NO proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ETRS could be disproven: (0) ETRS (1) RRRPoloETRSProof [EQUIVALENT, 292 ms] (2) ETRS (3) RRRPoloETRSProof [EQUIVALENT, 116 ms] (4) ETRS (5) RRRPoloETRSProof [EQUIVALENT, 124 ms] (6) ETRS (7) RRRPoloETRSProof [EQUIVALENT, 59 ms] (8) ETRS (9) RRRPoloETRSProof [EQUIVALENT, 55 ms] (10) ETRS (11) EquationalDependencyPairsProof [EQUIVALENT, 42 ms] (12) EDP (13) EDependencyGraphProof [EQUIVALENT, 0 ms] (14) AND (15) EDP (16) ESharpUsableEquationsProof [EQUIVALENT, 0 ms] (17) EDP (18) EUsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (19) EDP (20) EDPProblemToQDPProblemProof [EQUIVALENT, 13 ms] (21) QDP (22) TransformationProof [EQUIVALENT, 0 ms] (23) QDP (24) TransformationProof [EQUIVALENT, 0 ms] (25) QDP (26) TransformationProof [EQUIVALENT, 0 ms] (27) QDP (28) TransformationProof [EQUIVALENT, 0 ms] (29) QDP (30) NonTerminationLoopProof [COMPLETE, 0 ms] (31) NO (32) EDP ---------------------------------------- (0) Obligation: Equational rewrite system: The TRS R consists of the following rules: U11(tt, A, B, C) -> U12(tt, A, B, C) U12(tt, A, B, C) -> U13(tt, A, B, C) U13(tt, A, B, C) -> _xor_(_and_(A, B), _and_(A, C)) U21(tt, A, B) -> U22(tt, A, B) U22(tt, A, B) -> not_(_xor_(A, _and_(A, B))) U31(tt, U', U) -> U32(tt, U', U) U32(tt, U', U) -> U33(equal(_isNotEqualTo_(U, U'), true)) U33(tt) -> false U41(tt, U', U) -> U42(tt, U', U) U42(tt, U', U) -> if_then_else_fi(_isEqualTo_(U, U'), false, true) U51(tt, A, B) -> U52(tt, A, B) U52(tt, A, B) -> _xor_(_and_(A, B), _xor_(A, B)) U61(tt, B, U') -> U62(tt, B, U') U62(tt, B, U') -> U63(tt, B, U') U63(tt, B, U') -> U64(equal(_isNotEqualTo_(B, true), true), U') U64(tt, U') -> U' U71(tt, U) -> U72(tt, U) U72(tt, U) -> U _and_(A, A) -> A _and_(A, _xor_(B, C)) -> U11(tt, A, B, C) _and_(false, A) -> false _and_(true, A) -> A _implies_(A, B) -> U21(tt, A, B) _isEqualTo_(U, U') -> U31(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> U41(tt, U', U) _isNotEqualTo_(U, U) -> false _or_(A, B) -> U51(tt, A, B) _xor_(A, A) -> false _xor_(false, A) -> A equal(X, X) -> tt if_then_else_fi(B, U, U') -> U61(tt, B, U') if_then_else_fi(true, U, U') -> U71(tt, U) not_(A) -> _xor_(A, true) 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) RRRPoloETRSProof (EQUIVALENT) The following E TRS is given: Equational rewrite system: The TRS R consists of the following rules: U11(tt, A, B, C) -> U12(tt, A, B, C) U12(tt, A, B, C) -> U13(tt, A, B, C) U13(tt, A, B, C) -> _xor_(_and_(A, B), _and_(A, C)) U21(tt, A, B) -> U22(tt, A, B) U22(tt, A, B) -> not_(_xor_(A, _and_(A, B))) U31(tt, U', U) -> U32(tt, U', U) U32(tt, U', U) -> U33(equal(_isNotEqualTo_(U, U'), true)) U33(tt) -> false U41(tt, U', U) -> U42(tt, U', U) U42(tt, U', U) -> if_then_else_fi(_isEqualTo_(U, U'), false, true) U51(tt, A, B) -> U52(tt, A, B) U52(tt, A, B) -> _xor_(_and_(A, B), _xor_(A, B)) U61(tt, B, U') -> U62(tt, B, U') U62(tt, B, U') -> U63(tt, B, U') U63(tt, B, U') -> U64(equal(_isNotEqualTo_(B, true), true), U') U64(tt, U') -> U' U71(tt, U) -> U72(tt, U) U72(tt, U) -> U _and_(A, A) -> A _and_(A, _xor_(B, C)) -> U11(tt, A, B, C) _and_(false, A) -> false _and_(true, A) -> A _implies_(A, B) -> U21(tt, A, B) _isEqualTo_(U, U') -> U31(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> U41(tt, U', U) _isNotEqualTo_(U, U) -> false _or_(A, B) -> U51(tt, A, B) _xor_(A, A) -> false _xor_(false, A) -> A equal(X, X) -> tt if_then_else_fi(B, U, U') -> U61(tt, B, U') if_then_else_fi(true, U, U') -> U71(tt, U) not_(A) -> _xor_(A, true) 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)) The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering: U21(tt, A, B) -> U22(tt, A, B) _xor_(A, A) -> false _xor_(false, A) -> A not_(false) -> true not_(true) -> false Used ordering: Polynomial interpretation [POLO]: POL(U11(x_1, x_2, x_3, x_4)) = 1 + 3*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 3*x_1*x_2*x_3*x_4 + 2*x_1*x_2*x_4 + 2*x_1*x_3 + 3*x_1*x_3*x_4 + 2*x_1*x_4 + 2*x_2 + 3*x_2*x_3 + 3*x_2*x_4 + x_3 + x_4 POL(U12(x_1, x_2, x_3, x_4)) = 1 + 3*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 3*x_1*x_2*x_3*x_4 + 2*x_1*x_2*x_4 + 2*x_1*x_3 + 3*x_1*x_3*x_4 + 2*x_1*x_4 + 2*x_2 + 3*x_2*x_3 + 3*x_2*x_4 + x_3 + x_4 POL(U13(x_1, x_2, x_3, x_4)) = 1 + 3*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 3*x_1*x_2*x_3*x_4 + 2*x_1*x_2*x_4 + 2*x_1*x_3 + 3*x_1*x_3*x_4 + 2*x_1*x_4 + 2*x_2 + 3*x_2*x_3 + 3*x_2*x_4 + x_3 + x_4 POL(U21(x_1, x_2, x_3)) = 3 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 3*x_2 + 3*x_2*x_3 + 3*x_3 POL(U22(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + 3*x_2*x_3 + 2*x_3 POL(U31(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 POL(U32(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 POL(U33(x_1)) = x_1 POL(U41(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 POL(U42(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 POL(U51(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + 3*x_2*x_3 + 2*x_3 POL(U52(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + 3*x_2*x_3 + 2*x_3 POL(U61(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + 2*x_3 POL(U62(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + 2*x_3 POL(U63(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + 2*x_3 POL(U64(x_1, x_2)) = x_1 + x_2 POL(U71(x_1, x_2)) = 2*x_1 + 2*x_1*x_2 + x_2 POL(U72(x_1, x_2)) = 2*x_1 + 2*x_1*x_2 + x_2 POL(_and_(x_1, x_2)) = x_1 + 3*x_1*x_2 + x_2 POL(_implies_(x_1, x_2)) = 3 + 3*x_1 + 3*x_1*x_2 + 3*x_2 POL(_isEqualTo_(x_1, x_2)) = x_1 + x_2 POL(_isNotEqualTo_(x_1, x_2)) = x_1 + x_2 POL(_or_(x_1, x_2)) = 2 + 3*x_1 + 3*x_1*x_2 + 3*x_2 POL(_xor_(x_1, x_2)) = 1 + x_1 + x_2 POL(equal(x_1, x_2)) = x_1 + 2*x_2 POL(false) = 0 POL(if_then_else_fi(x_1, x_2, x_3)) = x_1 + 2*x_1*x_2*x_3 + 2*x_2 + 2*x_2*x_3 + 2*x_3 POL(not_(x_1)) = 1 + x_1 POL(true) = 0 POL(tt) = 0 ---------------------------------------- (2) Obligation: Equational rewrite system: The TRS R consists of the following rules: U11(tt, A, B, C) -> U12(tt, A, B, C) U12(tt, A, B, C) -> U13(tt, A, B, C) U13(tt, A, B, C) -> _xor_(_and_(A, B), _and_(A, C)) U22(tt, A, B) -> not_(_xor_(A, _and_(A, B))) U31(tt, U', U) -> U32(tt, U', U) U32(tt, U', U) -> U33(equal(_isNotEqualTo_(U, U'), true)) U33(tt) -> false U41(tt, U', U) -> U42(tt, U', U) U42(tt, U', U) -> if_then_else_fi(_isEqualTo_(U, U'), false, true) U51(tt, A, B) -> U52(tt, A, B) U52(tt, A, B) -> _xor_(_and_(A, B), _xor_(A, B)) U61(tt, B, U') -> U62(tt, B, U') U62(tt, B, U') -> U63(tt, B, U') U63(tt, B, U') -> U64(equal(_isNotEqualTo_(B, true), true), U') U64(tt, U') -> U' U71(tt, U) -> U72(tt, U) U72(tt, U) -> U _and_(A, A) -> A _and_(A, _xor_(B, C)) -> U11(tt, A, B, C) _and_(false, A) -> false _and_(true, A) -> A _implies_(A, B) -> U21(tt, A, B) _isEqualTo_(U, U') -> U31(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> U41(tt, U', U) _isNotEqualTo_(U, U) -> false _or_(A, B) -> U51(tt, A, B) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U61(tt, B, U') if_then_else_fi(true, U, U') -> U71(tt, U) not_(A) -> _xor_(A, true) 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)) ---------------------------------------- (3) RRRPoloETRSProof (EQUIVALENT) The following E TRS is given: Equational rewrite system: The TRS R consists of the following rules: U11(tt, A, B, C) -> U12(tt, A, B, C) U12(tt, A, B, C) -> U13(tt, A, B, C) U13(tt, A, B, C) -> _xor_(_and_(A, B), _and_(A, C)) U22(tt, A, B) -> not_(_xor_(A, _and_(A, B))) U31(tt, U', U) -> U32(tt, U', U) U32(tt, U', U) -> U33(equal(_isNotEqualTo_(U, U'), true)) U33(tt) -> false U41(tt, U', U) -> U42(tt, U', U) U42(tt, U', U) -> if_then_else_fi(_isEqualTo_(U, U'), false, true) U51(tt, A, B) -> U52(tt, A, B) U52(tt, A, B) -> _xor_(_and_(A, B), _xor_(A, B)) U61(tt, B, U') -> U62(tt, B, U') U62(tt, B, U') -> U63(tt, B, U') U63(tt, B, U') -> U64(equal(_isNotEqualTo_(B, true), true), U') U64(tt, U') -> U' U71(tt, U) -> U72(tt, U) U72(tt, U) -> U _and_(A, A) -> A _and_(A, _xor_(B, C)) -> U11(tt, A, B, C) _and_(false, A) -> false _and_(true, A) -> A _implies_(A, B) -> U21(tt, A, B) _isEqualTo_(U, U') -> U31(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> U41(tt, U', U) _isNotEqualTo_(U, U) -> false _or_(A, B) -> U51(tt, A, B) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U61(tt, B, U') if_then_else_fi(true, U, U') -> U71(tt, U) not_(A) -> _xor_(A, true) 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 following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering: U22(tt, A, B) -> not_(_xor_(A, _and_(A, B))) _implies_(A, B) -> U21(tt, A, B) Used ordering: Polynomial interpretation [POLO]: POL(U11(x_1, x_2, x_3, x_4)) = 1 + 3*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 3*x_1*x_2*x_3*x_4 + 2*x_1*x_2*x_4 + 2*x_1*x_3 + 3*x_1*x_3*x_4 + 2*x_1*x_4 + 2*x_2 + x_2*x_3 + x_2*x_4 + x_3 + x_4 POL(U12(x_1, x_2, x_3, x_4)) = 1 + 3*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 3*x_1*x_2*x_3*x_4 + 2*x_1*x_2*x_4 + 2*x_1*x_3 + 3*x_1*x_3*x_4 + 2*x_1*x_4 + 2*x_2 + x_2*x_3 + x_2*x_4 + x_3 + x_4 POL(U13(x_1, x_2, x_3, x_4)) = 1 + 3*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 3*x_1*x_2*x_3*x_4 + 2*x_1*x_2*x_4 + 2*x_1*x_3 + 3*x_1*x_3*x_4 + 2*x_1*x_4 + 2*x_2 + x_2*x_3 + x_2*x_4 + x_3 + x_4 POL(U21(x_1, x_2, x_3)) = 1 + 2*x_1 + 3*x_1*x_2 + 3*x_1*x_2*x_3 + 3*x_1*x_3 + 3*x_2 + 3*x_2*x_3 + 3*x_3 POL(U22(x_1, x_2, x_3)) = 3 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 3*x_2 + 2*x_2*x_3 + 2*x_3 POL(U31(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + x_3 POL(U32(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + x_3 POL(U33(x_1)) = x_1 POL(U41(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + x_3 POL(U42(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + x_3 POL(U51(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + x_2*x_3 + 2*x_3 POL(U52(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + x_2*x_3 + 2*x_3 POL(U61(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 POL(U62(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 POL(U63(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 POL(U64(x_1, x_2)) = x_1 + x_2 POL(U71(x_1, x_2)) = 2*x_1 + 2*x_1*x_2 + 2*x_2 POL(U72(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)) = 2 + 3*x_1 + 3*x_1*x_2 + 3*x_2 POL(_isEqualTo_(x_1, x_2)) = x_1 + 2*x_2 POL(_isNotEqualTo_(x_1, x_2)) = x_1 + 2*x_2 POL(_or_(x_1, x_2)) = 2 + 2*x_1 + x_1*x_2 + 2*x_2 