YES

Problem 1: 

(VAR A B C U U' V1 V2 X)
(THEORY 
(AC _and_ _or_ _xor_))
(RULES
U101(tt,A,B) -> U102(isBool(B),A,B)
U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
U11(tt,A) -> A
U111(tt) -> false
U121(tt,A) -> A
U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
U132(tt,B,U',U) -> U133(isS(U),B,U')
U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
U134(tt,U') -> U'
U141(tt,U) -> U142(isS(U),U)
U142(tt,U) -> U
U151(tt,V2) -> U152(isBool(V2))
U152(tt) -> tt
U161(tt,V2) -> U162(isBool(V2))
U162(tt) -> tt
U171(tt,V2) -> U172(isUniversal(V2))
U172(tt) -> tt
U181(tt,V2) -> U182(isUniversal(V2))
U182(tt) -> tt
U191(tt,V2) -> U192(isBool(V2))
U192(tt) -> tt
U201(tt,V2) -> U202(isBool(V2))
U202(tt) -> tt
U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
U211(tt) -> tt
U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
U221(tt,A) -> _xor_(A,true)
U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
U31(tt) -> false
U41(tt,A) -> A
U51(tt,A,B) -> U52(isBool(B),A,B)
U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
U61(tt,U',U) -> U62(isS(U),U',U)
U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
U63(tt) -> false
U71(tt) -> true
U81(tt,U',U) -> U82(isS(U),U',U)
U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
U91(tt) -> false
_and_(false,A) -> U31(isBool(A))
_and_(true,A) -> U41(isBool(A),A)
_and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
_and_(A,A) -> U11(isBool(A),A)
_implies_(A,B) -> U51(isBool(A),A,B)
_isEqualTo_(U,U) -> U71(isS(U))
_isEqualTo_(U,U') -> U61(isS(U'),U',U)
_isNotEqualTo_(U,U) -> U91(isS(U))
_isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
_or_(A,B) -> U101(isBool(A),A,B)
_xor_(false,A) -> U121(isBool(A),A)
_xor_(A,A) -> U111(isBool(A))
equal(X,X) -> tt
if_then_else_fi(true,U,U') -> U141(isS(U'),U)
if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
isBool(not_(V1)) -> U211(isBool(V1))
isBool(false) -> tt
isBool(true) -> tt
not_(false) -> true
not_(true) -> false
not_(A) -> U221(isBool(A),A)
)

Problem 1: 

Dependency Pairs Processor:
-> FAxioms:
 _AND_(_and_(x8,x9),x10) = _AND_(x8,_and_(x9,x10))
 _AND_(x8,x9) = _AND_(x9,x8)
 _OR_(_or_(x8,x9),x10) = _OR_(x8,_or_(x9,x10))
 _OR_(x8,x9) = _OR_(x9,x8)
 _XOR_(_xor_(x8,x9),x10) = _XOR_(x8,_xor_(x9,x10))
 _XOR_(x8,x9) = _XOR_(x9,x8)
-> Pairs:
 U101#(tt,A,B) -> U102#(isBool(B),A,B)
 U101#(tt,A,B) -> ISBOOL(B)
 U102#(tt,A,B) -> _AND_(A,B)
 U102#(tt,A,B) -> _XOR_(_and_(A,B),_xor_(A,B))
 U102#(tt,A,B) -> _XOR_(A,B)
 U131#(tt,B,U',U) -> U132#(isS(U'),B,U',U)
 U132#(tt,B,U',U) -> U133#(isS(U),B,U')
 U133#(tt,B,U') -> U134#(equal(_isNotEqualTo_(B,true),true),U')
 U133#(tt,B,U') -> _ISNOTEQUALTO_(B,true)
 U133#(tt,B,U') -> EQUAL(_isNotEqualTo_(B,true),true)
 U141#(tt,U) -> U142#(isS(U),U)
 U151#(tt,V2) -> U152#(isBool(V2))
 U151#(tt,V2) -> ISBOOL(V2)
 U161#(tt,V2) -> U162#(isBool(V2))
 U161#(tt,V2) -> ISBOOL(V2)
 U171#(tt,V2) -> U172#(isUniversal(V2))
 U181#(tt,V2) -> U182#(isUniversal(V2))
 U191#(tt,V2) -> U192#(isBool(V2))
 U191#(tt,V2) -> ISBOOL(V2)
 U201#(tt,V2) -> U202#(isBool(V2))
 U201#(tt,V2) -> ISBOOL(V2)
 U21#(tt,A,B,C) -> U22#(isBool(B),A,B,C)
 U21#(tt,A,B,C) -> ISBOOL(B)
 U22#(tt,A,B,C) -> U23#(isBool(C),A,B,C)
 U22#(tt,A,B,C) -> ISBOOL(C)
 U221#(tt,A) -> _XOR_(A,true)
 U23#(tt,A,B,C) -> _AND_(A,B)
 U23#(tt,A,B,C) -> _AND_(A,C)
 U23#(tt,A,B,C) -> _XOR_(_and_(A,B),_and_(A,C))
 U51#(tt,A,B) -> U52#(isBool(B),A,B)
 U51#(tt,A,B) -> ISBOOL(B)
 U52#(tt,A,B) -> _AND_(A,B)
 U52#(tt,A,B) -> _XOR_(A,_and_(A,B))
 U52#(tt,A,B) -> NOT_(_xor_(A,_and_(A,B)))
 U61#(tt,U',U) -> U62#(isS(U),U',U)
 U62#(tt,U',U) -> U63#(equal(_isNotEqualTo_(U,U'),true))
 U62#(tt,U',U) -> _ISNOTEQUALTO_(U,U')
 U62#(tt,U',U) -> EQUAL(_isNotEqualTo_(U,U'),true)
 U81#(tt,U',U) -> U82#(isS(U),U',U)
 U82#(tt,U',U) -> _ISEQUALTO_(U,U')
 U82#(tt,U',U) -> IF_THEN_ELSE_FI(_isEqualTo_(U,U'),false,true)
 _AND_(_and_(false,A),x8) -> U31#(isBool(A))
 _AND_(_and_(false,A),x8) -> _AND_(U31(isBool(A)),x8)
 _AND_(_and_(false,A),x8) -> ISBOOL(A)
 _AND_(_and_(true,A),x8) -> U41#(isBool(A),A)
 _AND_(_and_(true,A),x8) -> _AND_(U41(isBool(A),A),x8)
 _AND_(_and_(true,A),x8) -> ISBOOL(A)
 _AND_(_and_(A,_xor_(B,C)),x8) -> U21#(isBool(A),A,B,C)
 _AND_(_and_(A,_xor_(B,C)),x8) -> _AND_(U21(isBool(A),A,B,C),x8)
 _AND_(_and_(A,_xor_(B,C)),x8) -> ISBOOL(A)
 _AND_(_and_(A,A),x8) -> U11#(isBool(A),A)
 _AND_(_and_(A,A),x8) -> _AND_(U11(isBool(A),A),x8)
 _AND_(_and_(A,A),x8) -> ISBOOL(A)
 _AND_(false,A) -> U31#(isBool(A))
 _AND_(false,A) -> ISBOOL(A)
 _AND_(true,A) -> U41#(isBool(A),A)
 _AND_(true,A) -> ISBOOL(A)
 _AND_(A,_xor_(B,C)) -> U21#(isBool(A),A,B,C)
 _AND_(A,_xor_(B,C)) -> ISBOOL(A)
 _AND_(A,A) -> U11#(isBool(A),A)
 _AND_(A,A) -> ISBOOL(A)
 _IMPLIES_(A,B) -> U51#(isBool(A),A,B)
 _IMPLIES_(A,B) -> ISBOOL(A)
 _ISEQUALTO_(U,U) -> U71#(isS(U))
 _ISEQUALTO_(U,U') -> U61#(isS(U'),U',U)
 _ISNOTEQUALTO_(U,U) -> U91#(isS(U))
 _ISNOTEQUALTO_(U,U') -> U81#(isS(U'),U',U)
 _OR_(_or_(A,B),x8) -> U101#(isBool(A),A,B)
 _OR_(_or_(A,B),x8) -> _OR_(U101(isBool(A),A,B),x8)
 _OR_(_or_(A,B),x8) -> ISBOOL(A)
 _OR_(A,B) -> U101#(isBool(A),A,B)
 _OR_(A,B) -> ISBOOL(A)
 _XOR_(_xor_(false,A),x8) -> U121#(isBool(A),A)
 _XOR_(_xor_(false,A),x8) -> _XOR_(U121(isBool(A),A),x8)
 _XOR_(_xor_(false,A),x8) -> ISBOOL(A)
 _XOR_(_xor_(A,A),x8) -> U111#(isBool(A))
 _XOR_(_xor_(A,A),x8) -> _XOR_(U111(isBool(A)),x8)
 _XOR_(_xor_(A,A),x8) -> ISBOOL(A)
 _XOR_(false,A) -> U121#(isBool(A),A)
 _XOR_(false,A) -> ISBOOL(A)
 _XOR_(A,A) -> U111#(isBool(A))
 _XOR_(A,A) -> ISBOOL(A)
 IF_THEN_ELSE_FI(true,U,U') -> U141#(isS(U'),U)
 IF_THEN_ELSE_FI(B,U,U') -> U131#(isBool(B),B,U',U)
 IF_THEN_ELSE_FI(B,U,U') -> ISBOOL(B)
 ISBOOL(_and_(V1,V2)) -> U151#(isBool(V1),V2)
 ISBOOL(_and_(V1,V2)) -> ISBOOL(V1)
 ISBOOL(_implies_(V1,V2)) -> U161#(isBool(V1),V2)
 ISBOOL(_implies_(V1,V2)) -> ISBOOL(V1)
 ISBOOL(_isEqualTo_(V1,V2)) -> U171#(isUniversal(V1),V2)
 ISBOOL(_isNotEqualTo_(V1,V2)) -> U181#(isUniversal(V1),V2)
 ISBOOL(_or_(V1,V2)) -> U191#(isBool(V1),V2)
 ISBOOL(_or_(V1,V2)) -> ISBOOL(V1)
 ISBOOL(_xor_(V1,V2)) -> U201#(isBool(V1),V2)
 ISBOOL(_xor_(V1,V2)) -> ISBOOL(V1)
 ISBOOL(not_(V1)) -> U211#(isBool(V1))
 ISBOOL(not_(V1)) -> ISBOOL(V1)
 NOT_(A) -> U221#(isBool(A),A)
 NOT_(A) -> ISBOOL(A)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> SRules:
 _AND_(_and_(x8,x9),x10) -> _AND_(x8,x9)
 _AND_(x8,_and_(x9,x10)) -> _AND_(x9,x10)
 _OR_(_or_(x8,x9),x10) -> _OR_(x8,x9)
 _OR_(x8,_or_(x9,x10)) -> _OR_(x9,x10)
 _XOR_(_xor_(x8,x9),x10) -> _XOR_(x8,x9)
 _XOR_(x8,_xor_(x9,x10)) -> _XOR_(x9,x10)

