YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ETRS could be proven: (0) ETRS (1) EquationalDependencyPairsProof [EQUIVALENT, 88 ms] (2) EDP (3) EDependencyGraphProof [EQUIVALENT, 0 ms] (4) AND (5) EDP (6) ESharpUsableEquationsProof [EQUIVALENT, 0 ms] (7) EDP (8) EUsableRulesReductionPairsProof [EQUIVALENT, 31 ms] (9) EDP (10) EDependencyGraphProof [EQUIVALENT, 0 ms] (11) TRUE (12) EDP (13) ESharpUsableEquationsProof [EQUIVALENT, 0 ms] (14) EDP (15) EDPPoloProof [EQUIVALENT, 24 ms] (16) EDP (17) EDPPoloProof [EQUIVALENT, 21 ms] (18) EDP (19) PisEmptyProof [EQUIVALENT, 0 ms] (20) YES (21) EDP (22) ESharpUsableEquationsProof [EQUIVALENT, 0 ms] (23) EDP (24) EDPPoloProof [EQUIVALENT, 169 ms] (25) EDP (26) EDependencyGraphProof [EQUIVALENT, 0 ms] (27) EDP (28) EDPPoloProof [EQUIVALENT, 493 ms] (29) EDP (30) EDPPoloProof [EQUIVALENT, 496 ms] (31) EDP (32) EDPPoloProof [EQUIVALENT, 439 ms] (33) EDP (34) PisEmptyProof [EQUIVALENT, 0 ms] (35) YES (36) EDP (37) ESharpUsableEquationsProof [EQUIVALENT, 0 ms] (38) EDP (39) EDPPoloProof [EQUIVALENT, 603 ms] (40) EDP (41) PisEmptyProof [EQUIVALENT, 0 ms] (42) YES ---------------------------------------- (0) Obligation: Equational rewrite system: The TRS R consists of the following 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_(A, A) -> U11(isBool(A), A) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(false, A) -> U31(isBool(A)) _and_(true, A) -> U41(isBool(A), A) _implies_(A, B) -> U51(isBool(A), A, B) _isEqualTo_(U, U') -> U61(isS(U'), U', U) _isEqualTo_(U, U) -> U71(isS(U)) _isNotEqualTo_(U, U') -> U81(isS(U'), U', U) _isNotEqualTo_(U, U) -> U91(isS(U)) _or_(A, B) -> U101(isBool(A), A, B) _xor_(A, A) -> U111(isBool(A)) _xor_(false, A) -> U121(isBool(A), A) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U131(isBool(B), B, U', U) if_then_else_fi(true, U, U') -> U141(isS(U'), U) isBool(false) -> tt isBool(true) -> tt 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)) not_(A) -> U221(isBool(A), A) not_(false) -> true not_(true) -> false The set E consists of the following equations: _and_(x, y) == _and_(y, x) _or_(x, y) == _or_(y, x) _xor_(x, y) == _xor_(y, x) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _or_(_or_(x, y), z) == _or_(x, _or_(y, z)) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) ---------------------------------------- (1) EquationalDependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,DA_STEIN] we result in the following initial EDP problem: The TRS P consists of the following rules: U101^1(tt, A, B) -> U102^1(isBool(B), A, B) U101^1(tt, A, B) -> ISBOOL(B) U102^1(tt, A, B) -> _XOR_(_and_(A, B), _xor_(A, B)) U102^1(tt, A, B) -> _AND_(A, B) U102^1(tt, A, B) -> _XOR_(A, B) U131^1(tt, B, U', U) -> U132^1(isS(U'), B, U', U) U132^1(tt, B, U', U) -> U133^1(isS(U), B, U') U133^1(tt, B, U') -> U134^1(equal(_isNotEqualTo_(B, true), true), U') U133^1(tt, B, U') -> EQUAL(_isNotEqualTo_(B, true), true) U133^1(tt, B, U') -> _ISNOTEQUALTO_(B, true) U141^1(tt, U) -> U142^1(isS(U), U) U151^1(tt, V2) -> U152^1(isBool(V2)) U151^1(tt, V2) -> ISBOOL(V2) U161^1(tt, V2) -> U162^1(isBool(V2)) U161^1(tt, V2) -> ISBOOL(V2) U171^1(tt, V2) -> U172^1(isUniversal(V2)) U181^1(tt, V2) -> U182^1(isUniversal(V2)) U191^1(tt, V2) -> U192^1(isBool(V2)) U191^1(tt, V2) -> ISBOOL(V2) U201^1(tt, V2) -> U202^1(isBool(V2)) U201^1(tt, V2) -> ISBOOL(V2) U21^1(tt, A, B, C) -> U22^1(isBool(B), A, B, C) U21^1(tt, A, B, C) -> ISBOOL(B) U22^1(tt, A, B, C) -> U23^1(isBool(C), A, B, C) U22^1(tt, A, B, C) -> ISBOOL(C) U221^1(tt, A) -> _XOR_(A, true) U23^1(tt, A, B, C) -> _XOR_(_and_(A, B), _and_(A, C)) U23^1(tt, A, B, C) -> _AND_(A, B) U23^1(tt, A, B, C) -> _AND_(A, C) U51^1(tt, A, B) -> U52^1(isBool(B), A, B) U51^1(tt, A, B) -> ISBOOL(B) U52^1(tt, A, B) -> NOT_(_xor_(A, _and_(A, B))) U52^1(tt, A, B) -> _XOR_(A, _and_(A, B)) U52^1(tt, A, B) -> _AND_(A, B) U61^1(tt, U', U) -> U62^1(isS(U), U', U) U62^1(tt, U', U) -> U63^1(equal(_isNotEqualTo_(U, U'), true)) U62^1(tt, U', U) -> EQUAL(_isNotEqualTo_(U, U'), true) U62^1(tt, U', U) -> _ISNOTEQUALTO_(U, U') U81^1(tt, U', U) -> U82^1(isS(U), U', U) U82^1(tt, U', U) -> IF_THEN_ELSE_FI(_isEqualTo_(U, U'), false, true) U82^1(tt, U', U) -> _ISEQUALTO_(U, U') _AND_(A, A) -> U11^1(isBool(A), A) _AND_(A, A) -> ISBOOL(A) _AND_(A, _xor_(B, C)) -> U21^1(isBool(A), A, B, C) _AND_(A, _xor_(B, C)) -> ISBOOL(A) _AND_(false, A) -> U31^1(isBool(A)) _AND_(false, A) -> ISBOOL(A) _AND_(true, A) -> U41^1(isBool(A), A) _AND_(true, A) -> ISBOOL(A) _IMPLIES_(A, B) -> U51^1(isBool(A), A, B) _IMPLIES_(A, B) -> ISBOOL(A) _ISEQUALTO_(U, U') -> U61^1(isS(U'), U', U) _ISEQUALTO_(U, U) -> U71^1(isS(U)) _ISNOTEQUALTO_(U, U') -> U81^1(isS(U'), U', U) _ISNOTEQUALTO_(U, U) -> U91^1(isS(U)) _OR_(A, B) -> U101^1(isBool(A), A, B) _OR_(A, B) -> ISBOOL(A) _XOR_(A, A) -> U111^1(isBool(A)) _XOR_(A, A) -> ISBOOL(A) _XOR_(false, A) -> U121^1(isBool(A), A) _XOR_(false, A) -> ISBOOL(A) IF_THEN_ELSE_FI(B, U, U') -> U131^1(isBool(B), B, U', U) IF_THEN_ELSE_FI(B, U, U') -> ISBOOL(B) IF_THEN_ELSE_FI(true, U, U') -> U141^1(isS(U'), U) ISBOOL(_and_(V1, V2)) -> U151^1(isBool(V1), V2) ISBOOL(_and_(V1, V2)) -> ISBOOL(V1) ISBOOL(_implies_(V1, V2)) -> U161^1(isBool(V1), V2) ISBOOL(_implies_(V1, V2)) -> ISBOOL(V1) ISBOOL(_isEqualTo_(V1, V2)) -> U171^1(isUniversal(V1), V2) ISBOOL(_isNotEqualTo_(V1, V2)) -> U181^1(isUniversal(V1), V2) ISBOOL(_or_(V1, V2)) -> U191^1(isBool(V1), V2) ISBOOL(_or_(V1, V2)) -> ISBOOL(V1) ISBOOL(_xor_(V1, V2)) -> U201^1(isBool(V1), V2) ISBOOL(_xor_(V1, V2)) -> ISBOOL(V1) ISBOOL(not_(V1)) -> U211^1(isBool(V1)) ISBOOL(not_(V1)) -> ISBOOL(V1) NOT_(A) -> U221^1(isBool(A), A) NOT_(A) -> ISBOOL(A) _AND_(_and_(A, A), ext) -> _AND_(U11(isBool(A), A), ext) _AND_(_and_(A, A), ext) -> U11^1(isBool(A), A) _AND_(_and_(A, A), ext) -> ISBOOL(A) _AND_(_and_(A, _xor_(B, C)), ext) -> _AND_(U21(isBool(A), A, B, C), ext) _AND_(_and_(A, _xor_(B, C)), ext) -> U21^1(isBool(A), A, B, C) _AND_(_and_(A, _xor_(B, C)), ext) -> ISBOOL(A) _AND_(_and_(false, A), ext) -> _AND_(U31(isBool(A)), ext) _AND_(_and_(false, A), ext) -> U31^1(isBool(A)) _AND_(_and_(false, A), ext) -> ISBOOL(A) _AND_(_and_(true, A), ext) -> _AND_(U41(isBool(A), A), ext) _AND_(_and_(true, A), ext) -> U41^1(isBool(A), A) _AND_(_and_(true, A), ext) -> ISBOOL(A) _OR_(_or_(A, B), ext) -> _OR_(U101(isBool(A), A, B), ext) _OR_(_or_(A, B), ext) -> U101^1(isBool(A), A, B) _OR_(_or_(A, B), ext) -> ISBOOL(A) _XOR_(_xor_(A, A), ext) -> _XOR_(U111(isBool(A)), ext) _XOR_(_xor_(A, A), ext) -> U111^1(isBool(A)) _XOR_(_xor_(A, A), ext) -> ISBOOL(A) _XOR_(_xor_(false, A), ext) -> _XOR_(U121(isBool(A), A), ext) _XOR_(_xor_(false, A), ext) -> U121^1(isBool(A), A) _XOR_(_xor_(false, A), ext) -> ISBOOL(A) The TRS R consists of the following 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_(A, A) -> U11(isBool(A), A) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(false, A) -> U31(isBool(A)) _and_(true, A) -> U41(isBool(A), A) _implies_(A, B) -> U51(isBool(A), A, B) _isEqualTo_(U, U') -> U61(isS(U'), U', U) _isEqualTo_(U, U) -> U71(isS(U)) _isNotEqualTo_(U, U') -> U81(isS(U'), U', U) _isNotEqualTo_(U, U) -> U91(isS(U)) _or_(A, B) -> U101(isBool(A), A, B) _xor_(A, A) -> U111(isBool(A)) _xor_(false, A) -> U121(isBool(A), A) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U131(isBool(B), B, U', U) if_then_else_fi(true, U, U') -> U141(isS(U'), U) isBool(false) -> tt isBool(true) -> tt 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)) not_(A) -> U221(isBool(A), A) not_(false) -> true not_(true) -> false _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _or_(_or_(A, B), ext) -> _or_(U101(isBool(A), A, B), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) The set E consists of the following equations: _and_(x, y) == _and_(y, x) _or_(x, y) == _or_(y, x) _xor_(x, y) == _xor_(y, x) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _or_(_or_(x, y), z) == _or_(x, _or_(y, z)) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) The set E# consists of the following equations: _AND_(x, y) == _AND_(y, x) _OR_(x, y) == _OR_(y, x) _XOR_(x, y) == _XOR_(y, x) _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) _OR_(_or_(x, y), z) == _OR_(x, _or_(y, z)) _XOR_(_xor_(x, y), z) == _XOR_(x, _xor_(y, z)) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (2) Obligation: The TRS P consists of the following rules: U101^1(tt, A, B) -> U102^1(isBool(B), A, B) U101^1(tt, A, B) -> ISBOOL(B) U102^1(tt, A, B) -> _XOR_(_and_(A, B), _xor_(A, B)) U102^1(tt, A, B) -> _AND_(A, B) U102^1(tt, A, B) -> _XOR_(A, B) U131^1(tt, B, U', U) -> U132^1(isS(U'), B, U', U) U132^1(tt, B, U', U) -> U133^1(isS(U), B, U') U133^1(tt, B, U') -> U134^1(equal(_isNotEqualTo_(B, true), true), U') U133^1(tt, B, U') -> EQUAL(_isNotEqualTo_(B, true), true) U133^1(tt, B, U') -> _ISNOTEQUALTO_(B, true) U141^1(tt, U) -> U142^1(isS(U), U) U151^1(tt, V2) -> U152^1(isBool(V2)) U151^1(tt, V2) -> ISBOOL(V2) U161^1(tt, V2) -> U162^1(isBool(V2)) U161^1(tt, V2) -> ISBOOL(V2) U171^1(tt, V2) -> U172^1(isUniversal(V2)) U181^1(tt, V2) -> U182^1(isUniversal(V2)) U191^1(tt, V2) -> U192^1(isBool(V2)) U191^1(tt, V2) -> ISBOOL(V2) U201^1(tt, V2) -> U202^1(isBool(V2)) U201^1(tt, V2) -> ISBOOL(V2) U21^1(tt, A, B, C) -> U22^1(isBool(B), A, B, C) U21^1(tt, A, B, C) -> ISBOOL(B) U22^1(tt, A, B, C) -> U23^1(isBool(C), A, B, C) U22^1(tt, A, B, C) -> ISBOOL(C) U221^1(tt, A) -> _XOR_(A, true) U23^1(tt, A, B, C) -> _XOR_(_and_(A, B), _and_(A, C)) U23^1(tt, A, B, C) -> _AND_(A, B) U23^1(tt, A, B, C) -> _AND_(A, C) U51^1(tt, A, B) -> U52^1(isBool(B), A, B) U51^1(tt, A, B) -> ISBOOL(B) U52^1(tt, A, B) -> NOT_(_xor_(A, _and_(A, B))) U52^1(tt, A, B) -> _XOR_(A, _and_(A, B)) U52^1(tt, A, B) -> _AND_(A, B) U61^1(tt, U', U) -> U62^1(isS(U), U', U) U62^1(tt, U', U) -> U63^1(equal(_isNotEqualTo_(U, U'), true)) U62^1(tt, U', U) -> EQUAL(_isNotEqualTo_(U, U'), true) U62^1(tt, U', U) -> _ISNOTEQUALTO_(U, U') U81^1(tt, U', U) -> U82^1(isS(U), U', U) U82^1(tt, U', U) -> IF_THEN_ELSE_FI(_isEqualTo_(U, U'), false, true) U82^1(tt, U', U) -> _ISEQUALTO_(U, U') _AND_(A, A) -> U11^1(isBool(A), A) _AND_(A, A) -> ISBOOL(A) _AND_(A, _xor_(B, C)) -> U21^1(isBool(A), A, B, C) _AND_(A, _xor_(B, C)) -> ISBOOL(A) _AND_(false, A) -> U31^1(isBool(A)) _AND_(false, A) -> ISBOOL(A) _AND_(true, A) -> U41^1(isBool(A), A) _AND_(true, A) -> ISBOOL(A) _IMPLIES_(A, B) -> U51^1(isBool(A), A, B) _IMPLIES_(A, B) -> ISBOOL(A) _ISEQUALTO_(U, U') -> U61^1(isS(U'), U', U) _ISEQUALTO_(U, U) -> U71^1(isS(U)) _ISNOTEQUALTO_(U, U') -> U81^1(isS(U'), U', U) _ISNOTEQUALTO_(U, U) -> U91^1(isS(U)) _OR_(A, B) -> U101^1(isBool(A), A, B) _OR_(A, B) -> ISBOOL(A) _XOR_(A, A) -> U111^1(isBool(A)) _XOR_(A, A) -> ISBOOL(A) _XOR_(false, A) -> U121^1(isBool(A), A) _XOR_(false, A) -> ISBOOL(A) IF_THEN_ELSE_FI(B, U, U') -> U131^1(isBool(B), B, U', U) IF_THEN_ELSE_FI(B, U, U') -> ISBOOL(B) IF_THEN_ELSE_FI(true, U, U') -> U141^1(isS(U'), U) ISBOOL(_and_(V1, V2)) -> U151^1(isBool(V1), V2) ISBOOL(_and_(V1, V2)) -> ISBOOL(V1) ISBOOL(_implies_(V1, V2)) -> U161^1(isBool(V1), V2) ISBOOL(_implies_(V1, V2)) -> ISBOOL(V1) ISBOOL(_isEqualTo_(V1, V2)) -> U171^1(isUniversal(V1), V2) ISBOOL(_isNotEqualTo_(V1, V2)) -> U181^1(isUniversal(V1), V2) ISBOOL(_or_(V1, V2)) -> U191^1(isBool(V1), V2) ISBOOL(_or_(V1, V2)) -> ISBOOL(V1) ISBOOL(_xor_(V1, V2)) -> U201^1(isBool(V1), V2) ISBOOL(_xor_(V1, V2)) -> ISBOOL(V1) ISBOOL(not_(V1)) -> U211^1(isBool(V1)) ISBOOL(not_(V1)) -> ISBOOL(V1) NOT_(A) -> U221^1(isBool(A), A) NOT_(A) -> ISBOOL(A) _AND_(_and_(A, A), ext) -> _AND_(U11(isBool(A), A), ext) _AND_(_and_(A, A), ext) -> U11^1(isBool(A), A) _AND_(_and_(A, A), ext) -> ISBOOL(A) _AND_(_and_(A, _xor_(B, C)), ext) -> _AND_(U21(isBool(A), A, B, C), ext) _AND_(_and_(A, _xor_(B, C)), ext) -> U21^1(isBool(A), A, B, C) _AND_(_and_(A, _xor_(B, C)), ext) -> ISBOOL(A) _AND_(_and_(false, A), ext) -> _AND_(U31(isBool(A)), ext) _AND_(_and_(false, A), ext) -> U31^1(isBool(A)) _AND_(_and_(false, A), ext) -> ISBOOL(A) _AND_(_and_(true, A), ext) -> _AND_(U41(isBool(A), A), ext) _AND_(_and_(true, A), ext) -> U41^1(isBool(A), A) _AND_(_and_(true, A), ext) -> ISBOOL(A) _OR_(_or_(A, B), ext) -> _OR_(U101(isBool(A), A, B), ext) _OR_(_or_(A, B), ext) -> U101^1(isBool(A), A, B) _OR_(_or_(A, B), ext) -> ISBOOL(A) _XOR_(_xor_(A, A), ext) -> _XOR_(U111(isBool(A)), ext) _XOR_(_xor_(A, A), ext) -> U111^1(isBool(A)) _XOR_(_xor_(A, A), ext) -> ISBOOL(A) _XOR_(_xor_(false, A), ext) -> _XOR_(U121(isBool(A), A), ext) _XOR_(_xor_(false, A), ext) -> U121^1(isBool(A), A) _XOR_(_xor_(false, A), ext) -> ISBOOL(A) The TRS R consists of the following 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_(A, A) -> U11(isBool(A), A) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(false, A) -> U31(isBool(A)) _and_(true, A) -> U41(isBool(A), A) _implies_(A, B) -> U51(isBool(A), A, B) _isEqualTo_(U, U') -> U61(isS(U'), U', U) _isEqualTo_(U, U) -> U71(isS(U)) _isNotEqualTo_(U, U') -> U81(isS(U'), U', U) _isNotEqualTo_(U, U) -> U91(isS(U)) _or_(A, B) -> U101(isBool(A), A, B) _xor_(A, A) -> U111(isBool(A)) _xor_(false, A) -> U121(isBool(A), A) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U131(isBool(B), B, U', U) if_then_else_fi(true, U, U') -> U141(isS(U'), U) isBool(false) -> tt isBool(true) -> tt 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)) not_(A) -> U221(isBool(A), A) not_(false) -> true not_(true) -> false _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _or_(_or_(A, B), ext) -> _or_(U101(isBool(A), A, B), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) The set E consists of the following equations: _and_(x, y) == _and_(y, x) _or_(x, y) == _or_(y, x) _xor_(x, y) == _xor_(y, x) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _or_(_or_(x, y), z) == _or_(x, _or_(y, z)) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) The set E# consists of the following equations: _AND_(x, y) == _AND_(y, x) _OR_(x, y) == _OR_(y, x) _XOR_(x, y) == _XOR_(y, x) _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) _OR_(_or_(x, y), z) == _OR_(x, _or_(y, z)) _XOR_(_xor_(x, y), z) == _XOR_(x, _xor_(y, z)) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (3) EDependencyGraphProof (EQUIVALENT) The approximation of the Equational Dependency Graph [DA_STEIN] contains 4 SCCs with 73 less nodes. ---------------------------------------- (4) Complex Obligation (AND) ---------------------------------------- (5) Obligation: The TRS P consists of the following rules: U161^1(tt, V2) -> ISBOOL(V2) ISBOOL(_and_(V1, V2)) -> ISBOOL(V1) ISBOOL(_implies_(V1, V2)) -> U161^1(isBool(V1), V2) ISBOOL(_xor_(V1, V2)) -> U201^1(isBool(V1), V2) U151^1(tt, V2) -> ISBOOL(V2) ISBOOL(not_(V1)) -> ISBOOL(V1) ISBOOL(_or_(V1, V2)) -> ISBOOL(V1) ISBOOL(_implies_(V1, V2)) -> ISBOOL(V1) U201^1(tt, V2) -> ISBOOL(V2) ISBOOL(_or_(V1, V2)) -> U191^1(isBool(V1), V2) ISBOOL(_and_(V1, V2)) -> U151^1(isBool(V1), V2) U191^1(tt, V2) -> ISBOOL(V2) ISBOOL(_xor_(V1, V2)) -> ISBOOL(V1) The TRS R consists of the following 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_(A, A) -> U11(isBool(A), A) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(false, A) -> U31(isBool(A)) _and_(true, A) -> U41(isBool(A), A) _implies_(A, B) -> U51(isBool(A), A, B) _isEqualTo_(U, U') -> U61(isS(U'), U', U) _isEqualTo_(U, U) -> U71(isS(U)) _isNotEqualTo_(U, U') -> U81(isS(U'), U', U) _isNotEqualTo_(U, U) -> U91(isS(U)) _or_(A, B) -> U101(isBool(A), A, B) _xor_(A, A) -> U111(isBool(A)) _xor_(false, A) -> U121(isBool(A), A) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U131(isBool(B), B, U', U) if_then_else_fi(true, U, U') -> U141(isS(U'), U) isBool(false) -> tt isBool(true) -> tt 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)) not_(A) -> U221(isBool(A), A) not_(false) -> true not_(true) -> false _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _or_(_or_(A, B), ext) -> _or_(U101(isBool(A), A, B), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) The set E consists of the following equations: _and_(x, y) == _and_(y, x) _or_(x, y) == _or_(y, x) _xor_(x, y) == _xor_(y, x) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _or_(_or_(x, y), z) == _or_(x, _or_(y, z)) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) The set E# consists of the following equations: _AND_(x, y) == _AND_(y, x) _OR_(x, y) == _OR_(y, x) _XOR_(x, y) == _XOR_(y, x) _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) _OR_(_or_(x, y), z) == _OR_(x, _or_(y, z)) _XOR_(_xor_(x, y), z) == _XOR_(x, _xor_(y, z)) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (6) ESharpUsableEquationsProof (EQUIVALENT) We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]: _AND_(x, y) == _AND_(y, x) _OR_(x, y) == _OR_(y, x) _XOR_(x, y) == _XOR_(y, x) _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) _OR_(_or_(x, y), z) == _OR_(x, _or_(y, z)) _XOR_(_xor_(x, y), z) == _XOR_(x, _xor_(y, z)) ---------------------------------------- (7) Obligation: The TRS P consists of the following rules: U161^1(tt, V2) -> ISBOOL(V2) ISBOOL(_and_(V1, V2)) -> ISBOOL(V1) ISBOOL(_implies_(V1, V2)) -> U161^1(isBool(V1), V2) ISBOOL(_xor_(V1, V2)) -> U201^1(isBool(V1), V2) U151^1(tt, V2) -> ISBOOL(V2) ISBOOL(not_(V1)) -> ISBOOL(V1) ISBOOL(_or_(V1, V2)) -> ISBOOL(V1) ISBOOL(_implies_(V1, V2)) -> ISBOOL(V1) U201^1(tt, V2) -> ISBOOL(V2) ISBOOL(_or_(V1, V2)) -> U191^1(isBool(V1), V2) ISBOOL(_and_(V1, V2)) -> U151^1(isBool(V1), V2) U191^1(tt, V2) -> ISBOOL(V2) ISBOOL(_xor_(V1, V2)) -> ISBOOL(V1) The