POL(_xor_(x_1, x_2)) = 1 + x_1 + x_2 POL(equal(x_1, x_2)) = x_1 + 2*x_2 POL(false) = 0 POL(if_then_else_fi(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3 POL(not_(x_1)) = 1 + x_1 POL(true) = 0 POL(tt) = 0 ---------------------------------------- (4) Obligation: Equational rewrite system: The TRS R consists of the following rules: U11(tt, A, B, C) -> U12(tt, A, B, C) U12(tt, A, B, C) -> U13(tt, A, B, C) U13(tt, A, B, C) -> _xor_(_and_(A, B), _and_(A, C)) U31(tt, U', U) -> U32(tt, U', U) U32(tt, U', U) -> U33(equal(_isNotEqualTo_(U, U'), true)) U33(tt) -> false U41(tt, U', U) -> U42(tt, U', U) U42(tt, U', U) -> if_then_else_fi(_isEqualTo_(U, U'), false, true) U51(tt, A, B) -> U52(tt, A, B) U52(tt, A, B) -> _xor_(_and_(A, B), _xor_(A, B)) U61(tt, B, U') -> U62(tt, B, U') U62(tt, B, U') -> U63(tt, B, U') U63(tt, B, U') -> U64(equal(_isNotEqualTo_(B, true), true), U') U64(tt, U') -> U' U71(tt, U) -> U72(tt, U) U72(tt, U) -> U _and_(A, A) -> A _and_(A, _xor_(B, C)) -> U11(tt, A, B, C) _and_(false, A) -> false _and_(true, A) -> A _isEqualTo_(U, U') -> U31(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> U41(tt, U', U) _isNotEqualTo_(U, U) -> false _or_(A, B) -> U51(tt, A, B) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U61(tt, B, U') if_then_else_fi(true, U, U') -> U71(tt, U) not_(A) -> _xor_(A, true) 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)) ---------------------------------------- (5) RRRPoloETRSProof (EQUIVALENT) The following E TRS is given: Equational rewrite system: The TRS R consists of the following rules: U11(tt, A, B, C) -> U12(tt, A, B, C) U12(tt, A, B, C) -> U13(tt, A, B, C) U13(tt, A, B, C) -> _xor_(_and_(A, B), _and_(A, C)) U31(tt, U', U) -> U32(tt, U', U) U32(tt, U', U) -> U33(equal(_isNotEqualTo_(U, U'), true)) U33(tt) -> false U41(tt, U', U) -> U42(tt, U', U) U42(tt, U', U) -> if_then_else_fi(_isEqualTo_(U, U'), false, true) U51(tt, A, B) -> U52(tt, A, B) U52(tt, A, B) -> _xor_(_and_(A, B), _xor_(A, B)) U61(tt, B, U') -> U62(tt, B, U') U62(tt, B, U') -> U63(tt, B, U') U63(tt, B, U') -> U64(equal(_isNotEqualTo_(B, true), true), U') U64(tt, U') -> U' U71(tt, U) -> U72(tt, U) U72(tt, U) -> U _and_(A, A) -> A _and_(A, _xor_(B, C)) -> U11(tt, A, B, C) _and_(false, A) -> false _and_(true, A) -> A _isEqualTo_(U, U') -> U31(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> U41(tt, U', U) _isNotEqualTo_(U, U) -> false _or_(A, B) -> U51(tt, A, B) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U61(tt, B, U') if_then_else_fi(true, U, U') -> U71(tt, U) not_(A) -> _xor_(A, true) 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 following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering: U51(tt, A, B) -> U52(tt, A, B) not_(A) -> _xor_(A, true) Used ordering: Polynomial interpretation [POLO]: POL(U11(x_1, x_2, x_3, x_4)) = 1 + 3*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 3*x_1*x_2*x_3*x_4 + 2*x_1*x_2*x_4 + 2*x_1*x_3 + 3*x_1*x_3*x_4 + 2*x_1*x_4 + 2*x_2 + x_2*x_3 + x_2*x_4 + x_3 + x_4 POL(U12(x_1, x_2, x_3, x_4)) = 1 + 3*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 3*x_1*x_2*x_3*x_4 + 2*x_1*x_2*x_4 + 2*x_1*x_3 + 3*x_1*x_3*x_4 + 2*x_1*x_4 + 2*x_2 + x_2*x_3 + x_2*x_4 + x_3 + x_4 POL(U13(x_1, x_2, x_3, x_4)) = 1 + 3*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 3*x_1*x_2*x_3*x_4 + 2*x_1*x_2*x_4 + 2*x_1*x_3 + 3*x_1*x_3*x_4 + 2*x_1*x_4 + 2*x_2 + x_2*x_3 + x_2*x_4 + x_3 + x_4 POL(U31(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + x_3 POL(U32(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + x_3 POL(U33(x_1)) = x_1 POL(U41(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + x_3 POL(U42(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + x_3 POL(U51(x_1, x_2, x_3)) = 3 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + x_2*x_3 + 2*x_3 POL(U52(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + x_2*x_3 + 2*x_3 POL(U61(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 POL(U62(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 POL(U63(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 POL(U64(x_1, x_2)) = x_1 + x_2 POL(U71(x_1, x_2)) = 2*x_1 + 2*x_1*x_2 + x_2 POL(U72(x_1, x_2)) = 2*x_1 + 2*x_1*x_2 + x_2 POL(_and_(x_1, x_2)) = x_1 + x_1*x_2 + x_2 POL(_isEqualTo_(x_1, x_2)) = x_1 + 2*x_2 POL(_isNotEqualTo_(x_1, x_2)) = x_1 + 2*x_2 POL(_or_(x_1, x_2)) = 3 + 3*x_1 + 2*x_1*x_2 + 3*x_2 POL(_xor_(x_1, x_2)) = 1 + x_1 + x_2 POL(equal(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(if_then_else_fi(x_1, x_2, x_3)) = x_1 + 2*x_2 + x_3 POL(not_(x_1)) = 2 + 3*x_1 + 3*x_1^2 POL(true) = 0 POL(tt) = 0 ---------------------------------------- (6) Obligation: Equational rewrite system: The TRS R consists of the following rules: U11(tt, A, B, C) -> U12(tt, A, B, C) U12(tt, A, B, C) -> U13(tt, A, B, C) U13(tt, A, B, C) -> _xor_(_and_(A, B), _and_(A, C)) U31(tt, U', U) -> U32(tt, U', U) U32(tt, U', U) -> U33(equal(_isNotEqualTo_(U, U'), true)) U33(tt) -> false U41(tt, U', U) -> U42(tt, U', U) U42(tt, U', U) -> if_then_else_fi(_isEqualTo_(U, U'), false, true) U52(tt, A, B) -> _xor_(_and_(A, B), _xor_(A, B)) U61(tt, B, U') -> U62(tt, B, U') U62(tt, B, U') -> U63(tt, B, U') U63(tt, B, U') -> U64(equal(_isNotEqualTo_(B, true), true), U') U64(tt, U') -> U' U71(tt, U) -> U72(tt, U) U72(tt, U) -> U _and_(A, A) -> A _and_(A, _xor_(B, C)) -> U11(tt, A, B, C) _and_(false, A) -> false _and_(true, A) -> A _isEqualTo_(U, U') -> U31(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> U41(tt, U', U) _isNotEqualTo_(U, U) -> false _or_(A, B) -> U51(tt, A, B) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U61(tt, B, U') if_then_else_fi(true, U, U') -> U71(tt, U) 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)) ---------------------------------------- (7) RRRPoloETRSProof (EQUIVALENT) The following E TRS is given: Equational rewrite system: The TRS R consists of the following rules: U11(tt, A, B, C) -> U12(tt, A, B, C) U12(tt, A, B, C) -> U13(tt, A, B, C) U13(tt, A, B, C) -> _xor_(_and_(A, B), _and_(A, C)) U31(tt, U', U) -> U32(tt, U', U) U32(tt, U', U) -> U33(equal(_isNotEqualTo_(U, U'), true)) U33(tt) -> false U41(tt, U', U) -> U42(tt, U', U) U42(tt, U', U) -> if_then_else_fi(_isEqualTo_(U, U'), false, true) U52(tt, A, B) -> _xor_(_and_(A, B), _xor_(A, B)) U61(tt, B, U') -> U62(tt, B, U') U62(tt, B, U') -> U63(tt, B, U') U63(tt, B, U') -> U64(equal(_isNotEqualTo_(B, true), true), U') U64(tt, U') -> U' U71(tt, U) -> U72(tt, U) U72(tt, U) -> U _and_(A, A) -> A _and_(A, _xor_(B, C)) -> U11(tt, A, B, C) _and_(false, A) -> false _and_(true, A) -> A _isEqualTo_(U, U') -> U31(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> U41(tt, U', U) _isNotEqualTo_(U, U) -> false _or_(A, B) -> U51(tt, A, B) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U61(tt, B, U') if_then_else_fi(true, U, U') -> U71(tt, U) 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 following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering: _or_(A, B) -> U51(tt, A, B) Used ordering: Polynomial interpretation [POLO]: POL(U11(x_1, x_2, x_3, x_4)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 3*x_1*x_2*x_3*x_4 + 2*x_1*x_2*x_4 + 2*x_1*x_3 + 3*x_1*x_3*x_4 + 2*x_1*x_4 + 3*x_2 + 2*x_2*x_3 + 2*x_2*x_4 + x_3 + x_4 POL(U12(x_1, x_2, x_3, x_4)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 3*x_1*x_2*x_3*x_4 + 2*x_1*x_2*x_4 + 2*x_1*x_3 + 3*x_1*x_3*x_4 + 2*x_1*x_4 + 3*x_2 + 2*x_2*x_3 + 2*x_2*x_4 + x_3 + x_4 POL(U13(x_1, x_2, x_3, x_4)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 3*x_1*x_2*x_3*x_4 + 2*x_1*x_2*x_4 + 2*x_1*x_3 + 3*x_1*x_3*x_4 + 2*x_1*x_4 + 2*x_2 + 2*x_2*x_3 + 2*x_2*x_4 + x_3 + x_4 POL(U31(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 POL(U32(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 POL(U33(x_1)) = x_1 POL(U41(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 POL(U42(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 POL(U51(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 POL(U52(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 3*x_2 + 2*x_2*x_3 + 3*x_3 POL(U61(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + 2*x_3 POL(U62(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + 2*x_3 POL(U63(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + 2*x_3 POL(U64(x_1, x_2)) = x_1 + 2*x_2 POL(U71(x_1, x_2)) = 2*x_1 + 2*x_1*x_2 + x_2 POL(U72(x_1, x_2)) = 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(_isEqualTo_(x_1, x_2)) = x_1 + x_2 POL(_isNotEqualTo_(x_1, x_2)) = x_1 + x_2 POL(_or_(x_1, x_2)) = 1 + x_1 + x_2 POL(_xor_(x_1, x_2)) = 1 + x_1 + x_2 POL(equal(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(if_then_else_fi(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3 POL(true) = 0 POL(tt) = 0 ---------------------------------------- (8) Obligation: Equational rewrite system: The TRS R consists of the following rules: U11(tt, A, B, C) -> U12(tt, A, B, C) U12(tt, A, B, C) -> U13(tt, A, B, C) U13(tt, A, B, C) -> _xor_(_and_(A, B), _and_(A, C)) U31(tt, U', U) -> U32(tt, U', U) U32(tt, U', U) -> U33(equal(_isNotEqualTo_(U, U'), true)) U33(tt) -> false U41(tt, U', U) -> U42(tt, U', U) U42(tt, U', U) -> if_then_else_fi(_isEqualTo_(U, U'), false, true) U52(tt, A, B) -> _xor_(_and_(A, B), _xor_(A, B)) U61(tt, B, U') -> U62(tt, B, U') U62(tt, B, U') -> U63(tt, B, U') U63(tt, B, U') -> U64(equal(_isNotEqualTo_(B, true), true), U') U64(tt, U') -> U' U71(tt, U) -> U72(tt, U) U72(tt, U) -> U _and_(A, A) -> A _and_(A, _xor_(B, C)) -> U11(tt, A, B, C) _and_(false, A) -> false _and_(true, A) -> A _isEqualTo_(U, U') -> U31(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> U41(tt, U', U) _isNotEqualTo_(U, U) -> false equal(X, X) -> tt if_then_else_fi(B, U, U') -> U61(tt, B, U') if_then_else_fi(true, U, U') -> U71(tt, U) 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)) ---------------------------------------- (9) RRRPoloETRSProof (EQUIVALENT) The following E TRS is given: Equational rewrite system: The TRS R consists of the following rules: U11(tt, A, B, C) -> U12(tt, A, B, C) U12(tt, A, B, C) -> U13(tt, A, B, C) U13(tt, A, B, C) -> _xor_(_and_(A, B), _and_(A, C)) U31(tt, U', U) -> U32(tt, U', U) U32(tt, U', U) -> U33(equal(_isNotEqualTo_(U, U'), true)) U33(tt) -> false U41(tt, U', U) -> U42(tt, U', U) U42(tt, U', U) -> if_then_else_fi(_isEqualTo_(U, U'), false, true) U52(tt, A, B) -> _xor_(_and_(A, B), _xor_(A, B)) U61(tt, B, U') -> U62(tt, B, U') U62(tt, B, U') -> U63(tt, B, U') U63(tt, B, U') -> U64(equal(_isNotEqualTo_(B, true), true), U') U64(tt, U') -> U' U71(tt, U) -> U72(tt, U) U72(tt, U) -> U _and_(A, A) -> A _and_(A, _xor_(B, C)) -> U11(tt, A, B, C) _and_(false, A) -> false _and_(true, A) -> A _isEqualTo_(U, U') -> U31(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> U41(tt, U', U) _isNotEqualTo_(U, U) -> false equal(X, X) -> tt if_then_else_fi(B, U, U') -> U61(tt, B, U') if_then_else_fi(true, U, U') -> U71(tt, U) 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 following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering: U52(tt, A, B) -> _xor_(_and_(A, B), _xor_(A, B)) Used ordering: Polynomial interpretation [POLO]: POL(U11(x_1, x_2, x_3, x_4)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 3*x_1*x_2*x_3*x_4 + 2*x_1*x_2*x_4 + 2*x_1*x_3 + 3*x_1*x_3*x_4 + 2*x_1*x_4 + 2*x_2 + 3*x_2*x_3 + 3*x_2*x_4 + x_3 + x_4 POL(U12(x_1, x_2, x_3, x_4)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 3*x_1*x_2*x_3*x_4 + 2*x_1*x_2*x_4 + 2*x_1*x_3 + 3*x_1*x_3*x_4 + 2*x_1*x_4 + 2*x_2 + 3*x_2*x_3 + 3*x_2*x_4 + x_3 + x_4 POL(U13(x_1, x_2, x_3, x_4)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 3*x_1*x_2*x_3*x_4 + 2*x_1*x_2*x_4 + 2*x_1*x_3 + 3*x_1*x_3*x_4 + 2*x_1*x_4 + 2*x_2 + 3*x_2*x_3 + 3*x_2*x_4 + x_3 + x_4 POL(U31(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 POL(U32(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 POL(U33(x_1)) = x_1 POL(U41(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 POL(U42(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 POL(U52(x_1, x_2, x_3)) = 3 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 3*x_2 + 3*x_2*x_3 + 3*x_3 POL(U61(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + 2*x_3 POL(U62(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + 2*x_3 POL(U63(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + 2*x_3 POL(U64(x_1, x_2)) = x_1 + 2*x_2 POL(U71(x_1, x_2)) = 2*x_1 + 2*x_1*x_2 + 2*x_2 POL(U72(x_1, x_2)) = 2*x_1 + 2*x_1*x_2 + x_2 POL(_and_(x_1, x_2)) = x_1 + 3*x_1*x_2 + x_2 POL(_isEqualTo_(x_1, x_2)) = x_1 + x_2 POL(_isNotEqualTo_(x_1, x_2)) = x_1 + x_2 POL(_or_(x_1, x_2)) = 3 + 3*x_1 + 2*x_1*x_2 + 3*x_2 POL(_xor_(x_1, x_2)) = 1 + x_1 + x_2 POL(equal(x_1, x_2)) = x_1 + 2*x_2 POL(false) = 0 POL(if_then_else_fi(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3 POL(true) = 0 POL(tt) = 0 ---------------------------------------- (10) Obligation: Equational rewrite system: The TRS R consists of the following rules: U11(tt, A, B, C) -> U12(tt, A, B, C) U12(tt, A, B, C) -> U13(tt, A, B, C) U13(tt, A, B, C) -> _xor_(_and_(A, B), _and_(A, C)) U31(tt, U', U) -> U32(tt, U', U) U32(tt, U', U) -> U33(equal(_isNotEqualTo_(U, U'), true)) U33(tt) -> false U41(tt, U', U) -> U42(tt, U', U) U42(tt, U', U) -> if_then_else_fi(_isEqualTo_(U, U'), false, true) U61(tt, B, U') -> U62(tt, B, U') U62(tt, B, U') -> U63(tt, B, U') U63(tt, B, U') -> U64(equal(_isNotEqualTo_(B, true), true), U') U64(tt, U') -> U' U71(tt, U) -> U72(tt, U) U72(tt, U) -> U _and_(A, A) -> A _and_(A, _xor_(B, C)) -> U11(tt, A, B, C) _and_(false, A) -> false _and_(true, A) -> A _isEqualTo_(U, U') -> U31(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> U41(tt, U', U) _isNotEqualTo_(U, U) -> false equal(X, X) -> tt if_then_else_fi(B, U, U') -> U61(tt, B, U') if_then_else_fi(true, U, U') -> U71(tt, U) 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)) ---------------------------------------- (11) EquationalDependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,DA_STEIN] we result in the following initial EDP problem: The TRS P consists of the following rules: U11^1(tt, A, B, C) -> U12^1(tt, A, B, C) U12^1(tt, A, B, C) -> U13^1(tt, A, B, C) U13^1(tt, A, B, C) -> _AND_(A, B) U13^1(tt, A, B, C) -> _AND_(A, C) U31^1(tt, U', U) -> U32^1(tt, U', U) U32^1(tt, U', U) -> U33^1(equal(_isNotEqualTo_(U, U'), true)) U32^1(tt, U', U) -> EQUAL(_isNotEqualTo_(U, U'), true) U32^1(tt, U', U) -> _ISNOTEQUALTO_(U, U') U41^1(tt, U', U) -> U42^1(tt, U', U) U42^1(tt, U', U) -> IF_THEN_ELSE_FI(_isEqualTo_(U, U'), false, true) U42^1(tt, U', U) -> _ISEQUALTO_(U, U') U61^1(tt, B, U') -> U62^1(tt, B, U') U62^1(tt, B, U') -> U63^1(tt, B, U') U63^1(tt, B, U') -> U64^1(equal(_isNotEqualTo_(B, true), true), U') U63^1(tt, B, U') -> EQUAL(_isNotEqualTo_(B, true), true) U63^1(tt, B, U') -> _ISNOTEQUALTO_(B, true) U71^1(tt, U) -> U72^1(tt, U) _AND_(A, _xor_(B, C)) -> U11^1(tt, A, B, C) _ISEQUALTO_(U, U') -> U31^1(tt, U', U) _ISNOTEQUALTO_(U, U') -> U41^1(tt, U', U) IF_THEN_ELSE_FI(B, U, U') -> U61^1(tt, B, U') IF_THEN_ELSE_FI(true, U, U') -> U71^1(tt, U) _AND_(_and_(A, A), ext) -> _AND_(A, ext) _AND_(_and_(A, _xor_(B, C)), ext) -> _AND_(U11(tt, A, B, C), ext) _AND_(_and_(A, _xor_(B, C)), ext) -> U11^1(tt, A, B, C) _AND_(_and_(false, A), ext) -> _AND_(false, ext) The TRS R consists of the following rules: U11(tt, A, B, C) -> U12(tt, A, B, C) U12(tt, A, B, C) -> U13(tt, A, B, C) U13(tt, A, B, C) -> _xor_(_and_(A, B), _and_(A, C)) U31(tt, U', U) -> U32(tt, U', U) U32(tt, U', U) -> U33(equal(_isNotEqualTo_(U, U'), true)) U33(tt) -> false