Problem 1: 

SCC Processor:
-> FAxioms:
 _AND_(_and_(x8,x9),x10) = _AND_(x8,_and_(x9,x10))
 _AND_(x8,x9) = _AND_(x9,x8)
 _OR_(_or_(x8,x9),x10) = _OR_(x8,_or_(x9,x10))
 _OR_(x8,x9) = _OR_(x9,x8)
 _XOR_(_xor_(x8,x9),x10) = _XOR_(x8,_xor_(x9,x10))
 _XOR_(x8,x9) = _XOR_(x9,x8)
-> Pairs:
 U101#(tt,A,B) -> U102#(isBool(B),A,B)
 U101#(tt,A,B) -> ISBOOL(B)
 U102#(tt,A,B) -> _AND_(A,B)
 U102#(tt,A,B) -> _XOR_(_and_(A,B),_xor_(A,B))
 U102#(tt,A,B) -> _XOR_(A,B)
 U131#(tt,B,U',U) -> U132#(isS(U'),B,U',U)
 U132#(tt,B,U',U) -> U133#(isS(U),B,U')
 U133#(tt,B,U') -> U134#(equal(_isNotEqualTo_(B,true),true),U')
 U133#(tt,B,U') -> _ISNOTEQUALTO_(B,true)
 U133#(tt,B,U') -> EQUAL(_isNotEqualTo_(B,true),true)
 U141#(tt,U) -> U142#(isS(U),U)
 U151#(tt,V2) -> U152#(isBool(V2))
 U151#(tt,V2) -> ISBOOL(V2)
 U161#(tt,V2) -> U162#(isBool(V2))
 U161#(tt,V2) -> ISBOOL(V2)
 U171#(tt,V2) -> U172#(isUniversal(V2))
 U181#(tt,V2) -> U182#(isUniversal(V2))
 U191#(tt,V2) -> U192#(isBool(V2))
 U191#(tt,V2) -> ISBOOL(V2)
 U201#(tt,V2) -> U202#(isBool(V2))
 U201#(tt,V2) -> ISBOOL(V2)
 U21#(tt,A,B,C) -> U22#(isBool(B),A,B,C)
 U21#(tt,A,B,C) -> ISBOOL(B)
 U22#(tt,A,B,C) -> U23#(isBool(C),A,B,C)
 U22#(tt,A,B,C) -> ISBOOL(C)
 U221#(tt,A) -> _XOR_(A,true)
 U23#(tt,A,B,C) -> _AND_(A,B)
 U23#(tt,A,B,C) -> _AND_(A,C)
 U23#(tt,A,B,C) -> _XOR_(_and_(A,B),_and_(A,C))
 U51#(tt,A,B) -> U52#(isBool(B),A,B)
 U51#(tt,A,B) -> ISBOOL(B)
 U52#(tt,A,B) -> _AND_(A,B)
 U52#(tt,A,B) -> _XOR_(A,_and_(A,B))
 U52#(tt,A,B) -> NOT_(_xor_(A,_and_(A,B)))
 U61#(tt,U',U) -> U62#(isS(U),U',U)
 U62#(tt,U',U) -> U63#(equal(_isNotEqualTo_(U,U'),true))
 U62#(tt,U',U) -> _ISNOTEQUALTO_(U,U')
 U62#(tt,U',U) -> EQUAL(_isNotEqualTo_(U,U'),true)
 U81#(tt,U',U) -> U82#(isS(U),U',U)
 U82#(tt,U',U) -> _ISEQUALTO_(U,U')
 U82#(tt,U',U) -> IF_THEN_ELSE_FI(_isEqualTo_(U,U'),false,true)
 _AND_(_and_(false,A),x8) -> U31#(isBool(A))
 _AND_(_and_(false,A),x8) -> _AND_(U31(isBool(A)),x8)
 _AND_(_and_(false,A),x8) -> ISBOOL(A)
 _AND_(_and_(true,A),x8) -> U41#(isBool(A),A)
 _AND_(_and_(true,A),x8) -> _AND_(U41(isBool(A),A),x8)
 _AND_(_and_(true,A),x8) -> ISBOOL(A)
 _AND_(_and_(A,_xor_(B,C)),x8) -> U21#(isBool(A),A,B,C)
 _AND_(_and_(A,_xor_(B,C)),x8) -> _AND_(U21(isBool(A),A,B,C),x8)
 _AND_(_and_(A,_xor_(B,C)),x8) -> ISBOOL(A)
 _AND_(_and_(A,A),x8) -> U11#(isBool(A),A)
 _AND_(_and_(A,A),x8) -> _AND_(U11(isBool(A),A),x8)
 _AND_(_and_(A,A),x8) -> ISBOOL(A)
 _AND_(false,A) -> U31#(isBool(A))
 _AND_(false,A) -> ISBOOL(A)
 _AND_(true,A) -> U41#(isBool(A),A)
 _AND_(true,A) -> ISBOOL(A)
 _AND_(A,_xor_(B,C)) -> U21#(isBool(A),A,B,C)
 _AND_(A,_xor_(B,C)) -> ISBOOL(A)
 _AND_(A,A) -> U11#(isBool(A),A)
 _AND_(A,A) -> ISBOOL(A)
 _IMPLIES_(A,B) -> U51#(isBool(A),A,B)
 _IMPLIES_(A,B) -> ISBOOL(A)
 _ISEQUALTO_(U,U) -> U71#(isS(U))
 _ISEQUALTO_(U,U') -> U61#(isS(U'),U',U)
 _ISNOTEQUALTO_(U,U) -> U91#(isS(U))
 _ISNOTEQUALTO_(U,U') -> U81#(isS(U'),U',U)
 _OR_(_or_(A,B),x8) -> U101#(isBool(A),A,B)
 _OR_(_or_(A,B),x8) -> _OR_(U101(isBool(A),A,B),x8)
 _OR_(_or_(A,B),x8) -> ISBOOL(A)
 _OR_(A,B) -> U101#(isBool(A),A,B)
 _OR_(A,B) -> ISBOOL(A)
 _XOR_(_xor_(false,A),x8) -> U121#(isBool(A),A)
 _XOR_(_xor_(false,A),x8) -> _XOR_(U121(isBool(A),A),x8)
 _XOR_(_xor_(false,A),x8) -> ISBOOL(A)
 _XOR_(_xor_(A,A),x8) -> U111#(isBool(A))
 _XOR_(_xor_(A,A),x8) -> _XOR_(U111(isBool(A)),x8)
 _XOR_(_xor_(A,A),x8) -> ISBOOL(A)
 _XOR_(false,A) -> U121#(isBool(A),A)
 _XOR_(false,A) -> ISBOOL(A)
 _XOR_(A,A) -> U111#(isBool(A))
 _XOR_(A,A) -> ISBOOL(A)
 IF_THEN_ELSE_FI(true,U,U') -> U141#(isS(U'),U)
 IF_THEN_ELSE_FI(B,U,U') -> U131#(isBool(B),B,U',U)
 IF_THEN_ELSE_FI(B,U,U') -> ISBOOL(B)
 ISBOOL(_and_(V1,V2)) -> U151#(isBool(V1),V2)
 ISBOOL(_and_(V1,V2)) -> ISBOOL(V1)
 ISBOOL(_implies_(V1,V2)) -> U161#(isBool(V1),V2)
 ISBOOL(_implies_(V1,V2)) -> ISBOOL(V1)
 ISBOOL(_isEqualTo_(V1,V2)) -> U171#(isUniversal(V1),V2)
 ISBOOL(_isNotEqualTo_(V1,V2)) -> U181#(isUniversal(V1),V2)
 ISBOOL(_or_(V1,V2)) -> U191#(isBool(V1),V2)
 ISBOOL(_or_(V1,V2)) -> ISBOOL(V1)
 ISBOOL(_xor_(V1,V2)) -> U201#(isBool(V1),V2)
 ISBOOL(_xor_(V1,V2)) -> ISBOOL(V1)
 ISBOOL(not_(V1)) -> U211#(isBool(V1))
 ISBOOL(not_(V1)) -> ISBOOL(V1)
 NOT_(A) -> U221#(isBool(A),A)
 NOT_(A) -> ISBOOL(A)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> SRules:
 _AND_(_and_(x8,x9),x10) -> _AND_(x8,x9)
 _AND_(x8,_and_(x9,x10)) -> _AND_(x9,x10)
 _OR_(_or_(x8,x9),x10) -> _OR_(x8,x9)
 _OR_(x8,_or_(x9,x10)) -> _OR_(x9,x10)
 _XOR_(_xor_(x8,x9),x10) -> _XOR_(x8,x9)
 _XOR_(x8,_xor_(x9,x10)) -> _XOR_(x9,x10)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U151#(tt,V2) -> ISBOOL(V2)
 U161#(tt,V2) -> ISBOOL(V2)
 U191#(tt,V2) -> ISBOOL(V2)
 U201#(tt,V2) -> ISBOOL(V2)
 ISBOOL(_and_(V1,V2)) -> U151#(isBool(V1),V2)
 ISBOOL(_and_(V1,V2)) -> ISBOOL(V1)
 ISBOOL(_implies_(V1,V2)) -> U161#(isBool(V1),V2)
 ISBOOL(_implies_(V1,V2)) -> ISBOOL(V1)
 ISBOOL(_or_(V1,V2)) -> U191#(isBool(V1),V2)
 ISBOOL(_or_(V1,V2)) -> ISBOOL(V1)
 ISBOOL(_xor_(V1,V2)) -> U201#(isBool(V1),V2)
 ISBOOL(_xor_(V1,V2)) -> ISBOOL(V1)
 ISBOOL(not_(V1)) -> ISBOOL(V1)
-> FAxioms:
 _and_(_and_(x8,x9),x10) -> _and_(x8,_and_(x9,x10))
 _and_(x8,x9) -> _and_(x9,x8)
 _or_(_or_(x8,x9),x10) -> _or_(x8,_or_(x9,x10))
 _or_(x8,x9) -> _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) -> _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) -> _xor_(x9,x8)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
->->-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> SRules:
 Empty
->->Cycle:
->->-> Pairs:
 _XOR_(_xor_(false,A),x8) -> _XOR_(U121(isBool(A),A),x8)
 _XOR_(_xor_(A,A),x8) -> _XOR_(U111(isBool(A)),x8)
-> FAxioms:
 _and_(_and_(x8,x9),x10) -> _and_(x8,_and_(x9,x10))
 _and_(x8,x9) -> _and_(x9,x8)
 _or_(_or_(x8,x9),x10) -> _or_(x8,_or_(x9,x10))
 _or_(x8,x9) -> _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) -> _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) -> _xor_(x9,x8)
 _XOR_(_xor_(x8,x9),x10) -> _XOR_(x8,_xor_(x9,x10))
 _XOR_(x8,x9) -> _XOR_(x9,x8)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
->->-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> SRules:
 _XOR_(_xor_(x8,x9),x10) -> _XOR_(x8,x9)
 _XOR_(x8,_xor_(x9,x10)) -> _XOR_(x9,x10)
->->Cycle:
->->-> Pairs:
 U21#(tt,A,B,C) -> U22#(isBool(B),A,B,C)
 U22#(tt,A,B,C) -> U23#(isBool(C),A,B,C)
 U23#(tt,A,B,C) -> _AND_(A,B)
 U23#(tt,A,B,C) -> _AND_(A,C)
 _AND_(_and_(false,A),x8) -> _AND_(U31(isBool(A)),x8)
 _AND_(_and_(true,A),x8) -> _AND_(U41(isBool(A),A),x8)
 _AND_(_and_(A,_xor_(B,C)),x8) -> U21#(isBool(A),A,B,C)
 _AND_(_and_(A,_xor_(B,C)),x8) -> _AND_(U21(isBool(A),A,B,C),x8)
 _AND_(_and_(A,A),x8) -> _AND_(U11(isBool(A),A),x8)
 _AND_(A,_xor_(B,C)) -> U21#(isBool(A),A,B,C)
-> FAxioms:
 _and_(_and_(x8,x9),x10) -> _and_(x8,_and_(x9,x10))
 _and_(x8,x9) -> _and_(x9,x8)
 _or_(_or_(x8,x9),x10) -> _or_(x8,_or_(x9,x10))
 _or_(x8,x9) -> _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) -> _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) -> _xor_(x9,x8)
 _AND_(_and_(x8,x9),x10) -> _AND_(x8,_and_(x9,x10))
 _AND_(x8,x9) -> _AND_(x9,x8)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
->->-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> SRules:
 _AND_(_and_(x8,x9),x10) -> _AND_(x8,x9)
 _AND_(x8,_and_(x9,x10)) -> _AND_(x9,x10)
->->Cycle:
->->-> Pairs:
 _OR_(_or_(A,B),x8) -> _OR_(U101(isBool(A),A,B),x8)
-> FAxioms:
 _and_(_and_(x8,x9),x10) -> _and_(x8,_and_(x9,x10))
 _and_(x8,x9) -> _and_(x9,x8)
 _or_(_or_(x8,x9),x10) -> _or_(x8,_or_(x9,x10))
 _or_(x8,x9) -> _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) -> _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) -> _xor_(x9,x8)
 _OR_(_or_(x8,x9),x10) -> _OR_(x8,_or_(x9,x10))
 _OR_(x8,x9) -> _OR_(x9,x8)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
->->-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> SRules:
 _OR_(_or_(x8,x9),x10) -> _OR_(x8,x9)
 _OR_(x8,_or_(x9,x10)) -> _OR_(x9,x10)


