------------
op _<<_ : Term Substitution -> Term .
eq TL << none = TL .
eq C << Subst = C .
eq V << ((V <- T) ; Subst) = T .
eq V << Subst = V [owise] .
eq F[TL] << Subst = F[TL << Subst] .
op _<<_ : TermList Substitution -> TermList .
eq (T, NeTL) << Subst = (T << Subst, NeTL << Subst) .
eq empty << Subst = empty .
op _<<_ : Context Substitution -> Context .
eq Ct << none = Ct .
eq [] << Subst = [] .
eq F[CtL,NeTL] << Subst = F[CtL << Subst,NeTL << Subst] .
eq F[NeTL,CtL] << Subst = F[NeTL << Subst, CtL << Subst] .
eq F[Ct] << Subst = F[Ct << Subst] .
op _<<_ : Substitution Substitution -> Substitution .
eq S << (none).Substitution = S .
eq (none).Substitution << S = (none).Substitution .
eq ((V' <- T) ; S') << S
= (V' <- (T << S))
;
(S' << S) .
--- Combine Substitutions -------------------------------------------------
op _.._ : Substitution Substitution -> Substitution .
eq S .. S' = (S << S') ; S' .
--- Remove Substitutions -------------------------------------------------
op _\\_ : Substitution Substitution -> Substitution .
eq (S ; S') \\ S' = S .
eq S \\ S' = S [owise] .
--- Restrict Assignments to Variables in a Term ----------------------
op _|>_ : Substitution TermList -> Substitution .
eq Subst |> TL = Subst |>* Vars(TL) .
op _|>*_ : Substitution TermList -> Substitution .
--- eq noMatch |>* TL = noMatch .
eq Subst |>* TL = Subst |>** TL [none] .
op _|>**_[_] : Substitution TermList
Substitution -> Substitution .
eq none |>** TL [Subst']
= Subst' .
eq ((V <- V) ; Subst) |>** TL [Subst']
= Subst |>** TL [Subst'] .
eq ((V <- T') ; Subst) |>** TL [Subst']
= Subst |>** TL
[Subst' ; if any V in TL then (V <- T') else none fi] .
--- Remove bindings same variable -------------------
op removeSame : Substitution -> Substitution .
eq removeSame(V <- V ; Subst) = removeSame(Subst) .
eq removeSame(Subst) = Subst [owise] .
--- Remove Variables from list ----------------------
op _intersect_ : TermList TermList -> TermList .
eq (TL1,T,TL2) intersect (TL1',T,TL2')
= (T,((TL1,TL2) intersect (TL1',TL2'))) .
eq TL intersect TL' = empty [owise] .
op _intersectVar_ : TermList TermList -> TermList .
eq TL1 intersectVar TL2
= TL1 intersectVar* Vars(TL2) .
op _intersectVar*_ : TermList TermList -> TermList .
eq (T,TL1) intersectVar* TL2
= (if any Vars(T) in TL2 then T else empty fi,TL1 intersectVar* TL2) .
eq empty intersectVar* TL2
= empty .
--- Remove Variables from list ----------------------
op _setMinus_ : TermList TermList -> TermList .
eq (TL1,T,TL2) setMinus (TL1',T,TL2')
= (TL1,TL2) setMinus (TL1',T,TL2') .
eq TL setMinus TL' = TL [owise] .
--- Variables ---
op Vars : GTermList -> TermList .
eq Vars((T,TL:GTermList)) = VarsTerm(T),Vars(TL:GTermList) .
eq Vars((Ct,TL:GTermList)) = VarsTerm(Ct),Vars(TL:GTermList) .
eq Vars(empty) = empty .
op VarsTerm : Term -> TermList . ---warning memo
eq VarsTerm(V) = V .
eq VarsTerm(F[TL:TermList]) = Vars(TL:TermList) .
eq VarsTerm(C) = empty .
op VarsTerm : Context -> TermList . ---warning memo
eq VarsTerm(F[TL:GTermList]) = Vars(TL:GTermList) .
--- membership ---
op _in_ : Term TermList -> Bool .
eq T in (TL,T,TL') = true .
eq T in TL = false [owise] .
--- membership ---
op any_in_ : TermList TermList -> Bool . --- [memo] .
eq any empty in TL = false .
eq any (TL1,T,TL2) in (TL1',T,TL2') = true .
eq any TL in TL' = false [owise] .