TRS R consists of the following 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_(A, A) -> U11(isBool(A), A) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(false, A) -> U31(isBool(A)) _and_(true, A) -> U41(isBool(A), A) _implies_(A, B) -> U51(isBool(A), A, B) _isEqualTo_(U, U') -> U61(isS(U'), U', U) _isEqualTo_(U, U) -> U71(isS(U)) _isNotEqualTo_(U, U') -> U81(isS(U'), U', U) _isNotEqualTo_(U, U) -> U91(isS(U)) _or_(A, B) -> U101(isBool(A), A, B) _xor_(A, A) -> U111(isBool(A)) _xor_(false, A) -> U121(isBool(A), A) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U131(isBool(B), B, U', U) if_then_else_fi(true, U, U') -> U141(isS(U'), U) isBool(false) -> tt isBool(true) -> tt 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)) not_(A) -> U221(isBool(A), A) not_(false) -> true not_(true) -> false _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _or_(_or_(A, B), ext) -> _or_(U101(isBool(A), A, B), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) The set E consists of the following equations: _and_(x, y) == _and_(y, x) _or_(x, y) == _or_(y, x) _xor_(x, y) == _xor_(y, x) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _or_(_or_(x, y), z) == _or_(x, _or_(y, z)) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) E# is empty. We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (8) EUsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules and equations with reduction pair processor [DA_STEIN] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules can be oriented non-strictly, the usable equations and the esharp equations can be oriented equivalently. All non-usable rules and equations are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. The following dependency pairs can be deleted: ISBOOL(_and_(V1, V2)) -> ISBOOL(V1) ISBOOL(_implies_(V1, V2)) -> U161^1(isBool(V1), V2) ISBOOL(_xor_(V1, V2)) -> U201^1(isBool(V1), V2) ISBOOL(not_(V1)) -> ISBOOL(V1) ISBOOL(_or_(V1, V2)) -> ISBOOL(V1) ISBOOL(_implies_(V1, V2)) -> ISBOOL(V1) ISBOOL(_or_(V1, V2)) -> U191^1(isBool(V1), V2) ISBOOL(_and_(V1, V2)) -> U151^1(isBool(V1), V2) ISBOOL(_xor_(V1, V2)) -> ISBOOL(V1) The following rules are removed from R: 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 U171(tt, V2) -> U172(isUniversal(V2)) U201(tt, V2) -> U202(isBool(V2)) U21(tt, A, B, C) -> U22(isBool(B), A, B, C) 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_(A, A) -> U11(isBool(A), A) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(false, A) -> U31(isBool(A)) _and_(true, A) -> U41(isBool(A), A) _implies_(A, B) -> U51(isBool(A), A, B) _isEqualTo_(U, U') -> U61(isS(U'), U', U) _isEqualTo_(U, U) -> U71(isS(U)) _isNotEqualTo_(U, U') -> U81(isS(U'), U', U) _isNotEqualTo_(U, U) -> U91(isS(U)) _or_(A, B) -> U101(isBool(A), A, B) _xor_(A, A) -> U111(isBool(A)) _xor_(false, A) -> U121(isBool(A), A) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U131(isBool(B), B, U', U) if_then_else_fi(true, U, U') -> U141(isS(U'), U) isBool(false) -> tt isBool(true) -> tt 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)) not_(A) -> U221(isBool(A), A) not_(false) -> true not_(true) -> false _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _or_(_or_(A, B), ext) -> _or_(U101(isBool(A), A, B), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) The following equations are removed from E: _and_(x, y) == _and_(y, x) _or_(x, y) == _or_(y, x) _xor_(x, y) == _xor_(y, x) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _or_(_or_(x, y), z) == _or_(x, _or_(y, z)) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) Used ordering: POLO with Polynomial interpretation [POLO]: POL(ISBOOL(x_1)) = 2*x_1 POL(U151(x_1, x_2)) = 2*x_1 + 3*x_2 POL(U151^1(x_1, x_2)) = 2*x_1 + 3*x_2 POL(U152(x_1)) = x_1 POL(U161(x_1, x_2)) = 2*x_1 + 3*x_2 POL(U161^1(x_1, x_2)) = 2*x_1 + 3*x_2 POL(U162(x_1)) = x_1 POL(U171(x_1, x_2)) = 2 + 2*x_1 + 3*x_2 POL(U172(x_1)) = x_1 POL(U181(x_1, x_2)) = 2*x_1 + 3*x_2 POL(U182(x_1)) = x_1 POL(U191(x_1, x_2)) = 2*x_1 + 3*x_2 POL(U191^1(x_1, x_2)) = 2*x_1 + 3*x_2 POL(U192(x_1)) = x_1 POL(U201(x_1, x_2)) = 1 + 2*x_1 + 3*x_2 POL(U201^1(x_1, x_2)) = 2*x_1 + 3*x_2 POL(U202(x_1)) = x_1 POL(U211(x_1)) = 2*x_1 POL(_and_(x_1, x_2)) = 2 + 3*x_1 + 3*x_2 POL(_implies_(x_1, x_2)) = 3 + 3*x_1 + 3*x_2 POL(_isEqualTo_(x_1, x_2)) = 2 + 3*x_1 + 3*x_2 POL(_isNotEqualTo_(x_1, x_2)) = 1 + 3*x_1 + 3*x_2 POL(_or_(x_1, x_2)) = 3*x_1 + 3*x_2 POL(_xor_(x_1, x_2)) = 1 + 3*x_1 + 3*x_2 POL(false) = 0 POL(isBool(x_1)) = 2*x_1 POL(isUniversal(x_1)) = 2*x_1 POL(not_(x_1)) = 3 + 3*x_1 POL(true) = 0 POL(tt) = 0 ---------------------------------------- (9) Obligation: The TRS P consists of the following rules: U161^1(tt, V2) -> ISBOOL(V2) U151^1(tt, V2) -> ISBOOL(V2) U201^1(tt, V2) -> ISBOOL(V2) U191^1(tt, V2) -> ISBOOL(V2) The TRS R consists of the following rules: U162(tt) -> tt U202(tt) -> tt U181(tt, V2) -> U182(isUniversal(V2)) U161(tt, V2) -> U162(isBool(V2)) U151(tt, V2) -> U152(isBool(V2)) U152(tt) -> tt U172(tt) -> tt U192(tt) -> tt U182(tt) -> tt U211(tt) -> tt U191(tt, V2) -> U192(isBool(V2)) E is empty. E# is empty. We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (10) EDependencyGraphProof (EQUIVALENT) The approximation of the Equational Dependency Graph [DA_STEIN] contains 0 SCCs with 4 less nodes. ---------------------------------------- (11) TRUE ---------------------------------------- (12) Obligation: The TRS P consists of the following rules: _XOR_(_xor_(A, A), ext) -> _XOR_(U111(isBool(A)), ext) _XOR_(_xor_(false, A), ext) -> _XOR_(U121(isBool(A), A), ext) The TRS R consists of the following 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_(A, A) -> U11(isBool(A), A) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(false, A) -> U31(isBool(A)) _and_(true, A) -> U41(isBool(A), A) _implies_(A, B) -> U51(isBool(A), A, B) _isEqualTo_(U, U') -> U61(isS(U'), U', U) _isEqualTo_(U, U) -> U71(isS(U)) _isNotEqualTo_(U, U') -> U81(isS(U'), U', U) _isNotEqualTo_(U, U) -> U91(isS(U)) _or_(A, B) -> U101(isBool(A), A, B) _xor_(A, A) -> U111(isBool(A)) _xor_(false, A) -> U121(isBool(A), A) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U131(isBool(B), B, U', U) if_then_else_fi(true, U, U') -> U141(isS(U'), U) isBool(false) -> tt isBool(true) -> tt 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)) not_(A) -> U221(isBool(A), A) not_(false) -> true not_(true) -> false _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _or_(_or_(A, B), ext) -> _or_(U101(isBool(A), A, B), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) The set E consists of the following equations: _and_(x, y) == _and_(y, x) _or_(x, y) == _or_(y, x) _xor_(x, y) == _xor_(y, x) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _or_(_or_(x, y), z) == _or_(x, _or_(y, z)) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) The set E# consists of the following equations: _AND_(x, y) == _AND_(y, x) _OR_(x, y) == _OR_(y, x) _XOR_(x, y) == _XOR_(y, x) _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) _OR_(_or_(x, y), z) == _OR_(x, _or_(y, z)) _XOR_(_xor_(x, y), z) == _XOR_(x, _xor_(y, z)) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (13) ESharpUsableEquationsProof (EQUIVALENT) We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]: _AND_(x, y) == _AND_(y, x) _OR_(x, y) == _OR_(y, x) _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) _OR_(_or_(x, y), z) == _OR_(x, _or_(y, z)) ---------------------------------------- (14) Obligation: The TRS P consists of the following rules: _XOR_(_xor_(A, A), ext) -> _XOR_(U111(isBool(A)), ext) _XOR_(_xor_(false, A), ext) -> _XOR_(U121(isBool(A), A), ext) The TRS R consists of the following 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_(A, A) -> U11(isBool(A), A) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(false, A) -> U31(isBool(A)) _and_(true, A) -> U41(isBool(A), A) _implies_(A, B) -> U51(isBool(A), A, B) _isEqualTo_(U, U') -> U61(isS(U'), U', U) _isEqualTo_(U, U) -> U71(isS(U)) _isNotEqualTo_(U, U') -> U81(isS(U'), U', U) _isNotEqualTo_(U, U) -> U91(isS(U)) _or_(A, B) -> U101(isBool(A), A, B) _xor_(A, A) -> U111(isBool(A)) _xor_(false, A) -> U121(isBool(A), A) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U131(isBool(B), B, U', U) if_then_else_fi(true, U, U') -> U141(isS(U'), U) isBool(false) -> tt isBool(true) -> tt 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)) not_(A) -> U221(isBool(A), A) not_(false) -> true not_(true) -> false _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _or_(_or_(A, B), ext) -> _or_(U101(isBool(A), A, B), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) The