The problem is decomposed in 4 subproblems.

Problem 1.1: 

Subterm Processor:
-> FAxioms:
 Empty
-> Pairs:
 U151#(tt,V2) -> ISBOOL(V2)
 U161#(tt,V2) -> ISBOOL(V2)
 U191#(tt,V2) -> ISBOOL(V2)
 U201#(tt,V2) -> ISBOOL(V2)
 ISBOOL(_and_(V1,V2)) -> U151#(isBool(V1),V2)
 ISBOOL(_and_(V1,V2)) -> ISBOOL(V1)
 ISBOOL(_implies_(V1,V2)) -> U161#(isBool(V1),V2)
 ISBOOL(_implies_(V1,V2)) -> ISBOOL(V1)
 ISBOOL(_or_(V1,V2)) -> U191#(isBool(V1),V2)
 ISBOOL(_or_(V1,V2)) -> ISBOOL(V1)
 ISBOOL(_xor_(V1,V2)) -> U201#(isBool(V1),V2)
 ISBOOL(_xor_(V1,V2)) -> ISBOOL(V1)
 ISBOOL(not_(V1)) -> ISBOOL(V1)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> SRules:
 Empty
->Projection:
 pi(U151#) = [2]
 pi(U161#) = [2]
 pi(U191#) = [2]
 pi(U201#) = [2]
 pi(ISBOOL) = [1]

Problem 1.1: 