--- membership ---
op all_in_ : TermList TermList -> Bool . --- [memo] .
eq all empty in TL = true .
eq all (TL1,T,TL2) in (TL1',T,TL2') = all (TL1,TL2) in (TL1',T,TL2') .
eq all TL in TL' = false [owise] .
--- Occur check ---
op allVars_inVars_ : GTermList GTermList -> Bool .
eq allVars TL:GTermList inVars TL':GTermList
= all Vars(TL:GTermList) in Vars(TL':GTermList) .
op anyVars_inVars_ : GTermList GTermList -> Bool .
eq anyVars TL:GTermList inVars TL':GTermList
= any Vars(TL:GTermList) in Vars(TL':GTermList) .
op domainVars : Substitution -> TermList .
eq domainVars(V <- T ; Subst) = (V,domainVars(Subst)) .
eq domainVars(none) = empty .
op rangeVars : Substitution -> TermList .
eq rangeVars(V <- T ; Subst) = (Vars(T),rangeVars(Subst)) .
eq rangeVars(none) = empty .
op dom_inVars_ : Substitution TermList -> Bool .
eq dom Subst inVars TL = dom Subst in Vars(TL) .
op dom_in_ : Substitution TermList -> Bool .
eq dom (V <- T ; Subst) in (TL1,V,TL2) = true .
eq dom Subst in TL = false [owise] .
op dom_notInVars_ : Substitution TermList -> Bool .
eq dom Subst notInVars TL = dom Subst notIn Vars(TL) .
op dom_notIn_ : Substitution TermList -> Bool .
eq dom none notIn TL = true .
ceq dom (V <- T ; Subst) notIn TL = true if not (V in TL) .
eq dom Subst notIn TL = false [owise] .
op range_inVars_ : Substitution TermList -> Bool .
eq range Subst inVars TL = range Subst in Vars(TL) .
op range_in_ : Substitution TermList -> Bool .
eq range (V <- T ; Subst) in TL
= any Vars(T) in TL or-else range Subst in TL .
eq range none in TL
= false .
op valid-occur-check? : Substitution -> Bool .
eq valid-occur-check?(Subst)
= not (dom Subst inVars (rangeVars(Subst))) .
op extract-bindings : Substitution -> TermList .
eq extract-bindings(none) = empty .
eq extract-bindings(V <- T ; Subst) = (T,extract-bindings(Subst)) .
op isRenaming : Substitution -> Bool .
eq isRenaming(none)
= true .
eq isRenaming((V <- V') ; Subst)
= isRenaming(Subst) .
eq isRenaming((V <- T) ; Subst)
= false [owise] .
endfm
fmod TERMSET is
protecting META-LEVEL-MNPA .
protecting SUBSTITUTION-HANDLING .
sort TermSet .
subsort Term < TermSet .
op emptyTermSet : -> TermSet [ctor] .
op _|_ : TermSet TermSet -> TermSet
[ctor assoc comm id: emptyTermSet format (d n d d)] .
eq X:Term | X:Term = X:Term .
op _in_ : Term TermSet -> Bool .
eq T:Term in (T:Term | TS:TermSet) = true .
eq T:Term in TS:TermSet = false [owise] .
op TermSet : TermList -> TermSet .
eq TermSet(empty)
= emptyTermSet .
eq TermSet((T:Term,TL:TermList))
= T:Term | TermSet(TL:TermList) .
endfm
fmod RENAMING is
protecting META-TERM .
protecting META-LEVEL-MNPA .
protecting TERM-HANDLING .
protecting SUBSTITUTION-HANDLING .
protecting TERMSET .
protecting CONVERSION .
protecting QID .
protecting INT .
protecting UNIFICATIONTRIPLE .
var S S' Subst Subst' : Substitution .
var V V' : Variable .
var C C' : Constant .
var CtL : NeCTermList .
var Ct : Context .
var T T' T1 T2 T1' T2' T1'' T2'' : Term .
var F F' : Qid .
var TL TL' TL'' TL''' : TermList .
var Att : AttrSet .
var RLS : RuleSet .
var Rl : Rule .
var TP : Type .
var N N' : Nat .
var NeTL : NeTermList .
var Q Q' : Qid .
var IL : ImportList .
var SS : SortSet .
var SSDS : SubsortDeclSet .
var OPDS : OpDeclSet .
var MAS : MembAxSet .
var EQS : EquationSet .
var TPL : TypeList .