U41(tt, U', U) -> U42(tt, U', U) U42(tt, U', U) -> if_then_else_fi(_isEqualTo_(U, U'), false, true) U61(tt, B, U') -> U62(tt, B, U') U62(tt, B, U') -> U63(tt, B, U') U63(tt, B, U') -> U64(equal(_isNotEqualTo_(B, true), true), U') U64(tt, U') -> U' U71(tt, U) -> U72(tt, U) U72(tt, U) -> U _and_(A, A) -> A _and_(A, _xor_(B, C)) -> U11(tt, A, B, C) _and_(false, A) -> false _and_(true, A) -> A _isEqualTo_(U, U') -> U31(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> U41(tt, U', U) _isNotEqualTo_(U, U) -> false equal(X, X) -> tt if_then_else_fi(B, U, U') -> U61(tt, B, U') if_then_else_fi(true, U, U') -> U71(tt, U) _and_(_and_(A, A), ext) -> _and_(A, ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U11(tt, A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(false, 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) _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (12) Obligation: The TRS P consists of the following rules: U11^1(tt, A, B, C) -> U12^1(tt, A, B, C) U12^1(tt, A, B, C) -> U13^1(tt, A, B, C) U13^1(tt, A, B, C) -> _AND_(A, B) U13^1(tt, A, B, C) -> _AND_(A, C) U31^1(tt, U', U) -> U32^1(tt, U', U) U32^1(tt, U', U) -> U33^1(equal(_isNotEqualTo_(U, U'), true)) U32^1(tt, U', U) -> EQUAL(_isNotEqualTo_(U, U'), true) U32^1(tt, U', U) -> _ISNOTEQUALTO_(U, U') U41^1(tt, U', U) -> U42^1(tt, U', U) U42^1(tt, U', U) -> IF_THEN_ELSE_FI(_isEqualTo_(U, U'), false, true) U42^1(tt, U', U) -> _ISEQUALTO_(U, U') U61^1(tt, B, U') -> U62^1(tt, B, U') U62^1(tt, B, U') -> U63^1(tt, B, U') U63^1(tt, B, U') -> U64^1(equal(_isNotEqualTo_(B, true), true), U') U63^1(tt, B, U') -> EQUAL(_isNotEqualTo_(B, true), true) U63^1(tt, B, U') -> _ISNOTEQUALTO_(B, true) U71^1(tt, U) -> U72^1(tt, U) _AND_(A, _xor_(B, C)) -> U11^1(tt, A, B, C) _ISEQUALTO_(U, U') -> U31^1(tt, U', U) _ISNOTEQUALTO_(U, U') -> U41^1(tt, U', U) IF_THEN_ELSE_FI(B, U, U') -> U61^1(tt, B, U') IF_THEN_ELSE_FI(true, U, U') -> U71^1(tt, U) _AND_(_and_(A, A), ext) -> _AND_(A, ext) _AND_(_and_(A, _xor_(B, C)), ext) -> _AND_(U11(tt, A, B, C), ext) _AND_(_and_(A, _xor_(B, C)), ext) -> U11^1(tt, A, B, C) _AND_(_and_(false, A), ext) -> _AND_(false, ext) The TRS R consists of the following rules: U11(tt, A, B, C) -> U12(tt, A, B, C) U12(tt, A, B, C) -> U13(tt, A, B, C) U13(tt, A, B, C) -> _xor_(_and_(A, B), _and_(A, C)) U31(tt, U', U) -> U32(tt, U', U) U32(tt, U', U) -> U33(equal(_isNotEqualTo_(U, U'), true)) U33(tt) -> false U41(tt, U', U) -> U42(tt, U', U) U42(tt, U', U) -> if_then_else_fi(_isEqualTo_(U, U'), false, true) U61(tt, B, U') -> U62(tt, B, U') U62(tt, B, U') -> U63(tt, B, U') U63(tt, B, U') -> U64(equal(_isNotEqualTo_(B, true), true), U') U64(tt, U') -> U' U71(tt, U) -> U72(tt, U) U72(tt, U) -> U _and_(A, A) -> A _and_(A, _xor_(B, C)) -> U11(tt, A, B, C) _and_(false, A) -> false _and_(true, A) -> A _isEqualTo_(U, U') -> U31(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> U41(tt, U', U) _isNotEqualTo_(U, U) -> false equal(X, X) -> tt if_then_else_fi(B, U, U') -> U61(tt, B, U') if_then_else_fi(true, U, U') -> U71(tt, U) _and_(_and_(A, A), ext) -> _and_(A, ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U11(tt, A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(false, 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) _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (13) EDependencyGraphProof (EQUIVALENT) The approximation of the Equational Dependency Graph [DA_STEIN] contains 2 SCCs with 6 less nodes. ---------------------------------------- (14) Complex Obligation (AND) ---------------------------------------- (15) Obligation: The TRS P consists of the following rules: U32^1(tt, U', U) -> _ISNOTEQUALTO_(U, U') U31^1(tt, U', U) -> U32^1(tt, U', U) U63^1(tt, B, U') -> _ISNOTEQUALTO_(B, true) _ISNOTEQUALTO_(U, U') -> U41^1(tt, U', U) U61^1(tt, B, U') -> U62^1(tt, B, U') IF_THEN_ELSE_FI(B, U, U') -> U61^1(tt, B, U') U42^1(tt, U', U) -> _ISEQUALTO_(U, U') U62^1(tt, B, U') -> U63^1(tt, B, U') _ISEQUALTO_(U, U') -> U31^1(tt, U', U) U41^1(tt, U', U) -> U42^1(tt, U', U) U42^1(tt, U', U) -> IF_THEN_ELSE_FI(_isEqualTo_(U, U'), false, true) The TRS R consists of the following rules: U11(tt, A, B, C) -> U12(tt, A, B, C) U12(tt, A, B, C) -> U13(tt, A, B, C) U13(tt, A, B, C) -> _xor_(_and_(A, B), _and_(A, C)) U31(tt, U', U) -> U32(tt, U', U) U32(tt, U', U) -> U33(equal(_isNotEqualTo_(U, U'), true)) U33(tt) -> false U41(tt, U', U) -> U42(tt, U', U) U42(tt, U', U) -> if_then_else_fi(_isEqualTo_(U, U'), false, true) U61(tt, B, U') -> U62(tt, B, U') U62(tt, B, U') -> U63(tt, B, U') U63(tt, B, U') -> U64(equal(_isNotEqualTo_(B, true), true), U') U64(tt, U') -> U' U71(tt, U) -> U72(tt, U) U72(tt, U) -> U _and_(A, A) -> A _and_(A, _xor_(B, C)) -> U11(tt, A, B, C) _and_(false, A) -> false _and_(true, A) -> A _isEqualTo_(U, U') -> U31(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> U41(tt, U', U) _isNotEqualTo_(U, U) -> false equal(X, X) -> tt if_then_else_fi(B, U, U') -> U61(tt, B, U') if_then_else_fi(true, U, U') -> U71(tt, U) _and_(_and_(A, A), ext) -> _and_(A, ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U11(tt, A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(false, 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) _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (16) ESharpUsableEquationsProof (EQUIVALENT) We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]: _AND_(x, y) == _AND_(y, x) _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) ---------------------------------------- (17) Obligation: The TRS P consists of the following rules: U32^1(tt, U', U) -> _ISNOTEQUALTO_(U, U') U31^1(tt, U', U) -> U32^1(tt, U', U) U63^1(tt, B, U') -> _ISNOTEQUALTO_(B, true) _ISNOTEQUALTO_(U, U') -> U41^1(tt, U', U) U61^1(tt, B, U') -> U62^1(tt, B, U') IF_THEN_ELSE_FI(B, U, U') -> U61^1(tt, B, U') U42^1(tt, U', U) -> _ISEQUALTO_(U, U') U62^1(tt, B, U') -> U63^1(tt, B, U') _ISEQUALTO_(U, U') -> U31^1(tt, U', U) U41^1(tt, U', U) -> U42^1(tt, U', U) U42^1(tt, U', U) -> IF_THEN_ELSE_FI(_isEqualTo_(U, U'), false, true) The TRS R consists of the following rules: U11(tt, A, B, C) -> U12(tt, A, B, C) U12(tt, A, B, C) -> U13(tt, A, B, C) U13(tt, A, B, C) -> _xor_(_and_(A, B), _and_(A, C)) U31(tt, U', U) -> U32(tt, U', U) U32(tt, U', U) -> U33(equal(_isNotEqualTo_(U, U'), true)) U33(tt) -> false U41(tt, U', U) -> U42(tt, U', U) U42(tt, U', U) -> if_then_else_fi(_isEqualTo_(U, U'), false, true) U61(tt, B, U') -> U62(tt, B, U') U62(tt, B, U') -> U63(tt, B, U') U63(tt, B, U') -> U64(equal(_isNotEqualTo_(B, true), true), U') U64(tt, U') -> U' U71(tt, U) -> U72(tt, U) U72(tt, U) -> U _and_(A, A) -> A _and_(A, _xor_(B, C)) -> U11(tt, A, B, C) _and_(false, A) -> false _and_(true, A) -> A _isEqualTo_(U, U') -> U31(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> U41(tt, U', U) _isNotEqualTo_(U, U) -> false equal(X, X) -> tt if_then_else_fi(B, U, U') -> U61(tt, B, U') if_then_else_fi(true, U, U') -> U71(tt, U) _and_(_and_(A, A), ext) -> _and_(A, ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U11(tt, A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(false, 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 ---------------------------------------- (18) 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. No dependency pairs are removed. The following rules are removed from R: U11(tt, A, B, C) -> U12(tt, A, B, C) U12(tt, A, B, C) -> U13(tt, A, B, C) U13(tt, A, B, C) -> _xor_(_and_(A, B), _and_(A, C)) _and_(A, A) -> A _and_(A, _xor_(B, C)) -> U11(tt, A, B, C) _and_(false, A) -> false _and_(true, A) -> A _and_(_and_(A, A), ext) -> _and_(A, ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U11(tt, A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(false, 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(IF_THEN_ELSE_FI(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U31(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3 POL(U31^1(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U32(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3 POL(U32^1(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U33(x_1)) = x_1 POL(U41(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3 POL(U41^1(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U42(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3 POL(U42^1(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U61(x_1, x_2, x_3)) = 2*x_1 + x_2 + 2*x_3 POL(U61^1(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U62(x_1, x_2, x_3)) = 2*x_1 + x_2 + 2*x_3 POL(U62^1(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U63(x_1, x_2, x_3)) = 2*x_1 + x_2 + 2*x_3 POL(U63^1(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U64(x_1, x_2)) = x_1 + 2*x_2 POL(U71(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U72(x_1, x_2)) = 2*x_1 + 2*x_2 POL(_ISEQUALTO_(x_1, x_2)) = 2*x_1 + 2*x_2 POL(_ISNOTEQUALTO_(x_1, x_2)) = 2*x_1 + 2*x_2 POL(_isEqualTo_(x_1, x_2)) = x_1 + x_2 POL(_isNotEqualTo_(x_1, x_2)) = x_1 + x_2 POL(equal(x_1, x_2)) = x_1 + 2*x_2 POL(false) = 0 POL(if_then_else_fi(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3 POL(true) = 0 POL(tt) = 0 ---------------------------------------- (19) Obligation: The TRS P consists of the following rules: U32^1(tt, U', U) -> _ISNOTEQUALTO_(U, U') U31^1(tt, U', U) -> U32^1(tt, U', U) U63^1(tt, B, U') -> _ISNOTEQUALTO_(B, true) _ISNOTEQUALTO_(U, U') -> U41^1(tt, U', U) U61^1(tt, B, U') -> U62^1(tt, B, U') IF_THEN_ELSE_FI(B, U, U') -> U61^1(tt, B, U') U42^1(tt, U', U) -> _ISEQUALTO_(U, U') U62^1(tt, B, U') -> U63^1(tt, B, U') _ISEQUALTO_(U, U') -> U31^1(tt, U', U) U41^1(tt, U', U) -> U42^1(tt, U', U) U42^1(tt, U', U) -> IF_THEN_ELSE_FI(_isEqualTo_(U, U'), false, true) The TRS R consists of the following rules: U42(tt, U', U) -> if_then_else_fi(_isEqualTo_(U, U'), false, true) U32(tt, U', U) -> U33(equal(_isNotEqualTo_(U, U'), true)) U72(tt, U) -> U U41(tt, U', U) -> U42(tt, U', U) U33(tt) -> false U31(tt, U', U) -> U32(tt, U', U) U62(tt, B, U') -> U63(tt, B, U') _isNotEqualTo_(U, U') -> U41(tt, U', U) U63(tt, B, U') -> U64(equal(_isNotEqualTo_(B, true), true), U') equal(X, X) -> tt _isNotEqualTo_(U, U) -> false _isEqualTo_(U, U') -> U31(tt, U', U) if_then_else_fi(true, U, U') -> U71(tt, U) if_then_else_fi(B, U, U') -> U61(tt, B, U') U61(tt, B, U') -> U62(tt, B, U') U71(tt, U) -> U72(tt, U) _isEqualTo_(U, U) -> true U64(tt, U') -> U' E is empty. E# is empty. We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (20) EDPProblemToQDPProblemProof (EQUIVALENT) The EDP problem does not contain