SCC Processor:
-> FAxioms:
 Empty
-> Pairs:
 U151#(tt,V2) -> ISBOOL(V2)
 U161#(tt,V2) -> ISBOOL(V2)
 U191#(tt,V2) -> ISBOOL(V2)
 U201#(tt,V2) -> ISBOOL(V2)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> SRules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 _XOR_(_xor_(x8,x9),x10) = _XOR_(x8,_xor_(x9,x10))
 _XOR_(x8,x9) = _XOR_(x9,x8)
-> Pairs:
 _XOR_(_xor_(false,A),x8) -> _XOR_(U121(isBool(A),A),x8)
 _XOR_(_xor_(A,A),x8) -> _XOR_(U111(isBool(A)),x8)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Usable Equations:
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> Usable Rules:
 U111(tt) -> false
 U121(tt,A) -> A
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U211(tt) -> tt
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
-> SRules:
 _XOR_(_xor_(x8,x9),x10) -> _XOR_(x8,x9)
 _XOR_(x8,_xor_(x9,x10)) -> _XOR_(x9,x10)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U101](X1,X2,X3) = 0
[U102](X1,X2,X3) = 0
[U11](X1,X2) = 0
[U111](X) = 2
[U121](X1,X2) = X2 + 1
[U131](X1,X2,X3,X4) = 0
[U132](X1,X2,X3,X4) = 0
[U133](X1,X2,X3) = 0
[U134](X1,X2) = 0
[U141](X1,X2) = 0
[U142](X1,X2) = 0
[U151](X1,X2) = X1 + 2.X2 + 2
[U152](X) = 2.X
[U161](X1,X2) = 2.X1 + X2
[U162](X) = X
[U171](X1,X2) = 2.X2 + 1
[U172](X) = 2.X
[U181](X1,X2) = 2.X2 + 2
[U182](X) = X + 1
[U191](X1,X2) = 2.X1 + 2.X2
[U192](X) = 2
[U201](X1,X2) = X1 + X2
[U202](X) = 2
[U21](X1,X2,X3,X4) = 0
[U211](X) = X + 2
[U22](X1,X2,X3,X4) = 0
[U221](X1,X2) = 0
[U23](X1,X2,X3,X4) = 0
[U31](X) = 0
[U41](X1,X2) = 0
[U51](X1,X2,X3) = 0
[U52](X1,X2,X3) = 0
[U61](X1,X2,X3) = 0
[U62](X1,X2,X3) = 0
[U63](X) = 0
[U71](X) = 0
[U81](X1,X2,X3) = 0
[U82](X1,X2,X3) = 0
[U91](X) = 0
[_and_](X1,X2) = 2.X1 + 2.X2 + 2
[_implies_](X1,X2) = 2.X1 + X2 + 2
[_isEqualTo_](X1,X2) = X1 + 2.X2 + 2
[_isNotEqualTo_](X1,X2) = X1 + 2.X2
[_or_](X1,X2) = 2.X1 + 2.X2 + 2
[_xor_](X1,X2) = X1 + X2 + 2
[equal](X1,X2) = 0
[if_then_else_fi](X1,X2,X3) = 0
[isBool](X) = X + 2
[not_](X) = 2.X + 2
[false] = 0
[isS](X) = 0
[isUniversal](X) = X
[true] = 1
[tt] = 2
[U101#](X1,X2,X3) = 0
[U102#](X1,X2,X3) = 0
[U11#](X1,X2) = 0
[U111#](X) = 0
[U121#](X1,X2) = 0
[U131#](X1,X2,X3,X4) = 0
[U132#](X1,X2,X3,X4) = 0
[U133#](X1,X2,X3) = 0
[U134#](X1,X2) = 0
[U141#](X1,X2) = 0
[U142#](X1,X2) = 0
[U151#](X1,X2) = 0
[U152#](X) = 0
[U161#](X1,X2) = 0
[U162#](X) = 0
[U171#](X1,X2) = 0
[U172#](X) = 0
[U181#](X1,X2) = 0
[U182#](X) = 0
[U191#](X1,X2) = 0
[U192#](X) = 0
[U201#](X1,X2) = 0
[U202#](X) = 0
[U21#](X1,X2,X3,X4) = 0
[U211#](X) = 0
[U22#](X1,X2,X3,X4) = 0
[U221#](X1,X2) = 0
[U23#](X1,X2,X3,X4) = 0
[U31#](X) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U52#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U62#](X1,X2,X3) = 0
[U63#](X) = 0
[U71#](X) = 0
[U81#](X1,X2,X3) = 0
[U82#](X1,X2,X3) = 0
[U91#](X) = 0
[_AND_](X1,X2) = 0
[_IMPLIES_](X1,X2) = 0
[_ISEQUALTO_](X1,X2) = 0
[_ISNOTEQUALTO_](X1,X2) = 0
[_OR_](X1,X2) = 0
[_XOR_](X1,X2) = 2.X1 + 2.X2
[EQUAL](X1,X2) = 0
[IF_THEN_ELSE_FI](X1,X2,X3) = 0
[ISBOOL](X) = 0
[NOT_](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 _XOR_(_xor_(x8,x9),x10) = _XOR_(x8,_xor_(x9,x10))
 _XOR_(x8,x9) = _XOR_(x9,x8)
-> Pairs:
 _XOR_(_xor_(A,A),x8) -> _XOR_(U111(isBool(A)),x8)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> SRules:
 _XOR_(_xor_(x8,x9),x10) -> _XOR_(x8,x9)
 _XOR_(x8,_xor_(x9,x10)) -> _XOR_(x9,x10)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 _XOR_(_xor_(A,A),x8) -> _XOR_(U111(isBool(A)),x8)
-> FAxioms:
 _and_(_and_(x8,x9),x10) -> _and_(x8,_and_(x9,x10))
 _and_(x8,x9) -> _and_(x9,x8)
 _or_(_or_(x8,x9),x10) -> _or_(x8,_or_(x9,x10))
 _or_(x8,x9) -> _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) -> _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) -> _xor_(x9,x8)
 _XOR_(_xor_(x8,x9),x10) -> _XOR_(x8,_xor_(x9,x10))
 _XOR_(x8,x9) -> _XOR_(x9,x8)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
->->-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> SRules:
 _XOR_(_xor_(x8,x9),x10) -> _XOR_(x8,x9)
 _XOR_(x8,_xor_(x9,x10)) -> _XOR_(x9,x10)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 _XOR_(_xor_(x8,x9),x10) = _XOR_(x8,_xor_(x9,x10))
 _XOR_(x8,x9) = _XOR_(x9,x8)
-> Pairs:
 _XOR_(_xor_(A,A),x8) -> _XOR_(U111(isBool(A)),x8)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Usable Equations:
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> Usable Rules:
 U111(tt) -> false
 U121(tt,A) -> A
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U211(tt) -> tt
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
-> SRules:
 _XOR_(_xor_(x8,x9),x10) -> _XOR_(x8,x9)
 _XOR_(x8,_xor_(x9,x10)) -> _XOR_(x9,x10)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U101](X1,X2,X3) = 0
[U102](X1,X2,X3) = 0
[U11](X1,X2) = 0
[U111](X) = 1
[U121](X1,X2) = X2 + 2
[U131](X1,X2,X3,X4) = 0
[U132](X1,X2,X3,X4) = 0
[U133](X1,X2,X3) = 0
[U134](X1,X2) = 0
[U141](X1,X2) = 0
[U142](X1,X2) = 0
[U151](X1,X2) = X1 + 2
[U152](X) = 2
[U161](X1,X2) = X1 + X2 + 2
[U162](X) = 2
[U171](X1,X2) = X1 + 2.X2 + 2
[U172](X) = X + 2
[U181](X1,X2) = X1 + X2 + 1
[U182](X) = 2
[U191](X1,X2) = 2.X1 + 2.X2 + 2
[U192](X) = X + 2
[U201](X1,X2) = X1 + 2.X2
[U202](X) = X
[U21](X1,X2,X3,X4) = 0
[U211](X) = X + 2
[U22](X1,X2,X3,X4) = 0
[U221](X1,X2) = 0
[U23](X1,X2,X3,X4) = 0
[U31](X) = 0
[U41](X1,X2) = 0
[U51](X1,X2,X3) = 0
[U52](X1,X2,X3) = 0
[U61](X1,X2,X3) = 0
[U62](X1,X2,X3) = 0
[U63](X) = 0
[U71](X) = 0
[U81](X1,X2,X3) = 0
[U82](X1,X2,X3) = 0
[U91](X) = 0
[_and_](X1,X2) = X1 + 2.X2 + 1
[_implies_](X1,X2) = X1 + X2 + 1
[_isEqualTo_](X1,X2) = 2.X1 + 2.X2 + 2
[_isNotEqualTo_](X1,X2) = 2.X1 + X2 + 2
[_or_](X1,X2) = 2.X1 + 2.X2 + 2
[_xor_](X1,X2) = X1 + X2 + 2
[equal](X1,X2) = 0
[if_then_else_fi](X1,X2,X3) = 0
[isBool](X) = 2.X
[not_](X) = X + 2
[false] = 1
[isS](X) = 0
[isUniversal](X) = 2.X + 2
[true] = 2
[tt] = 2
[U101#](X1,X2,X3) = 0
[U102#](X1,X2,X3) = 0
[U11#](X1,X2) = 0
[U111#](X) = 0
[U121#](X1,X2) = 0
[U131#](X1,X2,X3,X4) = 0
[U132#](X1,X2,X3,X4) = 0
[U133#](X1,X2,X3) = 0
[U134#](X1,X2) = 0
[U141#](X1,X2) = 0
[U142#](X1,X2) = 0
[U151#](X1,X2) = 0
[U152#](X) = 0
[U161#](X1,X2) = 0
[U162#](X) = 0
[U171#](X1,X2) = 0
[U172#](X) = 0
[U181#](X1,X2) = 0
[U182#](X) = 0
[U191#](X1,X2) = 0
[U192#](X) = 0
[U201#](X1,X2) = 0
[U202#](X) = 0
[U21#](X1,X2,X3,X4) = 0
[U211#](X) = 0
[U22#](X1,X2,X3,X4) = 0
[U221#](X1,X2) = 0
[U23#](X1,X2,X3,X4) = 0
[U31#](X) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U52#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U62#](X1,X2,X3) = 0
[U63#](X) = 0
[U71#](X) = 0
[U81#](X1,X2,X3) = 0
[U82#](X1,X2,X3) = 0
[U91#](X) = 0
[_AND_](X1,X2) = 0
[_IMPLIES_](X1,X2) = 0
[_ISEQUALTO_](X1,X2) = 0
[_ISNOTEQUALTO_](X1,X2) = 0
[_OR_](X1,X2) = 0
[_XOR_](X1,X2) = 2.X1 + 2.X2
[EQUAL](X1,X2) = 0
[IF_THEN_ELSE_FI](X1,X2,X3) = 0
[ISBOOL](X) = 0
[NOT_](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 _XOR_(_xor_(x8,x9),x10) = _XOR_(x8,_xor_(x9,x10))
 _XOR_(x8,x9) = _XOR_(x9,x8)
-> Pairs:
 Empty
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> SRules:
 _XOR_(_xor_(x8,x9),x10) -> _XOR_(x8,x9)
 _XOR_(x8,_xor_(x9,x10)) -> _XOR_(x9,x10)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 _AND_(_and_(x8,x9),x10) = _AND_(x8,_and_(x9,x10))
 _AND_(x8,x9) = _AND_(x9,x8)
-> Pairs:
 U21#(tt,A,B,C) -> U22#(isBool(B),A,B,C)
 U22#(tt,A,B,C) -> U23#(isBool(C),A,B,C)
 U23#(tt,A,B,C) -> _AND_(A,B)
 U23#(tt,A,B,C) -> _AND_(A,C)
 _AND_(_and_(false,A),x8) -> _AND_(U31(isBool(A)),x8)
 _AND_(_and_(true,A),x8) -> _AND_(U41(isBool(A),A),x8)
 _AND_(_and_(A,_xor_(B,C)),x8) -> U21#(isBool(A),A,B,C)
 _AND_(_and_(A,_xor_(B,C)),x8) -> _AND_(U21(isBool(A),A,B,C),x8)
 _AND_(_and_(A,A),x8) -> _AND_(U11(isBool(A),A),x8)
 _AND_(A,_xor_(B,C)) -> U21#(isBool(A),A,B,C)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Usable Equations:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> Usable Rules:
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
-> SRules:
 _AND_(_and_(x8,x9),x10) -> _AND_(x8,x9)
 _AND_(x8,_and_(x9,x10)) -> _AND_(x9,x10)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[U101](X1,X2,X3) = 0
[U102](X1,X2,X3) = 0
[U11](X1,X2) = X2
[U111](X) = 1
[U121](X1,X2) = X1 + X2 + 1
[U131](X1,X2,X3,X4) = 0
[U132](X1,X2,X3,X4) = 0
[U133](X1,X2,X3) = 0
[U134](X1,X2) = 0
[U141](X1,X2) = 0
[U142](X1,X2) = 0
[U151](X1,X2) = X1
[U152](X) = 1
[U161](X1,X2) = X1
[U162](X) = X
[U171](X1,X2) = X1.X2 + X1
[U172](X) = X + 1
[U181](X1,X2) = X1.X2 + X1
[U182](X) = X.X + X
[U191](X1,X2) = X1
[U192](X) = X
[U201](X1,X2) = X1
[U202](X) = 1
[U21](X1,X2,X3,X4) = X1.X2.X3 + X1.X2.X4 + X1.X2 + X1 + X2 + X3 + X4
[U211](X) = X.X
[U22](X1,X2,X3,X4) = X1.X2.X3 + X1.X2 + X2.X4 + X1 + X2 + X3 + X4
[U221](X1,X2) = 0
[U23](X1,X2,X3,X4) = X1.X2.X3 + X1.X2 + X1.X3 + X1.X4 + X2.X4 + X2 + 1
[U31](X) = X.X
[U41](X1,X2) = X1.X2 + X2 + 1
[U51](X1,X2,X3) = 0
[U52](X1,X2,X3) = 0
[U61](X1,X2,X3) = 0
[U62](X1,X2,X3) = 0
[U63](X) = 0
[U71](X) = 0
[U81](X1,X2,X3) = 0
[U82](X1,X2,X3) = 0
[U91](X) = 0
[_and_](X1,X2) = X1.X2 + X1 + X2
[_implies_](X1,X2) = X1.X2 + X1 + X2 + 1
[_isEqualTo_](X1,X2) = X1.X2 + X2
[_isNotEqualTo_](X1,X2) = X1.X2 + 1
[_or_](X1,X2) = X1 + X2 + 1
[_xor_](X1,X2) = X1 + X2 + 1
[equal](X1,X2) = 0
[if_then_else_fi](X1,X2,X3) = 0
[isBool](X) = 1
[not_](X) = X.X
[false] = 1
[isS](X) = 0
[isUniversal](X) = 0
[true] = 1
[tt] = 1
[U101#](X1,X2,X3) = 0
[U102#](X1,X2,X3) = 0
[U11#](X1,X2) = 0
[U111#](X) = 0
[U121#](X1,X2) = 0
[U131#](X1,X2,X3,X4) = 0
[U132#](X1,X2,X3,X4) = 0
[U133#](X1,X2,X3) = 0
[U134#](X1,X2) = 0
[U141#](X1,X2) = 0
[U142#](X1,X2) = 0
[U151#](X1,X2) = 0
[U152#](X) = 0
[U161#](X1,X2) = 0
[U162#](X) = 0
[U171#](X1,X2) = 0
[U172#](X) = 0
[U181#](X1,X2) = 0
[U182#](X) = 0
[U191#](X1,X2) = 0
[U192#](X) = 0
[U201#](X1,X2) = 0
[U202#](X) = 0
[U21#](X1,X2,X3,X4) = X1.X2.X3 + X1.X2.X4 + X1.X2 + X1.X4 + X1 + X2 + X3 + 1
[U211#](X) = 0
[U22#](X1,X2,X3,X4) = X1.X2.X3 + X1.X2 + X2.X4 + X1 + X2 + X3 + X4
[U221#](X1,X2) = 0
[U23#](X1,X2,X3,X4) = X1.X2.X3 + X1.X2 + X1.X3 + X1.X4 + X2.X4 + X2 + 1
[U31#](X) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U52#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U62#](X1,X2,X3) = 0
[U63#](X) = 0
[U71#](X) = 0
[U81#](X1,X2,X3) = 0
[U82#](X1,X2,X3) = 0
[U91#](X) = 0
[_AND_](X1,X2) = X1.X2 + X1 + X2 + 1
[_IMPLIES_](X1,X2) = 0
[_ISEQUALTO_](X1,X2) = 0
[_ISNOTEQUALTO_](X1,X2) = 0
[_OR_](X1,X2) = 0
[_XOR_](X1,X2) = 0
[EQUAL](X1,X2) = 0
[IF_THEN_ELSE_FI](X1,X2,X3) = 0
[ISBOOL](X) = 0
[NOT_](X) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 _AND_(_and_(x8,x9),x10) = _AND_(x8,_and_(x9,x10))
 _AND_(x8,x9) = _AND_(x9,x8)
-> Pairs:
 U22#(tt,A,B,C) -> U23#(isBool(C),A,B,C)
 U23#(tt,A,B,C) -> _AND_(A,B)
 U23#(tt,A,B,C) -> _AND_(A,C)
 _AND_(_and_(false,A),x8) -> _AND_(U31(isBool(A)),x8)
 _AND_(_and_(true,A),x8) -> _AND_(U41(isBool(A),A),x8)
 _AND_(_and_(A,_xor_(B,C)),x8) -> U21#(isBool(A),A,B,C)
 _AND_(_and_(A,_xor_(B,C)),x8) -> _AND_(U21(isBool(A),A,B,C),x8)
 _AND_(_and_(A,A),x8) -> _AND_(U11(isBool(A),A),x8)
 _AND_(A,_xor_(B,C)) -> U21#(isBool(A),A,B,C)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> SRules:
 _AND_(_and_(x8,x9),x10) -> _AND_(x8,x9)
 _AND_(x8,_and_(x9,x10)) -> _AND_(x9,x10)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 _AND_(_and_(false,A),x8) -> _AND_(U31(isBool(A)),x8)
 _AND_(_and_(true,A),x8) -> _AND_(U41(isBool(A),A),x8)
 _AND_(_and_(A,_xor_(B,C)),x8) -> _AND_(U21(isBool(A),A,B,C),x8)
 _AND_(_and_(A,A),x8) -> _AND_(U11(isBool(A),A),x8)
-> FAxioms:
 _and_(_and_(x8,x9),x10) -> _and_(x8,_and_(x9,x10))
 _and_(x8,x9) -> _and_(x9,x8)
 _or_(_or_(x8,x9),x10) -> _or_(x8,_or_(x9,x10))
 _or_(x8,x9) -> _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) -> _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) -> _xor_(x9,x8)
 _AND_(_and_(x8,x9),x10) -> _AND_(x8,_and_(x9,x10))
 _AND_(x8,x9) -> _AND_(x9,x8)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
->->-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> SRules:
 _AND_(_and_(x8,x9),x10) -> _AND_(x8,x9)
 _AND_(x8,_and_(x9,x10)) -> _AND_(x9,x10)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 _AND_(_and_(x8,x9),x10) = _AND_(x8,_and_(x9,x10))
 _AND_(x8,x9) = _AND_(x9,x8)
-> Pairs:
 _AND_(_and_(false,A),x8) -> _AND_(U31(isBool(A)),x8)
 _AND_(_and_(true,A),x8) -> _AND_(U41(isBool(A),A),x8)
 _AND_(_and_(A,_xor_(B,C)),x8) -> _AND_(U21(isBool(A),A,B,C),x8)
 _AND_(_and_(A,A),x8) -> _AND_(U11(isBool(A),A),x8)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Usable Equations:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> Usable Rules:
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
-> SRules:
 _AND_(_and_(x8,x9),x10) -> _AND_(x8,x9)
 _AND_(x8,_and_(x9,x10)) -> _AND_(x9,x10)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[U101](X1,X2,X3) = 0
[U102](X1,X2,X3) = 0
[U11](X1,X2) = X1.X2 + X2
[U111](X) = 1
[U121](X1,X2) = X1.X2 + 1
[U131](X1,X2,X3,X4) = 0
[U132](X1,X2,X3,X4) = 0
[U133](X1,X2,X3) = 0
[U134](X1,X2) = 0
[U141](X1,X2) = 0
[U142](X1,X2) = 0
[U151](X1,X2) = 1
[U152](X) = 1
[U161](X1,X2) = X1
[U162](X) = 1
[U171](X1,X2) = 1
[U172](X) = X
[U181](X1,X2) = X1
[U182](X) = 1
[U191](X1,X2) = X1
[U192](X) = 1
[U201](X1,X2) = X1
[U202](X) = 1
[U21](X1,X2,X3,X4) = X1.X2.X4 + X1.X2 + X1.X4 + X2.X3 + X2 + X3 + 1
[U211](X) = 1
[U22](X1,X2,X3,X4) = X1.X2.X3 + X1.X2 + X2.X4 + X2 + X3 + X4 + 1
[U221](X1,X2) = 0
[U23](X1,X2,X3,X4) = X1.X2.X3 + X1.X2 + X1.X4 + X2.X4 + X1 + X2 + X3
[U31](X) = X.X
[U41](X1,X2) = X2
[U51](X1,X2,X3) = 0
[U52](X1,X2,X3) = 0
[U61](X1,X2,X3) = 0
[U62](X1,X2,X3) = 0
[U63](X) = 0
[U71](X) = 0
[U81](X1,X2,X3) = 0
[U82](X1,X2,X3) = 0
[U91](X) = 0
[_and_](X1,X2) = X1.X2 + X1 + X2
[_implies_](X1,X2) = X1.X2 + X2 + 1
[_isEqualTo_](X1,X2) = X2 + 1
[_isNotEqualTo_](X1,X2) = X2
[_or_](X1,X2) = X1.X2 + X1 + X2
[_xor_](X1,X2) = X1 + X2 + 1
[equal](X1,X2) = 0
[if_then_else_fi](X1,X2,X3) = 0
[isBool](X) = 1
[not_](X) = 0
[false] = 1
[isS](X) = 0
[isUniversal](X) = 1
[true] = 1
[tt] = 1
[U101#](X1,X2,X3) = 0
[U102#](X1,X2,X3) = 0
[U11#](X1,X2) = 0
[U111#](X) = 0
[U121#](X1,X2) = 0
[U131#](X1,X2,X3,X4) = 0
[U132#](X1,X2,X3,X4) = 0
[U133#](X1,X2,X3) = 0
[U134#](X1,X2) = 0
[U141#](X1,X2) = 0
[U142#](X1,X2) = 0
[U151#](X1,X2) = 0
[U152#](X) = 0
[U161#](X1,X2) = 0
[U162#](X) = 0
[U171#](X1,X2) = 0
[U172#](X) = 0
[U181#](X1,X2) = 0
[U182#](X) = 0
[U191#](X1,X2) = 0
[U192#](X) = 0
[U201#](X1,X2) = 0
[U202#](X) = 0
[U21#](X1,X2,X3,X4) = 0
[U211#](X) = 0
[U22#](X1,X2,X3,X4) = 0
[U221#](X1,X2) = 0
[U23#](X1,X2,X3,X4) = 0
[U31#](X) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U52#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U62#](X1,X2,X3) = 0
[U63#](X) = 0
[U71#](X) = 0
[U81#](X1,X2,X3) = 0
[U82#](X1,X2,X3) = 0
[U91#](X) = 0
[_AND_](X1,X2) = X1.X2 + X1 + X2
[_IMPLIES_](X1,X2) = 0
[_ISEQUALTO_](X1,X2) = 0
[_ISNOTEQUALTO_](X1,X2) = 0
[_OR_](X1,X2) = 0
[_XOR_](X1,X2) = 0
[EQUAL](X1,X2) = 0
[IF_THEN_ELSE_FI](X1,X2,X3) = 0
[ISBOOL](X) = 0
[NOT_](X) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 _AND_(_and_(x8,x9),x10) = _AND_(x8,_and_(x9,x10))
 _AND_(x8,x9) = _AND_(x9,x8)
-> Pairs:
 _AND_(_and_(false,A),x8) -> _AND_(U31(isBool(A)),x8)
 _AND_(_and_(A,_xor_(B,C)),x8) -> _AND_(U21(isBool(A),A,B,C),x8)
 _AND_(_and_(A,A),x8) -> _AND_(U11(isBool(A),A),x8)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> SRules:
 _AND_(_and_(x8,x9),x10) -> _AND_(x8,x9)
 _AND_(x8,_and_(x9,x10)) -> _AND_(x9,x10)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 _AND_(_and_(false,A),x8) -> _AND_(U31(isBool(A)),x8)
 _AND_(_and_(A,_xor_(B,C)),x8) -> _AND_(U21(isBool(A),A,B,C),x8)
 _AND_(_and_(A,A),x8) -> _AND_(U11(isBool(A),A),x8)
-> FAxioms:
 _and_(_and_(x8,x9),x10) -> _and_(x8,_and_(x9,x10))
 _and_(x8,x9) -> _and_(x9,x8)
 _or_(_or_(x8,x9),x10) -> _or_(x8,_or_(x9,x10))
 _or_(x8,x9) -> _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) -> _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) -> _xor_(x9,x8)
 _AND_(_and_(x8,x9),x10) -> _AND_(x8,_and_(x9,x10))
 _AND_(x8,x9) -> _AND_(x9,x8)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
->->-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> SRules:
 _AND_(_and_(x8,x9),x10) -> _AND_(x8,x9)
 _AND_(x8,_and_(x9,x10)) -> _AND_(x9,x10)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 _AND_(_and_(x8,x9),x10) = _AND_(x8,_and_(x9,x10))
 _AND_(x8,x9) = _AND_(x9,x8)
-> Pairs:
 _AND_(_and_(false,A),x8) -> _AND_(U31(isBool(A)),x8)
 _AND_(_and_(A,_xor_(B,C)),x8) -> _AND_(U21(isBool(A),A,B,C),x8)
 _AND_(_and_(A,A),x8) -> _AND_(U11(isBool(A),A),x8)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Usable Equations:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> Usable Rules:
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
-> SRules:
 _AND_(_and_(x8,x9),x10) -> _AND_(x8,x9)
 _AND_(x8,_and_(x9,x10)) -> _AND_(x9,x10)
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 3
->Interpretation:
 