--- Extra filter for substitutions ------
op _|>_ : Substitution Nat -> Substitution .
eq Subst |> N
= Subst |>* N [none] .
op _|>*_[_] : Substitution Nat Substitution -> Substitution .
eq none |>* N [Subst']
= Subst' .
eq ((V <- T') ; Subst) |>* N [Subst']
= Subst |>* N
[Subst' ; if highestVar(V) < N then (V <- T') else none fi ] .
--- instantiatesAbove -----------------------------------
op _instantiatesAbove_ : Substitution Nat -> Bool .
eq none instantiatesAbove N = false .
eq ((V <- T') ; Subst) instantiatesAbove N
= highestVar(V) >= N
or-else
Subst instantiatesAbove N .
----------------------------------------------
--- New Renaming Utilities -------------------
op highestVar : GTermList -> Nat .
eq highestVar(TL:GTermList)
= highestVar(TL:GTermList,0) .
op highestVarTerm : Term -> Nat . ---warning memo
op highestVarTerm : Context -> Nat . ---warning memo
eq highestVarTerm([]) = 0 .
eq highestVarTerm(C) = 0 .
eq highestVarTerm(V)
= if rfind(string(V), "#", length(string(V))) =/= notFound
and
rfind(string(V), ":", length(string(V))) =/= notFound
and
rat(substr(string(V),
rfind(string(V), "#", length(string(V))) + 1,
rfind(string(V), ":", length(string(V))) + (- 1))
,10)
:: Nat
then rat(substr(string(V),
rfind(string(V), "#", length(string(V))) + 1,
rfind(string(V), ":", length(string(V))) + (- 1))
,10)
else
if rfind(string(V), "%", length(string(V))) =/= notFound
and
rfind(string(V), ":", length(string(V))) =/= notFound
and
rat(substr(string(V),
rfind(string(V), "%", length(string(V))) + 1,
rfind(string(V), ":", length(string(V))) + (- 1))
,10)
:: Nat
then rat(substr(string(V),
rfind(string(V), "%", length(string(V))) + 1,
rfind(string(V), ":", length(string(V))) + (- 1))
,10)
else
if rfind(string(V), "@", length(string(V))) =/= notFound
and
rfind(string(V), ":", length(string(V))) =/= notFound
and
rat(substr(string(V),
rfind(string(V), "@", length(string(V))) + 1,
rfind(string(V), ":", length(string(V))) + (- 1))
,10)
:: Nat
then rat(substr(string(V),
rfind(string(V), "@", length(string(V))) + 1,
rfind(string(V), ":", length(string(V))) + (- 1))
,10)
else 0
fi
fi
fi .
eq highestVarTerm(F[TL:GTermList])
= highestVar(TL:GTermList,0) .
op highestVar : GTermList Nat -> Nat .
eq highestVar(empty,N)
= N .
eq highestVar((Ct,TL:GTermList),N)
= highestVar(TL:GTermList,
if highestVarTerm(Ct) > N then highestVarTerm(Ct) else N fi
) .
eq highestVar((T,TL:GTermList),N)
= highestVar(TL:GTermList,
if highestVarTerm(T) > N then highestVarTerm(T) else N fi
) .
--- For substitutions
op highestVar : Substitution -> Nat . --- [memo] .
eq highestVar(Subst)
= highestVar(Subst,0) .
op highestVar : Substitution Nat -> Nat .
eq highestVar((none).Substitution,N) = N .
eq highestVar(V <- T ; Subst,N)
= highestVar(Subst,highestVar((T,V),N)) .
--- Renaming ------------------------------------------------------
op newVar : Nat TypeList -> TermList .
eq newVar(N,nil) = empty .
eq newVar(N,TP TPL) = (newVar*(N,TP),newVar(s(N),TPL)) .
op newVar* : Nat Type -> Variable .
eq newVar*(N,TP)
= qid("#" + string(N,10) + ":" + string(TP)) .
op simplifyVars : TermList -> TermList .
eq simplifyVars(TL) = TL << 0 < .
op _<<`(_`)< : TermList GTermList -> TermList .
eq X:TermList <<(TL:GTermList)<
= X:TermList << highestVar(TL:GTermList) + 1 < .
op _<<_ : TermList UnificationPair -> TermList .
eq TL << {Subst,N} = TL << Subst .
op _<<_ : TermList UnificationTriple -> TermList .
eq TL << {Subst,Subst',N} = TL << (Subst ; Subst') .
op _<<_ : Substitution UnificationTriple -> Substitution .
eq S:Substitution << {Subst,Subst',N} = S:Substitution << (Subst ; Subst') .
op _<<_< : TermList Nat -> TermList .