set E consists of the following equations: _and_(x, y) == _and_(y, x) _or_(x, y) == _or_(y, x) _xor_(x, y) == _xor_(y, x) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _or_(_or_(x, y), z) == _or_(x, _or_(y, z)) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) The set E# consists of the following equations: _XOR_(x, y) == _XOR_(y, x) _XOR_(_xor_(x, y), z) == _XOR_(x, _xor_(y, z)) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (15) EDPPoloProof (EQUIVALENT) We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. The following set of Dependency Pairs of this DP problem can be strictly oriented. _XOR_(_xor_(false, A), ext) -> _XOR_(U121(isBool(A), A), ext) The remaining Dependency Pairs were at least non-strictly oriented. _XOR_(_xor_(A, A), ext) -> _XOR_(U111(isBool(A)), ext) With the implicit AFS we had to orient the following set of usable rules of R non-strictly. _xor_(false, A) -> U121(isBool(A), A) _xor_(A, A) -> U111(isBool(A)) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) U111(tt) -> false U181(tt, V2) -> U182(isUniversal(V2)) U161(tt, V2) -> U162(isBool(V2)) U192(tt) -> tt U191(tt, V2) -> U192(isBool(V2)) U151(tt, V2) -> U152(isBool(V2)) U162(tt) -> tt U121(tt, A) -> A U201(tt, V2) -> U202(isBool(V2)) U171(tt, V2) -> U172(isUniversal(V2)) U152(tt) -> tt U182(tt) -> tt U202(tt) -> tt U211(tt) -> tt isBool(_and_(V1, V2)) -> U151(isBool(V1), V2) isBool(true) -> tt isBool(not_(V1)) -> U211(isBool(V1)) isBool(false) -> tt isBool(_isEqualTo_(V1, V2)) -> U171(isUniversal(V1), V2) isBool(_or_(V1, V2)) -> U191(isBool(V1), V2) isBool(_xor_(V1, V2)) -> U201(isBool(V1), V2) isBool(_isNotEqualTo_(V1, V2)) -> U181(isUniversal(V1), V2) isBool(_implies_(V1, V2)) -> U161(isBool(V1), V2) U172(tt) -> tt We had to orient the following equations of E# equivalently. _XOR_(x, y) == _XOR_(y, x) _XOR_(_xor_(x, y), z) == _XOR_(x, _xor_(y, z)) With the implicit AFS we had to orient the following usable equations of E equivalently. _xor_(x, y) == _xor_(y, x) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) Used ordering: POLO with Polynomial interpretation [POLO]: POL(U111(x_1)) = 3 POL(U121(x_1, x_2)) = 1 + x_1 + x_2 POL(U151(x_1, x_2)) = 1 POL(U152(x_1)) = 0 POL(U161(x_1, x_2)) = 0 POL(U162(x_1)) = 0 POL(U171(x_1, x_2)) = 1 POL(U172(x_1)) = 0 POL(U181(x_1, x_2)) = 0 POL(U182(x_1)) = 0 POL(U191(x_1, x_2)) = 2 POL(U192(x_1)) = 0 POL(U201(x_1, x_2)) = 0 POL(U202(x_1)) = 0 POL(U211(x_1)) = 0 POL(_XOR_(x_1, x_2)) = 3*x_1 + 3*x_2 POL(_and_(x_1, x_2)) = 0 POL(_implies_(x_1, x_2)) = 0 POL(_isEqualTo_(x_1, x_2)) = 0 POL(_isNotEqualTo_(x_1, x_2)) = 0 POL(_or_(x_1, x_2)) = 0 POL(_xor_(x_1, x_2)) = 3 + x_1 + x_2 POL(false) = 3 POL(isBool(x_1)) = 3 POL(isUniversal(x_1)) = 0 POL(not_(x_1)) = 0 POL(true) = 0 POL(tt) = 0 ---------------------------------------- (16) Obligation: The TRS P consists of the following rules: _XOR_(_xor_(A, A), ext) -> _XOR_(U111(isBool(A)), ext) The TRS R consists of the following 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_(A, A) -> U11(isBool(A), A) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(false, A) -> U31(isBool(A)) _and_(true, A) -> U41(isBool(A), A) _implies_(A, B) -> U51(isBool(A), A, B) _isEqualTo_(U, U') -> U61(isS(U'), U', U) _isEqualTo_(U, U) -> U71(isS(U)) _isNotEqualTo_(U, U') -> U81(isS(U'), U', U) _isNotEqualTo_(U, U) -> U91(isS(U)) _or_(A, B) -> U101(isBool(A), A, B) _xor_(A, A) -> U111(isBool(A)) _xor_(false, A) -> U121(isBool(A), A) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U131(isBool(B), B, U', U) if_then_else_fi(true, U, U') -> U141(isS(U'), U) isBool(false) -> tt isBool(true) -> tt 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)) not_(A) -> U221(isBool(A), A) not_(false) -> true not_(true) -> false _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _or_(_or_(A, B), ext) -> _or_(U101(isBool(A), A, B), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) The set E consists of the following equations: _and_(x, y) == _and_(y, x) _or_(x, y) == _or_(y, x) _xor_(x, y) == _xor_(y, x) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _or_(_or_(x, y), z) == _or_(x, _or_(y, z)) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) The set E# consists of the following equations: _XOR_(x, y) == _XOR_(y, x) _XOR_(_xor_(x, y), z) == _XOR_(x, _xor_(y, z)) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (17) EDPPoloProof (EQUIVALENT) We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. All Dependency Pairs of this DP problem can be strictly oriented. _XOR_(_xor_(A, A), ext) -> _XOR_(U111(isBool(A)), ext) With the implicit AFS we had to orient the following set of usable rules of R non-strictly. U121(tt, A) -> A U111(tt) -> false _xor_(false, A) -> U121(isBool(A), A) _xor_(A, A) -> U111(isBool(A)) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) We had to orient the following equations of E# equivalently. _XOR_(x, y) == _XOR_(y, x) _XOR_(_xor_(x, y), z) == _XOR_(x, _xor_(y, z)) With the implicit AFS we had to orient the following usable equations of E equivalently. _xor_(x, y) == _xor_(y, x) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) Used ordering: POLO with Polynomial interpretation [POLO]: POL(U111(x_1)) = 0 POL(U121(x_1, x_2)) = x_2 POL(U151(x_1, x_2)) = 3 + 3*x_2 POL(U152(x_1)) = 3 POL(U161(x_1, x_2)) = 3 + 3*x_2 POL(U162(x_1)) = 3 POL(U171(x_1, x_2)) = 3 + 3*x_2 POL(U172(x_1)) = 3 POL(U181(x_1, x_2)) = 3 + 3*x_2 POL(U182(x_1)) = 3 POL(U191(x_1, x_2)) = 3 + 3*x_2 POL(U192(x_1)) = 3 POL(U201(x_1, x_2)) = 3 + 3*x_2 POL(U202(x_1)) = 3 POL(U211(x_1)) = 3 POL(_XOR_(x_1, x_2)) = 2*x_1 + 2*x_2 POL(_and_(x_1, x_2)) = 0 POL(_implies_(x_1, x_2)) = 0 POL(_isEqualTo_(x_1, x_2)) = 0 POL(_isNotEqualTo_(x_1, x_2)) = 0 POL(_or_(x_1, x_2)) = 0 POL(_xor_(x_1, x_2)) = 2 + x_1 + x_2 POL(false) = 0 POL(isBool(x_1)) = 0 POL(isUniversal(x_1)) = 0 POL(not_(x_1)) = 0 POL(true) = 0 POL(tt) = 0 ---------------------------------------- (18) Obligation: P is empty. The TRS R consists of the following 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_(A, A) -> U11(isBool(A), A) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(false, A) -> U31(isBool(A)) _and_(true, A) -> U41(isBool(A), A) _implies_(A, B) -> U51(isBool(A), A, B) _isEqualTo_(U, U') -> U61(isS(U'), U', U) _isEqualTo_(U, U) -> U71(isS(U)) _isNotEqualTo_(U, U') -> U81(isS(U'), U', U) _isNotEqualTo_(U, U) -> U91(isS(U)) _or_(A, B) -> U101(isBool(A), A, B) _xor_(A, A) -> U111(isBool(A)) _xor_(false, A) -> U121(isBool(A), A) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U131(isBool(B), B, U', U) if_then_else_fi(true, U, U') -> U141(isS(U'), U) isBool(false) -> tt isBool(true) -> tt 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)) not_(A) -> U221(isBool(A), A) not_(false) -> true not_(true) -> false _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _or_(_or_(A, B), ext) -> _or_(U101(isBool(A), A, B), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) The set E consists of the following equations: _and_(x, y) == _and_(y, x) _or_(x, y) == _or_(y, x) _xor_(x, y) == _xor_(y, x) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _or_(_or_(x, y), z) == _or_(x, _or_(y, z)) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) The set E# consists of the following equations: _XOR_(x, y) == _XOR_(y, x) _XOR_(_xor_(x, y), z) == _XOR_(x, _xor_(y, z)) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (19) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,E#,R,E) chain. ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: The TRS P consists of the following rules: _AND_(_and_(A, _xor_(B, C)), ext) -> _AND_(U21(isBool(A), A, B, C), ext) _AND_(_and_(A, A), ext) -> _AND_(U11(isBool(A), A), ext) _AND_(_and_(A, _xor_(B, C)), ext) -> U21^1(isBool(A), A, B, C) _AND_(_and_(false, A), ext) -> _AND_(U31(isBool(A)), ext) U23^1(tt, A, B, C) -> _AND_(A, C) _AND_(_and_(true, A), ext) -> _AND_(U41(isBool(A), A), ext) U21^1(tt, A, B, C) -> U22^1(isBool(B), A, B, C) U22^1(tt, A, B, C) -> U23^1(isBool(C), A, B, C) U23^1(tt, A, B, C) -> _AND_(A, B) _AND_(A, _xor_(B, C)) -> U21^1(isBool(A), A, B, C) The TRS R consists of the following 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_(A, A) -> U11(isBool(A), A) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(false, A) -> U31(isBool(A)) _and_(true, A) -> U41(isBool(A), A) _implies_(A, B) -> U51(isBool(A), A, B) _isEqualTo_(U, U') -> U61(isS(U'), U', U) _isEqualTo_(U, U) -> U71(isS(U)) _isNotEqualTo_(U, U') -> U81(isS(U'), U', U) _isNotEqualTo_(U, U) -> U91(isS(U)) _or_(A, B) -> U101(isBool(A), A, B) _xor_(A, A) -> U111(isBool(A)) _xor_(false, A) -> U121(isBool(A), A) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U131(isBool(B), B, U', U) if_then_else_fi(true, U, U') -> U141(isS(U'), U) isBool(false) -> tt isBool(true) -> tt 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)) not_(A) -> U221(isBool(A), A) not_(false) -> true not_(true) -> false _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _or_(_or_(A, B), ext) -> _or_(U101(isBool(A), A, B), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) The set E consists of the following equations: _and_(x, y) == _and_(y, x) _or_(x, y) == _or_(y, x) _xor_(x, y) == _xor_(y, x) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _or_(_or_(x, y), z) == _or_(x, _or_(y, z)) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) The set E# consists of the following equations: _AND_(x, y) == _AND_(y, x) _OR_(x, y) == _OR_(y, x) _XOR_(x, y) == _XOR_(y, x) _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) _OR_(_or_(x, y), z) == _OR_(x, _or_(y, z)) _XOR_(_xor_(x, y), z) == _XOR_(x, _xor_(y, z)) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (22) ESharpUsableEquationsProof (EQUIVALENT) We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]: _OR_(x, y) == _OR_(y, x) _XOR_(x, y) == _XOR_(y, x) _OR_(_or_(x, y), z) == _OR_(x, _or_(y, z)) _XOR_(_xor_(x, y), z) == _XOR_(x, _xor_(y, z)) ---------------------------------------- (23) Obligation: The TRS P consists of the following rules: _AND_(_and_(A, _xor_(B, C)), ext) -> _AND_(U21(isBool(A), A, B, C), ext) _AND_(_and_(A, A), ext) -> _AND_(U11(isBool(A), A), ext) _AND_(_and_(A, _xor_(B, C)), ext) -> U21^1(isBool(A), A, B, C) _AND_(_and_(false, A), ext) -> _AND_(U31(isBool(A)), ext) U23^1(tt, A, B, C) -> _AND_(A, C) _AND_(_and_(true, A), ext) -> _AND_(U41(isBool(A), A), ext) U21^1(tt, A, B, C) -> U22^1(isBool(B), A, B, C) U22^1(tt, A, B, C) -> U23^1(isBool(C), A, B, C) U23^1(tt, A, B, C) -> _AND_(A, B) _AND_(A, _xor_(B, C)) -> U21^1(isBool(A), A, B, C) The TRS R consists of the following 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_(A, A) -> U11(isBool(A), A) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(false, A) -> U31(isBool(A)) _and_(true, A) -> U41(isBool(A), A) _implies_(A, B) -> U51(isBool(A), A, B) _isEqualTo_(U, U') -> U61(isS(U'), U', U) _isEqualTo_(U, U) -> U71(isS(U)) _isNotEqualTo_(U, U') -> U81(isS(U'), U', U) _isNotEqualTo_(U, U) -> U91(isS(U)) _or_(A, B) -> U101(isBool(A), A, B) _xor_(A, A) -> U111(isBool(A)) _xor_(false, A) -> U121(isBool(A), A) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U131(isBool(B), B, U', U) if_then_else_fi(true, U, U') -> U141(isS(U'), U) isBool(false) -> tt isBool(true) -> tt 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)) not_(A) -> U221(isBool(A), A) not_(false) -> true not_(true) -> false _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _or_(_or_(A, B), ext) -> _or_(U101(isBool(A), A, B), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) The set E consists of the following equations: _and_(x, y) == _and_(y, x) _or_(x, y) == _or_(y, x) _xor_(x, y) == _xor_(y, x) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _or_(_or_(x, y), z) == _or_(x, _or_(y, z)) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) The set E# consists of the following equations: _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) _AND_(x, y) == _AND_(y, x) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (24) EDPPoloProof (EQUIVALENT) We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. The following set of Dependency Pairs of this DP problem can be strictly oriented. _AND_(_and_(A, _xor_(B, C)), ext) -> U21^1(isBool(A), A, B, C) _AND_(_and_(true, A), ext) -> _AND_(U41(isBool(A), A), ext) _AND_(A, _xor_(B, C)) -> U21^1(isBool(A), A, B, C) The remaining Dependency Pairs were at least non-strictly oriented. _AND_(_and_(A, _xor_(B, C)), ext) -> _AND_(U21(isBool(A), A, B, C), ext) _AND_(_and_(A, A), ext) -> _AND_(U11(isBool(A), A), ext) _AND_(_and_(false, A), ext) -> _AND_(U31(isBool(A)), ext) U23^1(tt, A, B, C) -> _AND_(A, C) U21^1(tt, A, B, C) -> U22^1(isBool(B), A, B, C) U22^1(tt, A, B, C) -> U23^1(isBool(C), A, B, C) U23^1(tt, A, B, C) -> _AND_(A, B) With the implicit AFS we had to orient the following set of usable rules of R non-strictly. U23(tt, A, B, C) -> _xor_(_and_(A, B), _and_(A, C)) U151(tt, V2) -> U152(isBool(V2)) U41(tt, A) -> A U171(tt, V2) -> U172(isUniversal(V2)) U202(tt) -> tt _xor_(false, A) -> U121(isBool(A), A) _xor_(A, A) -> U111(isBool(A)) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) U22(tt, A, B, C) -> U23(isBool(C), A, B, C) U21(tt, A, B, C) -> U22(isBool(B), A, B, C) U211(tt) -> tt U181(tt, V2) -> U182(isUniversal(V2)) U31(tt) -> false U172(tt) -> tt U161(tt, V2) -> U162(isBool(V2)) U121(tt, A) -> A U11(tt, A) -> A U192(tt) -> tt U182(tt) -> tt U111(tt) -> false isBool(_and_(V1, V2)) -> U151(isBool(V1), V2) isBool(true) -> tt isBool(not_(V1)) -> U211(isBool(V1)) isBool(false) -> tt isBool(_isEqualTo_(V1, V2)) -> U171(isUniversal(V1), V2) isBool(_or_(V1, V2)) -> U191(isBool(V1), V2) isBool(_xor_(V1, V2)) -> U201(isBool(V1), V2) isBool(_isNotEqualTo_(V1, V2)) -> U181(isUniversal(V1), V2) isBool(_implies_(V1, V2)) -> U161(isBool(V1), V2) U162(tt) -> tt U152(tt) -> tt U191(tt, V2) -> U192(isBool(V2)) _and_(false, A) -> U31(isBool(A)) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(A, A) -> U11(isBool(A), A) _and_(true, A) -> U41(isBool(A), A) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) U201(tt, V2) -> U202(isBool(V2)) We had to orient the following equations of E# equivalently. _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) _AND_(x, y) == _AND_(y, x) With the implicit AFS we had to orient the following usable equations of E equivalently. _xor_(x, y) == _xor_(y, x) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _and_(x, y) == _and_(y, x) Used ordering: POLO with Polynomial interpretation [POLO]: POL(U11(x_1, x_2)) = x_2 POL(U111(x_1)) = 0 POL(U121(x_1, x_2)) = x_1 + x_1*x_2 POL(U151(x_1, x_2)) = 1 POL(U152(x_1)) = x_1 POL(U161(x_1, x_2)) = 1 POL(U162(x_1)) = x_1^2 POL(U171(x_1, x_2)) = 1 + x_1 + x_1*x_2 POL(U172(x_1)) = 1 + x_1 + x_1^2 POL(U181(x_1, x_2)) = 1 + x_1 + x_1*x_2 POL(U182(x_1)) = 1 + x_1 + x_1^2 POL(U191(x_1, x_2)) = 1 POL(U192(x_1)) = x_1 POL(U201(x_1, x_2)) = x_1 POL(U202(x_1)) = x_1^2 POL(U21(x_1, x_2, x_3, x_4)) = 1 + x_1*x_2 + x_1*x_2*x_3 + x_2 + x_2*x_4 + x_3 + x_4 POL(U211(x_1)) = x_1^2 POL(U21^1(x_1, x_2, x_3, x_4)) = x_1*x_2 + x_1*x_2*x_3 + x_1*x_3 + x_1*x_4 + x_2*x_4 POL(U22(x_1, x_2, x_3, x_4)) = x_1 + x_1*x_2 + x_1*x_2*x_3 + x_1*x_3 + x_2 + x_2*x_4 + x_4 POL(U22^1(x_1, x_2, x_3, x_4)) = x_1*x_3 + x_2 + x_2*x_3 + x_2*x_4 + x_4 POL(U23(x_1, x_2, x_3, x_4)) = 1 + x_1*x_2 + x_2 + x_2*x_3 + x_2*x_4 + x_3 + x_4 POL(U23^1(x_1, x_2, x_3, x_4)) = x_1*x_2*x_3 + x_2 + x_2*x_4 + x_3 + x_4 POL(U31(x_1)) = 0 POL(U41(x_1, x_2)) = x_2 POL(_AND_(x_1, x_2)) = x_1 + x_1*x_2 + x_2 POL(_and_(x_1, x_2)) = x_1 + x_1*x_2 + x_2 POL(_implies_(x_1, x_2)) = 1 + x_1 + x_1*x_2 + x_2 POL(_isEqualTo_(x_1, x_2)) = 1 + x_1 + x_1*x_2 + x_2 POL(_isNotEqualTo_(x_1, x_2)) = 1 + x_1 + x_1*x_2 + x_2 POL(_or_(x_1, x_2)) = 1 + x_1 + x_1*x_2 + x_2 POL(_xor_(x_1, x_2)) = 1 + x_1 + x_2 POL(false) = 0 POL(isBool(x_1)) = 1 POL(isUniversal(x_1)) = 0 POL(not_(x_1)) = 0 POL(true) = 1 POL(tt) = 1 ---------------------------------------- (25) Obligation: The TRS P consists of the following rules: _AND_(_and_(A, _xor_(B, C)), ext) -> _AND_(U21(isBool(A), A, B, C), ext) _AND_(_and_(A, A), ext) -> _AND_(U11(isBool(A), A), ext) _AND_(_and_(false, A), ext) -> _AND_(U31(isBool(A)), ext) U23^1(tt, A, B, C) -> _AND_(A, C) U21^1(tt, A, B, C) -> U22^1(isBool(B), A, B, C) U22^1(tt, A, B, C) -> U23^1(isBool(C), A, B, C) U23^1(tt, A, B, C) -> _AND_(A, B) The TRS R consists of the following 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_(A, A) -> U11(isBool(A), A) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(false, A) -> U31(isBool(A)) _and_(true, A) -> U41(isBool(A), A) _implies_(A, B) -> U51(isBool(A), A, B) _isEqualTo_(U, U') -> U61(isS(U'), U', U) _isEqualTo_(U, U) -> U71(isS(U)) _isNotEqualTo_(U, U') -> U81(isS(U'), U', U) _isNotEqualTo_(U, U) -> U91(isS(U)) _or_(A, B) -> U101(isBool(A), A, B) _xor_(A, A) -> U111(isBool(A)) _xor_(false, A) -> U121(isBool(A), A) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U131(isBool(B), B, U', U) if_then_else_fi(true, U, U') -> U141(isS(U'), U) isBool(false) -> tt isBool(true) -> tt 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)) not_(A) -> U221(isBool(A), A) not_(false) -> true not_(true) -> false _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _or_(_or_(A, B), ext) -> _or_(U101(isBool(A), A, B), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) The set E consists of the following equations: _and_(x, y) == _and_(y, x) _or_(x, y) == _or_(y, x) _xor_(x, y) == _xor_(y, x) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _or_(_or_(x, y), z) == _or_(x, _or_(y, z)) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) The set E# consists of the following equations: _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) _AND_(x, y) == _AND_(y, x) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (26) EDependencyGraphProof (EQUIVALENT) The approximation of the Equational Dependency Graph [DA_STEIN] contains 1 SCC with 4 less nodes. ---------------------------------------- (27) Obligation: The TRS P consists of the following rules: _AND_(_and_(A, _xor_(B, C)), ext) -> _AND_(U21(isBool(A), A, B, C), ext) _AND_(_and_(A, A), ext) -> _AND_(U11(isBool(A), A), ext) _AND_(_and_(false, A), ext) -> _AND_(U31(isBool(A)), ext) The TRS R consists of the following 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_(A, A) -> U11(isBool(A), A) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(false, A) -> U31(isBool(A)) _and_(true, A) -> U41(isBool(A), A) _implies_(A, B) -> U51(isBool(A), A, B) _isEqualTo_(U, U') -> U61(isS(U'), U', U) _isEqualTo_(U, U) -> U71(isS(U)) _isNotEqualTo_(U, U') -> U81(isS(U'), U', U) _isNotEqualTo_(U, U) -> U91(isS(U)) _or_(A, B) -> U101(isBool(A), A, B) _xor_(A, A) -> U111(isBool(A)) _xor_(false, A) -> U121(isBool(A), A) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U131(isBool(B), B, U', U) if_then_else_fi(true, U, U') -> U141(isS(U'), U) isBool(false) -> tt isBool(true) -> tt 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)) not_(A) -> U221(isBool(A), A) not_(false) -> true not_(true) -> false _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _or_(_or_(A, B), ext) -> _or_(U101(isBool(A), A, B), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) The set E consists of the following equations: _and_(x, y) == _and_(y, x) _or_(x, y) == _or_(y, x) _xor_(x, y) == _xor_(y, x) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _or_(_or_(x, y), z) == _or_(x, _or_(y, z)) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) The set E# consists of the following equations: _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) _AND_(x, y) == _AND_(y, x) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (28) EDPPoloProof (EQUIVALENT) We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. The following set of Dependency Pairs of this DP problem can be strictly oriented. _AND_(_and_(false, A), ext) -> _AND_(U31(isBool(A)), ext) The remaining Dependency Pairs were at least non-strictly oriented. _AND_(_and_(A, _xor_(B, C)), ext) -> _AND_(U21(isBool(A), A, B, C), ext) _AND_(_and_(A, A), ext) -> _AND_(U11(isBool(A), A), ext) With the implicit AFS we had to orient the following set of usable rules of R non-strictly. U41(tt, A) -> A U152(tt) -> tt isBool(_and_(V1, V2)) -> U151(isBool(V1), V2) isBool(true) -> tt isBool(not_(V1)) -> U211(isBool(V1)) isBool(false) -> tt isBool(_isEqualTo_(V1, V2)) -> U171(isUniversal(V1), V2) isBool(_or_(V1, V2)) -> U191(isBool(V1), V2) isBool(_xor_(V1, V2)) -> U201(isBool(V1), V2) isBool(_isNotEqualTo_(V1, V2)) -> U181(isUniversal(V1), V2) isBool(_implies_(V1, V2)) -> U161(isBool(V1), V2) U11(tt, A) -> A U202(tt) -> tt U191(tt, V2) -> U192(isBool(V2)) U201(tt, V2) -> U202(isBool(V2)) _xor_(false, A) -> U121(isBool(A), A) _xor_(A, A) -> U111(isBool(A)) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) U172(tt) -> tt U111(tt) -> false U21(tt, A, B, C) -> U22(isBool(B), A, B, C) U211(tt) -> tt U162(tt) -> tt U23(tt, A, B, C) -> _xor_(_and_(A, B), _and_(A, C)) U121(tt, A) -> A U151(tt, V2) -> U152(isBool(V2)) U192(tt) -> tt _and_(false, A) -> U31(isBool(A)) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(A, A) -> U11(isBool(A), A) _and_(true, A) -> U41(isBool(A), A) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) U181(tt, V2) -> U182(isUniversal(V2)) U31(tt) -> false U22(tt, A, B, C) -> U23(isBool(C), A, B, C) U161(tt, V2) -> U162(isBool(V2)) U182(tt) -> tt U171(tt, V2) -> U172(isUniversal(V2)) We had to orient the following equations of E# equivalently. _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) _AND_(x, y) == _AND_(y, x) With the implicit AFS we had to orient the following usable equations of E equivalently. _xor_(x, y) == _xor_(y, x) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _and_(x, y) == _and_(y, x) Used ordering: POLO with Polynomial interpretation [POLO]: POL(U11(x_1, x_2)) = x_1*x_2 POL(U111(x_1)) = 1 + x_1 POL(U121(x_1, x_2)) = 2 + x_2 POL(U151(x_1, x_2)) = 2*x_1*x_2 POL(U152(x_1)) = 2*x_1 POL(U161(x_1, x_2)) = x_1 + 2*x_2 POL(U162(x_1)) = 2 POL(U171(x_1, x_2)) = 0 POL(U172(x_1)) = 2*x_1 POL(U181(x_1, x_2)) = 0 POL(U182(x_1)) = x_1^2 POL(U191(x_1, x_2)) = x_1*x_2 POL(U192(x_1)) = x_1 POL(U201(x_1, x_2)) = x_1 POL(U202(x_1)) = 2 POL(U21(x_1, x_2, x_3, x_4)) = 1 + 3*x_2 + 2*x_2*x_3 + 2*x_2*x_4 + x_3 + x_4 POL(U211(x_1)) = x_1^2 POL(U22(x_1, x_2, x_3, x_4)) = 1 + 3*x_2 + 2*x_2*x_3 + 2*x_2*x_4 + x_3 + x_4 POL(U23(x_1, x_2, x_3, x_4)) = 1 + 3*x_2 + 2*x_2*x_3 + 2*x_2*x_4 + x_3 + x_4 POL(U31(x_1)) = 2*x_1 POL(U41(x_1, x_2)) = 1 + x_2 POL(_AND_(x_1, x_2)) = x_1 + 2*x_1*x_2 + x_2 POL(_and_(x_1, x_2)) = x_1 + 2*x_1*x_2 + x_2 POL(_implies_(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(_isEqualTo_(x_1, x_2)) = 1 + 2*x_1 + x_1*x_2 + 3*x_2 POL(_isNotEqualTo_(x_1, x_2)) = x_1 + x_1*x_2 POL(_or_(x_1, x_2)) = x_1 + x_1*x_2 + 2*x_2 POL(_xor_(x_1, x_2)) = 1 + x_1 + x_2 POL(false) = 3 POL(isBool(x_1)) = 2*x_1 POL(isUniversal(x_1)) = 0 POL(not_(x_1)) = 3*x_1^2 POL(true) = 1 POL(tt) = 2 ---------------------------------------- (29) Obligation: The TRS P consists of the following rules: _AND_(_and_(A, _xor_(B, C)), ext) -> _AND_(U21(isBool(A), A, B, C), ext) _AND_(_and_(A, A), ext) -> _AND_(U11(isBool(A), A), ext) The TRS R consists of the following 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_(A, A) -> U11(isBool(A), A) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(false, A) -> U31(isBool(A)) _and_(true, A) -> U41(isBool(A), A) _implies_(A, B) -> U51(isBool(A), A, B) _isEqualTo_(U, U') -> U61(isS(U'), U', U) _isEqualTo_(U, U) -> U71(isS(U)) _isNotEqualTo_(U, U') -> U81(isS(U'), U', U) _isNotEqualTo_(U, U) -> U91(isS(U)) _or_(A, B) -> U101(isBool(A), A, B) _xor_(A, A) -> U111(isBool(A)) _xor_(false, A) -> U121(isBool(A), A) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U131(isBool(B), B, U', U) if_then_else_fi(true, U, U') -> U141(isS(U'), U) isBool(false) -> tt isBool(true) -> tt 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)) not_(A) -> U221(isBool(A), A) not_(false) -> true not_(true) -> false _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _or_(_or_(A, B), ext) -> _or_(U101(isBool(A), A, B), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) The set E consists of the following equations: _and_(x, y) == _and_(y, x) _or_(x, y) == _or_(y, x) _xor_(x, y) == _xor_(y, x) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _or_(_or_(x, y), z) == _or_(x, _or_(y, z)) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) The set E# consists of the following equations: _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) _AND_(x, y) == _AND_(y, x) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (30) EDPPoloProof (EQUIVALENT) We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. The following set of Dependency Pairs of this DP problem can be strictly oriented. _AND_(_and_(A, _xor_(B, C)), ext) -> _AND_(U21(isBool(A), A, B, C), ext) The remaining Dependency Pairs were at least non-strictly oriented. _AND_(_and_(A, A), ext) -> _AND_(U11(isBool(A), A), ext) With the implicit AFS we had to orient the following set of usable rules of R non-strictly. U161(tt, V2) -> U162(isBool(V2)) U201(tt, V2) -> U202(isBool(V2)) U31(tt) -> false U22(tt, A, B, C) -> U23(isBool(C), A, B, C) U121(tt, A) -> A U182(tt) -> tt U111(tt) -> false U151(tt, V2) -> U152(isBool(V2)) U41(tt, A) -> A U152(tt) -> tt U172(tt) -> tt U192(tt) -> tt U202(tt) -> tt U11(tt, A) -> A _xor_(false, A) -> U121(isBool(A), A) _xor_(A, A) -> U111(isBool(A)) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) isBool(_and_(V1, V2)) -> U151(isBool(V1), V2) isBool(true) -> tt isBool(not_(V1)) -> U211(isBool(V1)) isBool(false) -> tt