[U101](X1,X2,X3) = 0
[U102](X1,X2,X3) = 0
[U11](X1,X2) = 2.X2 + 1/2
[U111](X) = 2/3
[U121](X1,X2) = X2 + 3/2
[U131](X1,X2,X3,X4) = 0
[U132](X1,X2,X3,X4) = 0
[U133](X1,X2,X3) = 0
[U134](X1,X2) = 0
[U141](X1,X2) = 0
[U142](X1,X2) = 0
[U151](X1,X2) = 1/2.X1 + 3/2.X2 + 2/3
[U152](X) = 2/3
[U161](X1,X2) = 1/2.X1 + 3/2.X2 + 1/2
[U162](X) = 0
[U171](X1,X2) = 3.X1.X2 + 1/3.X2 + 3/2
[U172](X) = 1/3
[U181](X1,X2) = 2/3.X1 + X2 + 1/2
[U182](X) = 2/3.X.X
[U191](X1,X2) = 1
[U192](X) = 0
[U201](X1,X2) = 3/2.X2 + 3
[U202](X) = 1/2
[U21](X1,X2,X3,X4) = 3/2.X2.X3 + 3/2.X2.X4 + 3.X2 + 3/2.X3 + 3/2.X4 + 3
[U211](X) = 1/3.X
[U22](X1,X2,X3,X4) = 3/2.X2.X3 + 3/2.X2.X4 + 3.X2 + 3/2.X3 + 3/2.X4 + 3
[U221](X1,X2) = 0
[U23](X1,X2,X3,X4) = 3/2.X2.X3 + 3/2.X2.X4 + 3.X2 + 3/2.X3 + 3/2.X4 + 3
[U31](X) = 1/3
[U41](X1,X2) = 3/2.X2 + 2/3
[U51](X1,X2,X3) = 0
[U52](X1,X2,X3) = 0
[U61](X1,X2,X3) = 0
[U62](X1,X2,X3) = 0
[U63](X) = 0
[U71](X) = 0
[U81](X1,X2,X3) = 0
[U82](X1,X2,X3) = 0
[U91](X) = 0
[_and_](X1,X2) = 3/2.X1.X2 + 3/2.X1 + 3/2.X2 + 1/2
[_implies_](X1,X2) = 2.X1.X2 + 3.X1 + 3.X2 + 3/2
[_isEqualTo_](X1,X2) = 2/3.X1.X2 + 2.X1 + 1/3.X2 + 2/3
[_isNotEqualTo_](X1,X2) = 3/2.X1.X2 + 2.X2 + 2/3
[_or_](X1,X2) = 2/3.X1.X2 + 3.X1 + X2 + 1/2
[_xor_](X1,X2) = X1 + X2 + 2
[equal](X1,X2) = 0
[if_then_else_fi](X1,X2,X3) = 0
[isBool](X) = 1/3.X.X + X + 2
[not_](X) = 1/2.X.X + 2/3.X + 2/3
[false] = 0
[isS](X) = 0
[isUniversal](X) = 0
[true] = 3
[tt] = 0
[U101#](X1,X2,X3) = 0
[U102#](X1,X2,X3) = 0
[U11#](X1,X2) = 0
[U111#](X) = 0
[U121#](X1,X2) = 0
[U131#](X1,X2,X3,X4) = 0
[U132#](X1,X2,X3,X4) = 0
[U133#](X1,X2,X3) = 0
[U134#](X1,X2) = 0
[U141#](X1,X2) = 0
[U142#](X1,X2) = 0
[U151#](X1,X2) = 0
[U152#](X) = 0
[U161#](X1,X2) = 0
[U162#](X) = 0
[U171#](X1,X2) = 0
[U172#](X) = 0
[U181#](X1,X2) = 0
[U182#](X) = 0
[U191#](X1,X2) = 0
[U192#](X) = 0
[U201#](X1,X2) = 0
[U202#](X) = 0
[U21#](X1,X2,X3,X4) = 0
[U211#](X) = 0
[U22#](X1,X2,X3,X4) = 0
[U221#](X1,X2) = 0
[U23#](X1,X2,X3,X4) = 0
[U31#](X) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U52#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U62#](X1,X2,X3) = 0
[U63#](X) = 0
[U71#](X) = 0
[U81#](X1,X2,X3) = 0
[U82#](X1,X2,X3) = 0
[U91#](X) = 0
[_AND_](X1,X2) = 3/2.X1.X2 + 3/2.X1 + 3/2.X2
[_IMPLIES_](X1,X2) = 0
[_ISEQUALTO_](X1,X2) = 0
[_ISNOTEQUALTO_](X1,X2) = 0
[_OR_](X1,X2) = 0
[_XOR_](X1,X2) = 0
[EQUAL](X1,X2) = 0
[IF_THEN_ELSE_FI](X1,X2,X3) = 0
[ISBOOL](X) = 0
[NOT_](X) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 _AND_(_and_(x8,x9),x10) = _AND_(x8,_and_(x9,x10))
 _AND_(x8,x9) = _AND_(x9,x8)
-> Pairs:
 _AND_(_and_(A,_xor_(B,C)),x8) -> _AND_(U21(isBool(A),A,B,C),x8)
 _AND_(_and_(A,A),x8) -> _AND_(U11(isBool(A),A),x8)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> SRules:
 _AND_(_and_(x8,x9),x10) -> _AND_(x8,x9)
 _AND_(x8,_and_(x9,x10)) -> _AND_(x9,x10)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 _AND_(_and_(A,_xor_(B,C)),x8) -> _AND_(U21(isBool(A),A,B,C),x8)
 _AND_(_and_(A,A),x8) -> _AND_(U11(isBool(A),A),x8)
-> FAxioms:
 _and_(_and_(x8,x9),x10) -> _and_(x8,_and_(x9,x10))
 _and_(x8,x9) -> _and_(x9,x8)
 _or_(_or_(x8,x9),x10) -> _or_(x8,_or_(x9,x10))
 _or_(x8,x9) -> _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) -> _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) -> _xor_(x9,x8)
 _AND_(_and_(x8,x9),x10) -> _AND_(x8,_and_(x9,x10))
 _AND_(x8,x9) -> _AND_(x9,x8)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
->->-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> SRules:
 _AND_(_and_(x8,x9),x10) -> _AND_(x8,x9)
 _AND_(x8,_and_(x9,x10)) -> _AND_(x9,x10)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 _AND_(_and_(x8,x9),x10) = _AND_(x8,_and_(x9,x10))
 _AND_(x8,x9) = _AND_(x9,x8)
-> Pairs:
 _AND_(_and_(A,_xor_(B,C)),x8) -> _AND_(U21(isBool(A),A,B,C),x8)
 _AND_(_and_(A,A),x8) -> _AND_(U11(isBool(A),A),x8)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Usable Equations:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> Usable Rules:
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
-> SRules:
 _AND_(_and_(x8,x9),x10) -> _AND_(x8,x9)
 _AND_(x8,_and_(x9,x10)) -> _AND_(x9,x10)
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 3
->Interpretation:
 