eq TL << N < = TL << (TL << { none, N } <) .
op _<<_< : TermList UnificationPair -> UnificationPair . ***Huge [memo] .
eq C << {S,N} < = {S,N} .
eq F[TL] << {S,N} < = TL << {S,N} < .
eq V << {S,N} <
= if not (dom S inVars V)
then {S ; V <- newVar(N,getType(V)), N + 1}
else {S,N}
fi .
eq (T,TL:NeTermList) << {S,N} <
= TL:NeTermList << (T << {S,N} < ) < .
eq empty << {S,N} <
= {S,N} .
endfm
fmod SUBSTITUTIONSET is
protecting SUBSTITUTION-HANDLING .
protecting META-LEVEL-MNPA .
protecting TERMSET .
protecting RENAMING .
sort SubstitutionSet NeSubstitutionSet .
subsort Substitution < NeSubstitutionSet < SubstitutionSet .
op empty : -> SubstitutionSet [ctor] .
op _|_ : SubstitutionSet SubstitutionSet -> SubstitutionSet
[ctor assoc comm id: empty format (d n d d)] .
op _|_ : NeSubstitutionSet SubstitutionSet -> NeSubstitutionSet
[ctor ditto] .
eq X:Substitution | X:Substitution = X:Substitution .
vars SS SS' : SubstitutionSet .
vars S S' Subst : Substitution .
vars T T' : Term .
vars TL TL' : TermList .
vars N N' : Nat .
var V : Variable .
op _<<_ : Substitution SubstitutionSet -> SubstitutionSet .
eq S << empty = empty .
ceq S << (S' | SS') = (S << S') | (S << SS') if SS' =/= empty .
op _..._ : SubstitutionSet [SubstitutionSet]
-> SubstitutionSet [strat (1) gather (e E)] .
eq empty ... SS':[SubstitutionSet] = empty .
eq (S | SS) ... SS':[SubstitutionSet]
= (S ...' SS':[SubstitutionSet])
|
(SS ... SS':[SubstitutionSet]) .
op _...'_ : Substitution SubstitutionSet -> SubstitutionSet .
eq S ...' empty
= empty .
eq S ...' (S' | SS')
= (S .. S')
|
(S ...' SS') .
op _|>_ : SubstitutionSet TermList -> SubstitutionSet .
eq (empty).SubstitutionSet |> TL = empty .
eq (S | SS:NeSubstitutionSet) |> TL
= (S |> TL) | (SS:NeSubstitutionSet |> TL) .
op _|>_ : SubstitutionSet Nat -> SubstitutionSet .
eq SS:NeSubstitutionSet |> N
= SS:NeSubstitutionSet |> (0,N) .
op _|>`(_,_`) : SubstitutionSet Nat Nat -> SubstitutionSet .
eq (empty).SubstitutionSet |> (N,N') = empty .
eq (S | SS:NeSubstitutionSet) |> (N,N')
= (S |> (N,N')) | (SS:NeSubstitutionSet |> (N,N')) .
op _|>`(_,_`) : Substitution Nat Nat -> Substitution .
eq none |> (N,N') = none .
eq ((V <- T') ; Subst) |> (N,N')
= if N <= highestVar(V) and highestVar(V) <= N'
then (V <- T')
else none
fi
; (Subst |> (N,N')) .
op filter_by!InVars_ : SubstitutionSet TermList -> SubstitutionSet .
eq filter (empty).SubstitutionSet by!InVars TL
= (empty).SubstitutionSet .
eq filter (S | SS) by!InVars TL
= if dom S inVars TL
then empty
else S
fi
| filter SS by!InVars TL .
op _==* none : SubstitutionSet -> Bool .
eq (none | SS) ==* none = SS ==* none .
eq (empty).SubstitutionSet ==* none = true .
eq SS ==* none = false [owise] .
op |_| : SubstitutionSet -> Nat .
eq | (empty).SubstitutionSet | = 0 .
eq | (S | SS) | = s(| SS |) .
--- Remove bindings same variable -------------------
op removeSame : SubstitutionSet -> SubstitutionSet .
ceq removeSame(Subst | SS) = removeSame(Subst) | removeSame(SS) if SS =/= empty .
eq removeSame(empty) = empty .
endfm
fmod UNIFICATIONPAIRSET is
protecting SUBSTITUTIONSET .
protecting RENAMING .
protecting UNIFICATIONTRIPLE .
vars V V' : Variable .
vars U U' : UnificationPair .
vars US US' : UnificationPairSet .
vars S S' S1 S1' S2 S2' : Substitution .
var SS : SubstitutionSet .
vars N N' N1 N2 : Nat .
vars T T' : Term .
var TL : TermList .
var M : Module .