isBool(_isEqualTo_(V1, V2)) -> U171(isUniversal(V1), V2) isBool(_or_(V1, V2)) -> U191(isBool(V1), V2) isBool(_xor_(V1, V2)) -> U201(isBool(V1), V2) isBool(_isNotEqualTo_(V1, V2)) -> U181(isUniversal(V1), V2) isBool(_implies_(V1, V2)) -> U161(isBool(V1), V2) U171(tt, V2) -> U172(isUniversal(V2)) _and_(false, A) -> U31(isBool(A)) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(A, A) -> U11(isBool(A), A) _and_(true, A) -> U41(isBool(A), A) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) U181(tt, V2) -> U182(isUniversal(V2)) U162(tt) -> tt U211(tt) -> tt U191(tt, V2) -> U192(isBool(V2)) U21(tt, A, B, C) -> U22(isBool(B), A, B, C) U23(tt, A, B, C) -> _xor_(_and_(A, B), _and_(A, C)) We had to orient the following equations of E# equivalently. _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) _AND_(x, y) == _AND_(y, x) With the implicit AFS we had to orient the following usable equations of E equivalently. _xor_(x, y) == _xor_(y, x) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _and_(x, y) == _and_(y, x) Used ordering: POLO with Polynomial interpretation [POLO]: POL(U11(x_1, x_2)) = 2*x_2 POL(U111(x_1)) = 2 POL(U121(x_1, x_2)) = 2 + x_2 POL(U151(x_1, x_2)) = x_1*x_2 + 2*x_2 POL(U152(x_1)) = 2*x_1 POL(U161(x_1, x_2)) = 2*x_1 POL(U162(x_1)) = 2 POL(U171(x_1, x_2)) = 0 POL(U172(x_1)) = x_1 POL(U181(x_1, x_2)) = 0 POL(U182(x_1)) = 2*x_1^2 POL(U191(x_1, x_2)) = 2*x_2 POL(U192(x_1)) = x_1 POL(U201(x_1, x_2)) = 2*x_2 POL(U202(x_1)) = x_1 POL(U21(x_1, x_2, x_3, x_4)) = 1 + x_1 + 2*x_2 + x_2*x_3 + x_2*x_4 + x_3 + x_4 POL(U211(x_1)) = 2*x_1 POL(U22(x_1, x_2, x_3, x_4)) = 3 + 2*x_2 + x_2*x_3 + x_2*x_4 + x_3 + x_4 POL(U23(x_1, x_2, x_3, x_4)) = 3 + 2*x_2 + x_2*x_3 + x_2*x_4 + x_3 + x_4 POL(U31(x_1)) = x_1 POL(U41(x_1, x_2)) = 1 + x_2 POL(_AND_(x_1, x_2)) = 2*x_1 + 2*x_1*x_2 + 2*x_2 POL(_and_(x_1, x_2)) = x_1 + x_1*x_2 + x_2 POL(_implies_(x_1, x_2)) = 1 + 3*x_1 + x_1*x_2 + 3*x_2 POL(_isEqualTo_(x_1, x_2)) = 3*x_1 + 3*x_1*x_2 + 3*x_2 POL(_isNotEqualTo_(x_1, x_2)) = 3*x_1 + 3*x_2 POL(_or_(x_1, x_2)) = 1 + x_1 + x_1*x_2 + 2*x_2 POL(_xor_(x_1, x_2)) = 3 + x_1 + x_2 POL(false) = 2 POL(isBool(x_1)) = 2*x_1 POL(isUniversal(x_1)) = 0 POL(not_(x_1)) = 2*x_1 + 2*x_1^2 POL(true) = 1 POL(tt) = 2 ---------------------------------------- (31) Obligation: The TRS P consists of the following rules: _AND_(_and_(A, A), ext) -> _AND_(U11(isBool(A), A), ext) The TRS R consists of the following 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_(A, A) -> U11(isBool(A), A) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(false, A) -> U31(isBool(A)) _and_(true, A) -> U41(isBool(A), A) _implies_(A, B) -> U51(isBool(A), A, B) _isEqualTo_(U, U') -> U61(isS(U'), U', U) _isEqualTo_(U, U) -> U71(isS(U)) _isNotEqualTo_(U, U') -> U81(isS(U'), U', U) _isNotEqualTo_(U, U) -> U91(isS(U)) _or_(A, B) -> U101(isBool(A), A, B) _xor_(A, A) -> U111(isBool(A)) _xor_(false, A) -> U121(isBool(A), A) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U131(isBool(B), B, U', U) if_then_else_fi(true, U, U') -> U141(isS(U'), U) isBool(false) -> tt isBool(true) -> tt 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)) not_(A) -> U221(isBool(A), A) not_(false) -> true not_(true) -> false _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _or_(_or_(A, B), ext) -> _or_(U101(isBool(A), A, B), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) The set E consists of the following equations: _and_(x, y) == _and_(y, x) _or_(x, y) == _or_(y, x) _xor_(x, y) == _xor_(y, x) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _or_(_or_(x, y), z) == _or_(x, _or_(y, z)) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) The set E# consists of the following equations: _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) _AND_(x, y) == _AND_(y, x) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (32) EDPPoloProof (EQUIVALENT) We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. All Dependency Pairs of this DP problem can be strictly oriented. _AND_(_and_(A, A), ext) -> _AND_(U11(isBool(A), A), ext) With the implicit AFS we had to orient the following set of usable rules of R non-strictly. U152(tt) -> tt U192(tt) -> tt U182(tt) -> tt U22(tt, A, B, C) -> U23(isBool(C), A, B, C) U201(tt, V2) -> U202(isBool(V2)) U21(tt, A, B, C) -> U22(isBool(B), A, B, C) U202(tt) -> tt U191(tt, V2) -> U192(isBool(V2)) U211(tt) -> tt U11(tt, A) -> A _and_(false, A) -> U31(isBool(A)) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(A, A) -> U11(isBool(A), A) _and_(true, A) -> U41(isBool(A), A) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) U23(tt, A, B, C) -> _xor_(_and_(A, B), _and_(A, C)) U162(tt) -> tt U161(tt, V2) -> U162(isBool(V2)) U41(tt, A) -> A U151(tt, V2) -> U152(isBool(V2)) U171(tt, V2) -> U172(isUniversal(V2)) U172(tt) -> tt _xor_(false, A) -> U121(isBool(A), A) _xor_(A, A) -> U111(isBool(A)) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) U31(tt) -> false U111(tt) -> false U181(tt, V2) -> U182(isUniversal(V2)) U121(tt, A) -> A isBool(_and_(V1, V2)) -> U151(isBool(V1), V2) isBool(true) -> tt isBool(not_(V1)) -> U211(isBool(V1)) isBool(false) -> tt isBool(_isEqualTo_(V1, V2)) -> U171(isUniversal(V1), V2) isBool(_or_(V1, V2)) -> U191(isBool(V1), V2) isBool(_xor_(V1, V2)) -> U201(isBool(V1), V2) isBool(_isNotEqualTo_(V1, V2)) -> U181(isUniversal(V1), V2) isBool(_implies_(V1, V2)) -> U161(isBool(V1), V2) We had to orient the following equations of E# equivalently. _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) _AND_(x, y) == _AND_(y, x) With the implicit AFS we had to orient the following usable equations of E equivalently. _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _and_(x, y) == _and_(y, x) _xor_(x, y) == _xor_(y, x) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) Used ordering: POLO with Polynomial interpretation [POLO]: POL(U11(x_1, x_2)) = 2*x_2 POL(U111(x_1)) = 2 POL(U121(x_1, x_2)) = x_2 POL(U151(x_1, x_2)) = 2 POL(U152(x_1)) = x_1 POL(U161(x_1, x_2)) = 2 POL(U162(x_1)) = x_1 POL(U171(x_1, x_2)) = 0 POL(U172(x_1)) = 2*x_1 POL(U181(x_1, x_2)) = 2*x_1 POL(U182(x_1)) = 2 POL(U191(x_1, x_2)) = x_1 POL(U192(x_1)) = x_1 POL(U201(x_1, x_2)) = x_1 POL(U202(x_1)) = x_1 POL(U21(x_1, x_2, x_3, x_4)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_2*x_3 + 2*x_2*x_4 + 2*x_3 + 2*x_4 POL(U211(x_1)) = x_1 POL(U22(x_1, x_2, x_3, x_4)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_2*x_3 + 2*x_2*x_4 + 2*x_3 + 2*x_4 POL(U23(x_1, x_2, x_3, x_4)) = 2*x_1 + 2*x_1*x_2 + 2*x_2*x_3 + 2*x_2*x_4 + 2*x_3 + 2*x_4 POL(U31(x_1)) = 0 POL(U41(x_1, x_2)) = 2*x_2 POL(_AND_(x_1, x_2)) = x_1 + x_1*x_2 + x_2 POL(_and_(x_1, x_2)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_2 POL(_implies_(x_1, x_2)) = 0 POL(_isEqualTo_(x_1, x_2)) = 0 POL(_isNotEqualTo_(x_1, x_2)) = 0 POL(_or_(x_1, x_2)) = 0 POL(_xor_(x_1, x_2)) = 2 + x_1 + x_2 POL(false) = 0 POL(isBool(x_1)) = 2 POL(isUniversal(x_1)) = 0 POL(not_(x_1)) = 0 POL(true) = 3 POL(tt) = 2 ---------------------------------------- (33) Obligation: P is empty. The TRS R consists of the following 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_(A, A) -> U11(isBool(A), A) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(false, A) -> U31(isBool(A)) _and_(true, A) -> U41(isBool(A), A) _implies_(A, B) -> U51(isBool(A), A, B) _isEqualTo_(U, U') -> U61(isS(U'), U', U) _isEqualTo_(U, U) -> U71(isS(U)) _isNotEqualTo_(U, U') -> U81(isS(U'), U', U) _isNotEqualTo_(U, U) -> U91(isS(U)) _or_(A, B) -> U101(isBool(A), A, B) _xor_(A, A) -> U111(isBool(A)) _xor_(false, A) -> U121(isBool(A), A) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U131(isBool(B), B, U', U) if_then_else_fi(true, U, U') -> U141(isS(U'), U) isBool(false) -> tt isBool(true) -> tt 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)) not_(A) -> U221(isBool(A), A) not_(false) -> true not_(true) -> false _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _or_(_or_(A, B), ext) -> _or_(U101(isBool(A), A, B), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) The set E consists of the following equations: _and_(x, y) == _and_(y, x) _or_(x, y) == _or_(y, x) _xor_(x, y) == _xor_(y, x) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _or_(_or_(x, y), z) == _or_(x, _or_(y, z)) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) The set E# consists of the following equations: _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) _AND_(x, y) == _AND_(y, x) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (34) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,E#,R,E) chain. ---------------------------------------- (35) YES ---------------------------------------- (36) Obligation: The TRS P consists of the following rules: _OR_(_or_(A, B), ext) -> _OR_(U101(isBool(A), A, B), ext) The TRS R consists of the following 