[U101](X1,X2,X3) = 0
[U102](X1,X2,X3) = 0
[U11](X1,X2) = 3/2.X1.X2 + X2
[U111](X) = 2
[U121](X1,X2) = X2 + 2/3
[U131](X1,X2,X3,X4) = 0
[U132](X1,X2,X3,X4) = 0
[U133](X1,X2,X3) = 0
[U134](X1,X2) = 0
[U141](X1,X2) = 0
[U142](X1,X2) = 0
[U151](X1,X2) = X1 + 1/3.X2
[U152](X) = 1/3.X + 1/2
[U161](X1,X2) = 1/2.X1.X2 + 1/3.X2 + 1/2
[U162](X) = X
[U171](X1,X2) = 3.X1.X2 + 3/2.X1 + 2/3.X2 + 2/3
[U172](X) = 1
[U181](X1,X2) = X1.X2 + 2.X1 + 1/3.X2 + 2
[U182](X) = 2.X.X + 1/3.X + 3
[U191](X1,X2) = 2/3.X1.X2 + 3/2.X2 + 2
[U192](X) = 1/2.X + 2/3
[U201](X1,X2) = X1 + 2/3.X2 + 3/2
[U202](X) = 2/3.X + 1
[U21](X1,X2,X3,X4) = 3.X2.X3 + 3.X2.X4 + 3.X1 + 2.X2 + X3 + X4 + 1
[U211](X) = 2/3.X.X + 1/2
[U22](X1,X2,X3,X4) = 3.X2.X3 + 3.X2.X4 + 2.X2 + X3 + X4 + 3
[U221](X1,X2) = 0
[U23](X1,X2,X3,X4) = 3.X2.X3 + 3.X2.X4 + 2.X2 + X3 + X4 + 3
[U31](X) = 3/2.X + 1
[U41](X1,X2) = X2 + 3
[U51](X1,X2,X3) = 0
[U52](X1,X2,X3) = 0
[U61](X1,X2,X3) = 0
[U62](X1,X2,X3) = 0
[U63](X) = 0
[U71](X) = 0
[U81](X1,X2,X3) = 0
[U82](X1,X2,X3) = 0
[U91](X) = 0
[_and_](X1,X2) = 3.X1.X2 + X1 + X2
[_implies_](X1,X2) = 3.X1.X2 + 2.X1 + 2.X2 + 1/2
[_isEqualTo_](X1,X2) = 2.X1 + X2 + 2/3
[_isNotEqualTo_](X1,X2) = 1/3.X1.X2 + 2/3.X1 + 2/3.X2 + 3
[_or_](X1,X2) = 3.X1.X2 + X1 + 3.X2 + 3
[_xor_](X1,X2) = X1 + X2 + 3
[equal](X1,X2) = 0
[if_then_else_fi](X1,X2,X3) = 0
[isBool](X) = 2/3.X + 1/2
[not_](X) = 1/2.X.X + 3.X + 1/3
[false] = 2
[isS](X) = 0
[isUniversal](X) = 0
[true] = 3
[tt] = 2/3
[U101#](X1,X2,X3) = 0
[U102#](X1,X2,X3) = 0
[U11#](X1,X2) = 0
[U111#](X) = 0
[U121#](X1,X2) = 0
[U131#](X1,X2,X3,X4) = 0
[U132#](X1,X2,X3,X4) = 0
[U133#](X1,X2,X3) = 0
[U134#](X1,X2) = 0
[U141#](X1,X2) = 0
[U142#](X1,X2) = 0
[U151#](X1,X2) = 0
[U152#](X) = 0
[U161#](X1,X2) = 0
[U162#](X) = 0
[U171#](X1,X2) = 0
[U172#](X) = 0
[U181#](X1,X2) = 0
[U182#](X) = 0
[U191#](X1,X2) = 0
[U192#](X) = 0
[U201#](X1,X2) = 0
[U202#](X) = 0
[U21#](X1,X2,X3,X4) = 0
[U211#](X) = 0
[U22#](X1,X2,X3,X4) = 0
[U221#](X1,X2) = 0
[U23#](X1,X2,X3,X4) = 0
[U31#](X) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U52#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U62#](X1,X2,X3) = 0
[U63#](X) = 0
[U71#](X) = 0
[U81#](X1,X2,X3) = 0
[U82#](X1,X2,X3) = 0
[U91#](X) = 0
[_AND_](X1,X2) = X1.X2 + 1/3.X1 + 1/3.X2
[_IMPLIES_](X1,X2) = 0
[_ISEQUALTO_](X1,X2) = 0
[_ISNOTEQUALTO_](X1,X2) = 0
[_OR_](X1,X2) = 0
[_XOR_](X1,X2) = 0
[EQUAL](X1,X2) = 0
[IF_THEN_ELSE_FI](X1,X2,X3) = 0
[ISBOOL](X) = 0
[NOT_](X) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 _AND_(_and_(x8,x9),x10) = _AND_(x8,_and_(x9,x10))
 _AND_(x8,x9) = _AND_(x9,x8)
-> Pairs:
 _AND_(_and_(A,A),x8) -> _AND_(U11(isBool(A),A),x8)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> SRules:
 _AND_(_and_(x8,x9),x10) -> _AND_(x8,x9)
 _AND_(x8,_and_(x9,x10)) -> _AND_(x9,x10)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 _AND_(_and_(A,A),x8) -> _AND_(U11(isBool(A),A),x8)
-> FAxioms:
 _and_(_and_(x8,x9),x10) -> _and_(x8,_and_(x9,x10))
 _and_(x8,x9) -> _and_(x9,x8)
 _or_(_or_(x8,x9),x10) -> _or_(x8,_or_(x9,x10))
 _or_(x8,x9) -> _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) -> _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) -> _xor_(x9,x8)
 _AND_(_and_(x8,x9),x10) -> _AND_(x8,_and_(x9,x10))
 _AND_(x8,x9) -> _AND_(x9,x8)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
->->-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> SRules:
 _AND_(_and_(x8,x9),x10) -> _AND_(x8,x9)
 _AND_(x8,_and_(x9,x10)) -> _AND_(x9,x10)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 _AND_(_and_(x8,x9),x10) = _AND_(x8,_and_(x9,x10))
 _AND_(x8,x9) = _AND_(x9,x8)
-> Pairs:
 _AND_(_and_(A,A),x8) -> _AND_(U11(isBool(A),A),x8)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Usable Equations:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> Usable Rules:
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
-> SRules:
 _AND_(_and_(x8,x9),x10) -> _AND_(x8,x9)
 _AND_(x8,_and_(x9,x10)) -> _AND_(x9,x10)
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 3
->Interpretation:
 