--- Combine UnificationPair ---------------------------------------------
op _.._ : UnificationPair UnificationPair -> UnificationPair .
eq {S,N} .. {S',N'} = {S .. S',max(N,N')} .
--- Detect used variables ----------------------------------------------
op dom_inVars_ : UnificationPair TermList -> Bool . --- [memo] .
eq dom {S,N} inVars TL = dom S inVars TL .
--- UnificationPairSet --------------------------------------------------
sort UnificationPairSet .
subsort UnificationPair < UnificationPairSet .
op empty : -> UnificationPairSet [ctor] .
op _|_ : UnificationPairSet UnificationPairSet -> UnificationPairSet
[ctor assoc comm id: empty format (d n d d)] .
eq X:UnificationPair | X:UnificationPair = X:UnificationPair .
op _..._ : UnificationPairSet [UnificationPairSet]
-> UnificationPairSet [strat (1) gather (e E)] .
eq (empty).UnificationPairSet ... US':[UnificationPairSet]
= (empty).UnificationPairSet .
eq (U | US) ... US':[UnificationPairSet]
= (U ...' US':[UnificationPairSet])
|
(US ... US':[UnificationPairSet]) .
op _...'_ : UnificationPair UnificationPairSet -> UnificationPairSet .
eq U ...' (empty).UnificationPairSet
= (empty).UnificationPairSet .
eq U ...' (U' | US')
= (U .. U')
|
(U ...' US') .
--- Restriction -----------------------
op _|>_ : UnificationPairSet TermList -> UnificationPairSet .
eq (empty).UnificationPairSet |> TL = empty .
eq ({S,N} | US) |> TL = {(S |> TL),N} | (US |> TL) .
op filter_by!InVars_ : UnificationPairSet TermList
-> UnificationPairSet .
eq filter (empty).UnificationPairSet by!InVars TL
= (empty).UnificationPairSet .
eq filter (U | US) by!InVars TL
= if dom U inVars TL
then empty
else U
fi
| filter US by!InVars TL .
op toUnificationPair[_]`(_`) : Nat SubstitutionSet -> UnificationPairSet .
eq toUnificationPair[N](empty)
= empty .
eq toUnificationPair[N](S | SS)
= {S,highestVar(S,N)}
| toUnificationPair[N](SS) .
op toSubstitution : UnificationPairSet -> SubstitutionSet .
eq toSubstitution((empty).UnificationPairSet)
= empty .
eq toSubstitution({S,N} | US)
= S | toSubstitution(US) .
op _in_ : UnificationPair UnificationPairSet -> Bool .
eq X:UnificationPair in (X:UnificationPair | XS:UnificationPairSet) = true .
eq X:UnificationPair in XS:UnificationPairSet = false [owise] .
endfm
fmod UNIFICATIONTRIPLESET is
protecting SUBSTITUTIONSET .
protecting RENAMING .
protecting UNIFICATIONPAIRSET .
vars V V' : Variable .
var C : Constant .
var F : Qid .
vars U U' : UnificationTriple .
vars US US' : UnificationTripleSet .
vars S S' S1 S1' S2 S2' : Substitution .
var SS : SubstitutionSet .
var SSe : NeSubstitutionSet .
vars N N' N1 N2 NextVar : Nat .
vars T T' : Term .
var TL : TermList .
var NeTL : NeTermList .
var M : Module .
var UPS : UnificationPairSet .
--- Combine UnificationPair ---------------------------------------------
op _.._ : UnificationTriple UnificationTriple -> UnificationTriple .
eq {S1,S1',N1} .. {S2,S2',N2} = {S1 .. S2,S1' .. S2',max(N1,N2)} .
--- UnificationPairSet --------------------------------------------------
sort UnificationTripleSet .
subsort UnificationTriple < UnificationTripleSet .
op empty : -> UnificationTripleSet [ctor] .
op _|_ : UnificationTripleSet UnificationTripleSet
-> UnificationTripleSet
[ctor assoc comm id: empty format (d n d d)] .
eq X:UnificationTriple | X:UnificationTriple = X:UnificationTriple .
op _..._ : UnificationTripleSet [UnificationTripleSet]
-> UnificationTripleSet [strat (1) gather (e E)] .
eq (empty).UnificationTripleSet ... US':[UnificationTripleSet]
= (empty).UnificationTripleSet .
eq (U | US) ... US':[UnificationTripleSet]
= (U ...' US':[UnificationTripleSet])
|
(US ... US':[UnificationTripleSet]) .
op _...'_ : UnificationTriple UnificationTripleSet
-> UnificationTripleSet .
eq U ...' (empty).UnificationTripleSet
= (empty).UnificationTripleSet .
eq U ...' (U' | US')
= (U .. U')
|
(U ...' US') .