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_(A, A) -> U11(isBool(A), A) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(false, A) -> U31(isBool(A)) _and_(true, A) -> U41(isBool(A), A) _implies_(A, B) -> U51(isBool(A), A, B) _isEqualTo_(U, U') -> U61(isS(U'), U', U) _isEqualTo_(U, U) -> U71(isS(U)) _isNotEqualTo_(U, U') -> U81(isS(U'), U', U) _isNotEqualTo_(U, U) -> U91(isS(U)) _or_(A, B) -> U101(isBool(A), A, B) _xor_(A, A) -> U111(isBool(A)) _xor_(false, A) -> U121(isBool(A), A) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U131(isBool(B), B, U', U) if_then_else_fi(true, U, U') -> U141(isS(U'), U) isBool(false) -> tt isBool(true) -> tt 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)) not_(A) -> U221(isBool(A), A) not_(false) -> true not_(true) -> false _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _or_(_or_(A, B), ext) -> _or_(U101(isBool(A), A, B), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) The set E consists of the following equations: _and_(x, y) == _and_(y, x) _or_(x, y) == _or_(y, x) _xor_(x, y) == _xor_(y, x) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _or_(_or_(x, y), z) == _or_(x, _or_(y, z)) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) The set E# consists of the following equations: _AND_(x, y) == _AND_(y, x) _OR_(x, y) == _OR_(y, x) _XOR_(x, y) == _XOR_(y, x) _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) _OR_(_or_(x, y), z) == _OR_(x, _or_(y, z)) _XOR_(_xor_(x, y), z) == _XOR_(x, _xor_(y, z)) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (37) ESharpUsableEquationsProof (EQUIVALENT) We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]: _AND_(x, y) == _AND_(y, x) _XOR_(x, y) == _XOR_(y, x) _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) _XOR_(_xor_(x, y), z) == _XOR_(x, _xor_(y, z)) ---------------------------------------- (38) Obligation: The TRS P consists of the following rules: _OR_(_or_(A, B), ext) -> _OR_(U101(isBool(A), A, B), ext) The TRS R consists of the following 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_(A, A) -> U11(isBool(A), A) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(false, A) -> U31(isBool(A)) _and_(true, A) -> U41(isBool(A), A) _implies_(A, B) -> U51(isBool(A), A, B) _isEqualTo_(U, U') -> U61(isS(U'), U', U) _isEqualTo_(U, U) -> U71(isS(U)) _isNotEqualTo_(U, U') -> U81(isS(U'), U', U) _isNotEqualTo_(U, U) -> U91(isS(U)) _or_(A, B) -> U101(isBool(A), A, B) _xor_(A, A) -> U111(isBool(A)) _xor_(false, A) -> U121(isBool(A), A) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U131(isBool(B), B, U', U) if_then_else_fi(true, U, U') -> U141(isS(U'), U) isBool(false) -> tt isBool(true) -> tt 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)) not_(A) -> U221(isBool(A), A) not_(false) -> true not_(true) -> false _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _or_(_or_(A, B), ext) -> _or_(U101(isBool(A), A, B), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) The set E consists of the following equations: _and_(x, y) == _and_(y, x) _or_(x, y) == _or_(y, x) _xor_(x, y) == _xor_(y, x) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _or_(_or_(x, y), z) == _or_(x, _or_(y, z)) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) The set E# consists of the following equations: _OR_(x, y) == _OR_(y, x) _OR_(_or_(x, y), z) == _OR_(x, _or_(y, z)) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (39) EDPPoloProof (EQUIVALENT) We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. All Dependency Pairs of this DP problem can be strictly oriented. _OR_(_or_(A, B), ext) -> _OR_(U101(isBool(A), A, B), ext) With the implicit AFS we had to orient the following set of usable rules of R non-strictly. _or_(_or_(A, B), ext) -> _or_(U101(isBool(A), A, B), ext) _or_(A, B) -> U101(isBool(A), A, B) U102(tt, A, B) -> _xor_(_and_(A, B), _xor_(A, B)) U22(tt, A, B, C) -> U23(isBool(C), A, B, C) U171(tt, V2) -> U172(isUniversal(V2)) isBool(_and_(V1, V2)) -> U151(isBool(V1), V2) isBool(true) -> tt isBool(not_(V1)) -> U211(isBool(V1)) isBool(false) -> tt isBool(_isEqualTo_(V1, V2)) -> U171(isUniversal(V1), V2) isBool(_or_(V1, V2)) -> U191(isBool(V1), V2) isBool(_xor_(V1, V2)) -> U201(isBool(V1), V2) isBool(_isNotEqualTo_(V1, V2)) -> U181(isUniversal(V1), V2) isBool(_implies_(V1, V2)) -> U161(isBool(V1), V2) U181(tt, V2) -> U182(isUniversal(V2)) U101(tt, A, B) -> U102(isBool(B), A, B) U162(tt) -> tt U11(tt, A) -> A U182(tt) -> tt U121(tt, A) -> A U152(tt) -> tt U191(tt, V2) -> U192(isBool(V2)) _and_(false, A) -> U31(isBool(A)) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(A, A) -> U11(isBool(A), A) _and_(true, A) -> U41(isBool(A), A) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) _xor_(false, A) -> U121(isBool(A), A) _xor_(A, A) -> U111(isBool(A)) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) U31(tt) -> false U211(tt) -> tt U41(tt, A) -> A U23(tt, A, B, C) -> _xor_(_and_(A, B), _and_(A, C)) U151(tt, V2) -> U152(isBool(V2)) U201(tt, V2) -> U202(isBool(V2)) U111(tt) -> false U161(tt, V2) -> U162(isBool(V2)) U192(tt) -> tt U202(tt) -> tt U172(tt) -> tt U21(tt, A, B, C) -> U22(isBool(B), A, B, C) We had to orient the following equations of E# equivalently. _OR_(x, y) == _OR_(y, x) _OR_(_or_(x, y), z) == _OR_(x, _or_(y, z)) With the implicit AFS we had to orient the following usable equations of E equivalently. _or_(x, y) == _or_(y, x) _or_(_or_(x, y), z) == _or_(x, _or_(y, z)) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _and_(x, y) == _and_(y, x) _xor_(x, y) == _xor_(y, x) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) Used ordering: POLO with Polynomial interpretation [POLO]: POL(U101(x_1, x_2, x_3)) = 2*x_2 + 2*x_2*x_3 + x_3 POL(U102(x_1, x_2, x_3)) = x_2 + 2*x_2*x_3 + x_3 POL(U11(x_1, x_2)) = x_1*x_2 POL(U111(x_1)) = x_1 POL(U121(x_1, x_2)) = x_2 POL(U151(x_1, x_2)) = x_1*x_2 POL(U152(x_1)) = x_1 POL(U161(x_1, x_2)) = 3*x_1*x_2 + 2*x_2 POL(U162(x_1)) = 2*x_1 POL(U171(x_1, x_2)) = x_1*x_2 + 2*x_2 POL(U172(x_1)) = 2*x_1^2 POL(U181(x_1, x_2)) = 2*x_1 + 2*x_1*x_2 + 2*x_2 POL(U182(x_1)) = x_1^2 POL(U191(x_1, x_2)) = x_1 + 2*x_2 POL(U192(x_1)) = 2 + x_1 POL(U201(x_1, x_2)) = x_1 + x_2 POL(U202(x_1)) = 2 POL(U21(x_1, x_2, x_3, x_4)) = 2*x_2*x_3 + 2*x_2*x_4 POL(U211(x_1)) = 2 POL(U22(x_1, x_2, x_3, x_4)) = 2*x_2*x_3 + 2*x_2*x_4 POL(U23(x_1, x_2, x_3, x_4)) = 2*x_2*x_3 + 2*x_2*x_4 POL(U31(x_1)) = x_1 POL(U41(x_1, x_2)) = 2*x_2 POL(_OR_(x_1, x_2)) = x_1 + x_1*x_2 + x_2 POL(_and_(x_1, x_2)) = 2*x_1*x_2 POL(_implies_(x_1, x_2)) = 3*x_1 + 3*x_1*x_2 + 3*x_2 POL(_isEqualTo_(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(_isNotEqualTo_(x_1, x_2)) = 3 + 2*x_1 + x_1*x_2 + 2*x_2 POL(_or_(x_1, x_2)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_2 POL(_xor_(x_1, x_2)) = x_1 + x_2 POL(false) = 1 POL(isBool(x_1)) = 2*x_1 POL(isUniversal(x_1)) = 0 POL(not_(x_1)) = 1 + x_1^2 POL(true) = 1 POL(tt) = 2 ---------------------------------------- (40) Obligation: P is empty. The TRS R consists of the following 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_(A, A) -> U11(isBool(A), A) _and_(A, _xor_(B, C)) -> U21(isBool(A), A, B, C) _and_(false, A) -> U31(isBool(A)) _and_(true, A) -> U41(isBool(A), A) _implies_(A, B) -> U51(isBool(A), A, B) _isEqualTo_(U, U') -> U61(isS(U'), U', U) _isEqualTo_(U, U) -> U71(isS(U)) _isNotEqualTo_(U, U') -> U81(isS(U'), U', U) _isNotEqualTo_(U, U) -> U91(isS(U)) _or_(A, B) -> U101(isBool(A), A, B) _xor_(A, A) -> U111(isBool(A)) _xor_(false, A) -> U121(isBool(A), A) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U131(isBool(B), B, U', U) if_then_else_fi(true, U, U') -> U141(isS(U'), U) isBool(false) -> tt isBool(true) -> tt 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)) not_(A) -> U221(isBool(A), A) not_(false) -> true not_(true) -> false _and_(_and_(A, A), ext) -> _and_(U11(isBool(A), A), ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U21(isBool(A), A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(U31(isBool(A)), ext) _and_(_and_(true, A), ext) -> _and_(U41(isBool(A), A), ext) _or_(_or_(A, B), ext) -> _or_(U101(isBool(A), A, B), ext) _xor_(_xor_(A, A), ext) -> _xor_(U111(isBool(A)), ext) _xor_(_xor_(false, A), ext) -> _xor_(U121(isBool(A), A), ext) The set E consists of the following equations: _and_(x, y) == _and_(y, x) _or_(x, y) == _or_(y, x) _xor_(x, y) == _xor_(y, x) _and_(_and_(x, y), z) == _and_(x, _and_(y, z)) _or_(_or_(x, y), z) == _or_(x, _or_(y, z)) _xor_(_xor_(x, y), z) == _xor_(x, _xor_(y, z)) The set E# consists of the following equations: _OR_(x, y) == _OR_(y, x) _OR_(_or_(x, y), z) == _OR_(x, _or_(y, z)) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (41) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,E#,R,E) chain. ---------------------------------------- (42) YES