[U101](X1,X2,X3) = 0
[U102](X1,X2,X3) = 0
[U11](X1,X2) = X2 + 1/3
[U111](X) = 1/3
[U121](X1,X2) = X2 + 1
[U131](X1,X2,X3,X4) = 0
[U132](X1,X2,X3,X4) = 0
[U133](X1,X2,X3) = 0
[U134](X1,X2) = 0
[U141](X1,X2) = 0
[U142](X1,X2) = 0
[U151](X1,X2) = X1.X2 + 1/2.X1 + 2.X2 + 2/3
[U152](X) = 2/3
[U161](X1,X2) = 1/3.X2 + 2/3
[U162](X) = 0
[U171](X1,X2) = 2/3.X2 + 1/2
[U172](X) = 1/2
[U181](X1,X2) = X2 + 1
[U182](X) = 1
[U191](X1,X2) = 1/2.X1 + 2/3.X2 + 2
[U192](X) = 0
[U201](X1,X2) = 1/3.X1 + 1/2.X2 + 1
[U202](X) = 2/3
[U21](X1,X2,X3,X4) = 3/2.X2.X3 + 3/2.X2.X4 + 3.X2 + 3/2.X3 + 3/2.X4 + 3
[U211](X) = 1/3.X
[U22](X1,X2,X3,X4) = 3/2.X2.X3 + 3/2.X2.X4 + 3.X2 + 3/2.X3 + 3/2.X4 + 3
[U221](X1,X2) = 0
[U23](X1,X2,X3,X4) = 3/2.X2.X3 + 3/2.X2.X4 + 3.X2 + 3/2.X3 + 3/2.X4 + 3
[U31](X) = 1/3
[U41](X1,X2) = 2.X2 + 1
[U51](X1,X2,X3) = 0
[U52](X1,X2,X3) = 0
[U61](X1,X2,X3) = 0
[U62](X1,X2,X3) = 0
[U63](X) = 0
[U71](X) = 0
[U81](X1,X2,X3) = 0
[U82](X1,X2,X3) = 0
[U91](X) = 0
[_and_](X1,X2) = 3/2.X1.X2 + 3/2.X1 + 3/2.X2 + 1/2
[_implies_](X1,X2) = 1/2.X1.X2 + 3.X2 + 2/3
[_isEqualTo_](X1,X2) = X1.X2 + 1/3.X1 + 2.X2 + 1/3
[_isNotEqualTo_](X1,X2) = 1/3.X1 + 2/3.X2 + 2/3
[_or_](X1,X2) = 3.X1 + X2 + 3/2
[_xor_](X1,X2) = X1 + X2 + 2
[equal](X1,X2) = 0
[if_then_else_fi](X1,X2,X3) = 0
[isBool](X) = 2.X.X + 1/3
[not_](X) = 1/2.X.X + 1/3
[false] = 1/3
[isS](X) = 0
[isUniversal](X) = 2.X
[true] = 1/3
[tt] = 0
[U101#](X1,X2,X3) = 0
[U102#](X1,X2,X3) = 0
[U11#](X1,X2) = 0
[U111#](X) = 0
[U121#](X1,X2) = 0
[U131#](X1,X2,X3,X4) = 0
[U132#](X1,X2,X3,X4) = 0
[U133#](X1,X2,X3) = 0
[U134#](X1,X2) = 0
[U141#](X1,X2) = 0
[U142#](X1,X2) = 0
[U151#](X1,X2) = 0
[U152#](X) = 0
[U161#](X1,X2) = 0
[U162#](X) = 0
[U171#](X1,X2) = 0
[U172#](X) = 0
[U181#](X1,X2) = 0
[U182#](X) = 0
[U191#](X1,X2) = 0
[U192#](X) = 0
[U201#](X1,X2) = 0
[U202#](X) = 0
[U21#](X1,X2,X3,X4) = 0
[U211#](X) = 0
[U22#](X1,X2,X3,X4) = 0
[U221#](X1,X2) = 0
[U23#](X1,X2,X3,X4) = 0
[U31#](X) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U52#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U62#](X1,X2,X3) = 0
[U63#](X) = 0
[U71#](X) = 0
[U81#](X1,X2,X3) = 0
[U82#](X1,X2,X3) = 0
[U91#](X) = 0
[_AND_](X1,X2) = 1/3.X1.X2 + 1/3.X1 + 1/3.X2
[_IMPLIES_](X1,X2) = 0
[_ISEQUALTO_](X1,X2) = 0
[_ISNOTEQUALTO_](X1,X2) = 0
[_OR_](X1,X2) = 0
[_XOR_](X1,X2) = 0
[EQUAL](X1,X2) = 0
[IF_THEN_ELSE_FI](X1,X2,X3) = 0
[ISBOOL](X) = 0
[NOT_](X) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 _AND_(_and_(x8,x9),x10) = _AND_(x8,_and_(x9,x10))
 _AND_(x8,x9) = _AND_(x9,x8)
-> Pairs:
 Empty
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> SRules:
 _AND_(_and_(x8,x9),x10) -> _AND_(x8,x9)
 _AND_(x8,_and_(x9,x10)) -> _AND_(x9,x10)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Reduction Pairs Processor:
-> FAxioms:
 _OR_(_or_(x8,x9),x10) = _OR_(x8,_or_(x9,x10))
 _OR_(x8,x9) = _OR_(x9,x8)
-> Pairs:
 _OR_(_or_(A,B),x8) -> _OR_(U101(isBool(A),A,B),x8)
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Usable Equations:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> Usable Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
-> SRules:
 _OR_(_or_(x8,x9),x10) -> _OR_(x8,x9)
 _OR_(x8,_or_(x9,x10)) -> _OR_(x9,x10)
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 3
->Interpretation:
 