--- convert -----------------------------------------------------
op split : UnificationPair Nat -> UnificationTriple .
eq split({none,N},N') = {none,none,N} .
eq split({(V <- T') ; S,N},N')
= if highestVar(V) < N'
then {(V <- T'),none,N}
else {none,(V <- T'),N}
fi
.. split({S,N},N') .
op split : UnificationPairSet Term Term -> UnificationTripleSet .
eq split(empty,T,T') = empty .
eq split({S,N} | UPS,T,T') = {S |> T, S |> T',N} | split(UPS,T,T') .
op split : UnificationPairSet Term -> UnificationTripleSet .
eq split(empty,T) = empty .
eq split({S,N} | UPS,T) = {S |> T, S \\ (S |> T),N} | split(UPS,T) .
op toUnificationTriple[_]`(_`) :
Nat SubstitutionSet -> UnificationTripleSet .
eq toUnificationTriple[N](SS)
= toUnificationTriple*[N](SS,empty) .
op toUnificationTriple*[_]`(_,_`) :
Nat SubstitutionSet
UnificationTripleSet -> UnificationTripleSet .
eq toUnificationTriple*[N](empty,US)
= US .
eq toUnificationTriple*[N](S | SS,US)
= toUnificationTriple*[N](SS, US | {none,S,highestVar(S,N)}) .
op toUnificationTriple[_,_]`(_`) :
Nat Nat SubstitutionSet -> UnificationTripleSet .
eq toUnificationTriple[NextVar,N](SS)
= toUnificationTriple*[NextVar,N](SS,empty) .
op toUnificationTriple*[_,_]`(_,_`) :
Nat Nat SubstitutionSet
UnificationTripleSet -> UnificationTripleSet .
eq toUnificationTriple*[NextVar,N](empty,US)
= US .
eq toUnificationTriple*[NextVar,N](S | SS,US)
= toUnificationTriple*[NextVar,N](SS,
US | split({S,highestVar(S,N)},NextVar)) .
op toUnificationTriple[_,_,_]`(_`) :
Term Term Nat SubstitutionSet -> UnificationTripleSet .
eq toUnificationTriple[T,T',N](SS)
= toUnificationTriple*[T,T',N](SS,empty) .
op toUnificationTriple*[_,_,_]`(_,_`) :
Term Term Nat SubstitutionSet
UnificationTripleSet -> UnificationTripleSet .
eq toUnificationTriple*[T,T',N](empty,US)
= US .
eq toUnificationTriple*[T,T',N](S | SS,US)
= toUnificationTriple*[T,T',N](SS, US | {S |> T,S |> T',highestVar(S,N)}) .
op toSubstitution : UnificationTripleSet -> SubstitutionSet .
eq toSubstitution(US)
= toSubstitution*(US,empty) .
op toSubstitution* : UnificationTripleSet
SubstitutionSet -> SubstitutionSet .
eq toSubstitution*((empty).UnificationTripleSet,SS)
= SS .
eq toSubstitution*({S,S',N} | US,SS)
= toSubstitution*(US,SS | (S ; S')) .
op _in_ : UnificationTriple UnificationTripleSet -> Bool .
eq X:UnificationTriple
in (X:UnificationTriple | XS:UnificationTripleSet) = true .
eq X:UnificationTriple in XS:UnificationTripleSet = false [owise] .
--- restriction ---------------------------------------------------
op _|>_ : UnificationTripleSet TermList -> UnificationTripleSet .
eq US |> TL
= US *|> TL [empty] .
op _*|>_[_] : UnificationTripleSet TermList
UnificationTripleSet -> UnificationTripleSet .
eq (empty).UnificationTripleSet *|> TL [US']
= US' .
eq ({S,S',N} | US) *|> TL [US']
= US *|> TL [US' | {(S |> TL),(S' |> TL),N} ] .
op _filterBy_ : UnificationTripleSet Nat -> UnificationTripleSet .
eq US filterBy NextVar
= US filterBy* NextVar [empty] .