[U101](X1,X2,X3) = 3.X2.X3 + 3.X2 + 3.X3 + 1
[U102](X1,X2,X3) = 3.X2.X3 + 3.X2 + 3.X3 + 2/3
[U11](X1,X2) = X2
[U111](X) = 1/3
[U121](X1,X2) = X2 + 2/3
[U131](X1,X2,X3,X4) = 0
[U132](X1,X2,X3,X4) = 0
[U133](X1,X2,X3) = 0
[U134](X1,X2) = 0
[U141](X1,X2) = 0
[U142](X1,X2) = 0
[U151](X1,X2) = 3/2.X1.X2 + X1
[U152](X) = 1/3
[U161](X1,X2) = 1/3.X1.X2 + X1 + 2/3.X2 + 2
[U162](X) = 1/3
[U171](X1,X2) = 3.X1.X2 + 3/2.X1 + X2
[U172](X) = 1/2
[U181](X1,X2) = 3.X2 + 1
[U182](X) = 1
[U191](X1,X2) = X1.X2 + 2/3.X1 + X2 + 2/3
[U192](X) = 1/3
[U201](X1,X2) = 3.X2 + 3
[U202](X) = 3
[U21](X1,X2,X3,X4) = 3.X2.X3 + 3.X2.X4 + 2.X2 + X3 + X4 + 1/3
[U211](X) = 1/3
[U22](X1,X2,X3,X4) = 3.X2.X3 + 3.X2.X4 + 2.X2 + X3 + X4 + 1/3
[U221](X1,X2) = 0
[U23](X1,X2,X3,X4) = 3.X2.X3 + 3.X2.X4 + 2.X2 + X3 + X4 + 1/3
[U31](X) = 1/3
[U41](X1,X2) = 3.X2 + 2/3
[U51](X1,X2,X3) = 0
[U52](X1,X2,X3) = 0
[U61](X1,X2,X3) = 0
[U62](X1,X2,X3) = 0
[U63](X) = 0
[U71](X) = 0
[U81](X1,X2,X3) = 0
[U82](X1,X2,X3) = 0
[U91](X) = 0
[_and_](X1,X2) = 3.X1.X2 + X1 + X2
[_implies_](X1,X2) = 1/3.X1.X2 + 2.X1 + 1/2.X2 + 2
[_isEqualTo_](X1,X2) = 2.X1.X2 + 2.X1 + 1/2.X2
[_isNotEqualTo_](X1,X2) = 1/2.X1.X2 + 1/3.X1 + 3.X2 + 2/3
[_or_](X1,X2) = 3.X1.X2 + 3.X1 + 3.X2 + 2
[_xor_](X1,X2) = X1 + X2 + 1/3
[equal](X1,X2) = 0
[if_then_else_fi](X1,X2,X3) = 0
[isBool](X) = 1/3.X.X + 3.X + 2
[not_](X) = 2.X.X + 1/2
[false] = 1/3
[isS](X) = 0
[isUniversal](X) = 1/2.X.X
[true] = 1
[tt] = 1/3
[U101#](X1,X2,X3) = 0
[U102#](X1,X2,X3) = 0
[U11#](X1,X2) = 0
[U111#](X) = 0
[U121#](X1,X2) = 0
[U131#](X1,X2,X3,X4) = 0
[U132#](X1,X2,X3,X4) = 0
[U133#](X1,X2,X3) = 0
[U134#](X1,X2) = 0
[U141#](X1,X2) = 0
[U142#](X1,X2) = 0
[U151#](X1,X2) = 0
[U152#](X) = 0
[U161#](X1,X2) = 0
[U162#](X) = 0
[U171#](X1,X2) = 0
[U172#](X) = 0
[U181#](X1,X2) = 0
[U182#](X) = 0
[U191#](X1,X2) = 0
[U192#](X) = 0
[U201#](X1,X2) = 0
[U202#](X) = 0
[U21#](X1,X2,X3,X4) = 0
[U211#](X) = 0
[U22#](X1,X2,X3,X4) = 0
[U221#](X1,X2) = 0
[U23#](X1,X2,X3,X4) = 0
[U31#](X) = 0
[U41#](X1,X2) = 0
[U51#](X1,X2,X3) = 0
[U52#](X1,X2,X3) = 0
[U61#](X1,X2,X3) = 0
[U62#](X1,X2,X3) = 0
[U63#](X) = 0
[U71#](X) = 0
[U81#](X1,X2,X3) = 0
[U82#](X1,X2,X3) = 0
[U91#](X) = 0
[_AND_](X1,X2) = 0
[_IMPLIES_](X1,X2) = 0
[_ISEQUALTO_](X1,X2) = 0
[_ISNOTEQUALTO_](X1,X2) = 0
[_OR_](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2
[_XOR_](X1,X2) = 0
[EQUAL](X1,X2) = 0
[IF_THEN_ELSE_FI](X1,X2,X3) = 0
[ISBOOL](X) = 0
[NOT_](X) = 0

Problem 1.4: 

SCC Processor:
-> FAxioms:
 _OR_(_or_(x8,x9),x10) = _OR_(x8,_or_(x9,x10))
 _OR_(x8,x9) = _OR_(x9,x8)
-> Pairs:
 Empty
-> EAxioms:
 _and_(_and_(x8,x9),x10) = _and_(x8,_and_(x9,x10))
 _and_(x8,x9) = _and_(x9,x8)
 _or_(_or_(x8,x9),x10) = _or_(x8,_or_(x9,x10))
 _or_(x8,x9) = _or_(x9,x8)
 _xor_(_xor_(x8,x9),x10) = _xor_(x8,_xor_(x9,x10))
 _xor_(x8,x9) = _xor_(x9,x8)
-> Rules:
 U101(tt,A,B) -> U102(isBool(B),A,B)
 U102(tt,A,B) -> _xor_(_and_(A,B),_xor_(A,B))
 U11(tt,A) -> A
 U111(tt) -> false
 U121(tt,A) -> A
 U131(tt,B,U',U) -> U132(isS(U'),B,U',U)
 U132(tt,B,U',U) -> U133(isS(U),B,U')
 U133(tt,B,U') -> U134(equal(_isNotEqualTo_(B,true),true),U')
 U134(tt,U') -> U'
 U141(tt,U) -> U142(isS(U),U)
 U142(tt,U) -> U
 U151(tt,V2) -> U152(isBool(V2))
 U152(tt) -> tt
 U161(tt,V2) -> U162(isBool(V2))
 U162(tt) -> tt
 U171(tt,V2) -> U172(isUniversal(V2))
 U172(tt) -> tt
 U181(tt,V2) -> U182(isUniversal(V2))
 U182(tt) -> tt
 U191(tt,V2) -> U192(isBool(V2))
 U192(tt) -> tt
 U201(tt,V2) -> U202(isBool(V2))
 U202(tt) -> tt
 U21(tt,A,B,C) -> U22(isBool(B),A,B,C)
 U211(tt) -> tt
 U22(tt,A,B,C) -> U23(isBool(C),A,B,C)
 U221(tt,A) -> _xor_(A,true)
 U23(tt,A,B,C) -> _xor_(_and_(A,B),_and_(A,C))
 U31(tt) -> false
 U41(tt,A) -> A
 U51(tt,A,B) -> U52(isBool(B),A,B)
 U52(tt,A,B) -> not_(_xor_(A,_and_(A,B)))
 U61(tt,U',U) -> U62(isS(U),U',U)
 U62(tt,U',U) -> U63(equal(_isNotEqualTo_(U,U'),true))
 U63(tt) -> false
 U71(tt) -> true
 U81(tt,U',U) -> U82(isS(U),U',U)
 U82(tt,U',U) -> if_then_else_fi(_isEqualTo_(U,U'),false,true)
 U91(tt) -> false
 _and_(false,A) -> U31(isBool(A))
 _and_(true,A) -> U41(isBool(A),A)
 _and_(A,_xor_(B,C)) -> U21(isBool(A),A,B,C)
 _and_(A,A) -> U11(isBool(A),A)
 _implies_(A,B) -> U51(isBool(A),A,B)
 _isEqualTo_(U,U) -> U71(isS(U))
 _isEqualTo_(U,U') -> U61(isS(U'),U',U)
 _isNotEqualTo_(U,U) -> U91(isS(U))
 _isNotEqualTo_(U,U') -> U81(isS(U'),U',U)
 _or_(A,B) -> U101(isBool(A),A,B)
 _xor_(false,A) -> U121(isBool(A),A)
 _xor_(A,A) -> U111(isBool(A))
 equal(X,X) -> tt
 if_then_else_fi(true,U,U') -> U141(isS(U'),U)
 if_then_else_fi(B,U,U') -> U131(isBool(B),B,U',U)
 isBool(_and_(V1,V2)) -> U151(isBool(V1),V2)
 isBool(_implies_(V1,V2)) -> U161(isBool(V1),V2)
 isBool(_isEqualTo_(V1,V2)) -> U171(isUniversal(V1),V2)
 isBool(_isNotEqualTo_(V1,V2)) -> U181(isUniversal(V1),V2)
 isBool(_or_(V1,V2)) -> U191(isBool(V1),V2)
 isBool(_xor_(V1,V2)) -> U201(isBool(V1),V2)
 isBool(not_(V1)) -> U211(isBool(V1))
 isBool(false) -> tt
 isBool(true) -> tt
 not_(false) -> true
 not_(true) -> false
 not_(A) -> U221(isBool(A),A)
-> SRules:
 _OR_(_or_(x8,x9),x10) -> _OR_(x8,x9)
 _OR_(x8,_or_(x9,x10)) -> _OR_(x9,x10)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.