equations anymore, so we can transform it with the EDP to QDP problem processor [DA_STEIN] into a QDP problem. ---------------------------------------- (21) Obligation: Q DP problem: The TRS P consists of the following rules: U32^1(tt, U', U) -> _ISNOTEQUALTO_(U, U') U31^1(tt, U', U) -> U32^1(tt, U', U) U63^1(tt, B, U') -> _ISNOTEQUALTO_(B, true) _ISNOTEQUALTO_(U, U') -> U41^1(tt, U', U) U61^1(tt, B, U') -> U62^1(tt, B, U') IF_THEN_ELSE_FI(B, U, U') -> U61^1(tt, B, U') U42^1(tt, U', U) -> _ISEQUALTO_(U, U') U62^1(tt, B, U') -> U63^1(tt, B, U') _ISEQUALTO_(U, U') -> U31^1(tt, U', U) U41^1(tt, U', U) -> U42^1(tt, U', U) U42^1(tt, U', U) -> IF_THEN_ELSE_FI(_isEqualTo_(U, U'), false, true) The TRS R consists of the following rules: U42(tt, U', U) -> if_then_else_fi(_isEqualTo_(U, U'), false, true) U32(tt, U', U) -> U33(equal(_isNotEqualTo_(U, U'), true)) U72(tt, U) -> U U41(tt, U', U) -> U42(tt, U', U) U33(tt) -> false U31(tt, U', U) -> U32(tt, U', U) U62(tt, B, U') -> U63(tt, B, U') _isNotEqualTo_(U, U') -> U41(tt, U', U) U63(tt, B, U') -> U64(equal(_isNotEqualTo_(B, true), true), U') equal(X, X) -> tt _isNotEqualTo_(U, U) -> false _isEqualTo_(U, U') -> U31(tt, U', U) if_then_else_fi(true, U, U') -> U71(tt, U) if_then_else_fi(B, U, U') -> U61(tt, B, U') U61(tt, B, U') -> U62(tt, B, U') U71(tt, U) -> U72(tt, U) _isEqualTo_(U, U) -> true U64(tt, U') -> U' Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (22) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF_THEN_ELSE_FI(B, U, U') -> U61^1(tt, B, U') we obtained the following new rules [LPAR04]: (IF_THEN_ELSE_FI(y_2, false, true) -> U61^1(tt, y_2, true),IF_THEN_ELSE_FI(y_2, false, true) -> U61^1(tt, y_2, true)) ---------------------------------------- (23) Obligation: Q DP problem: The TRS P consists of the following rules: U32^1(tt, U', U) -> _ISNOTEQUALTO_(U, U') U31^1(tt, U', U) -> U32^1(tt, U', U) U63^1(tt, B, U') -> _ISNOTEQUALTO_(B, true) _ISNOTEQUALTO_(U, U') -> U41^1(tt, U', U) U61^1(tt, B, U') -> U62^1(tt, B, U') U42^1(tt, U', U) -> _ISEQUALTO_(U, U') U62^1(tt, B, U') -> U63^1(tt, B, U') _ISEQUALTO_(U, U') -> U31^1(tt, U', U) U41^1(tt, U', U) -> U42^1(tt, U', U) U42^1(tt, U', U) -> IF_THEN_ELSE_FI(_isEqualTo_(U, U'), false, true) IF_THEN_ELSE_FI(y_2, false, true) -> U61^1(tt, y_2, true) The TRS R consists of the following rules: U42(tt, U', U) -> if_then_else_fi(_isEqualTo_(U, U'), false, true) U32(tt, U', U) -> U33(equal(_isNotEqualTo_(U, U'), true)) U72(tt, U) -> U U41(tt, U', U) -> U42(tt, U', U) U33(tt) -> false U31(tt, U', U) -> U32(tt, U', U) U62(tt, B, U') -> U63(tt, B, U') _isNotEqualTo_(U, U') -> U41(tt, U', U) U63(tt, B, U') -> U64(equal(_isNotEqualTo_(B, true), true), U') equal(X, X) -> tt _isNotEqualTo_(U, U) -> false _isEqualTo_(U, U') -> U31(tt, U', U) if_then_else_fi(true, U, U') -> U71(tt, U) if_then_else_fi(B, U, U') -> U61(tt, B, U') U61(tt, B, U') -> U62(tt, B, U') U71(tt, U) -> U72(tt, U) _isEqualTo_(U, U) -> true U64(tt, U') -> U' Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (24) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U61^1(tt, B, U') -> U62^1(tt, B, U') we obtained the following new rules [LPAR04]: (U61^1(tt, y_0, true) -> U62^1(tt, y_0, true),U61^1(tt, y_0, true) -> U62^1(tt, y_0, true)) ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: U32^1(tt, U', U) -> _ISNOTEQUALTO_(U, U') U31^1(tt, U', U) -> U32^1(tt, U', U) U63^1(tt, B, U') -> _ISNOTEQUALTO_(B, true) _ISNOTEQUALTO_(U, U') -> U41^1(tt, U', U) U42^1(tt, U', U) -> _ISEQUALTO_(U, U') U62^1(tt, B, U') -> U63^1(tt, B, U') _ISEQUALTO_(U, U') -> U31^1(tt, U', U) U41^1(tt, U', U) -> U42^1(tt, U', U) U42^1(tt, U', U) -> IF_THEN_ELSE_FI(_isEqualTo_(U, U'), false, true) IF_THEN_ELSE_FI(y_2, false, true) -> U61^1(tt, y_2, true) U61^1(tt, y_0, true) -> U62^1(tt, y_0, true) The TRS R consists of the following rules: U42(tt, U', U) -> if_then_else_fi(_isEqualTo_(U, U'), false, true) U32(tt, U', U) -> U33(equal(_isNotEqualTo_(U, U'), true)) U72(tt, U) -> U U41(tt, U', U) -> U42(tt, U', U) U33(tt) -> false U31(tt, U', U) -> U32(tt, U', U) U62(tt, B, U') -> U63(tt, B, U') _isNotEqualTo_(U, U') -> U41(tt, U', U) U63(tt, B, U') -> U64(equal(_isNotEqualTo_(B, true), true), U') equal(X, X) -> tt _isNotEqualTo_(U, U) -> false _isEqualTo_(U, U') -> U31(tt, U', U) if_then_else_fi(true, U, U') -> U71(tt, U) if_then_else_fi(B, U, U') -> U61(tt, B, U') U61(tt, B, U') -> U62(tt, B, U') U71(tt, U) -> U72(tt, U) _isEqualTo_(U, U) -> true U64(tt, U') -> U' Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (26) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U62^1(tt, B, U') -> U63^1(tt, B, U') we obtained the following new rules [LPAR04]: (U62^1(tt, y_0, true) -> U63^1(tt, y_0, true),U62^1(tt, y_0, true) -> U63^1(tt, y_0, true)) ---------------------------------------- (27) Obligation: Q DP problem: The TRS P consists of the following rules: U32^1(tt, U', U) -> _ISNOTEQUALTO_(U, U') U31^1(tt, U', U) -> U32^1(tt, U', U) U63^1(tt, B, U') -> _ISNOTEQUALTO_(B, true) _ISNOTEQUALTO_(U, U') -> U41^1(tt, U', U) U42^1(tt, U', U) -> _ISEQUALTO_(U, U') _ISEQUALTO_(U, U') -> U31^1(tt, U', U) U41^1(tt, U', U) -> U42^1(tt, U', U) U42^1(tt, U', U) -> IF_THEN_ELSE_FI(_isEqualTo_(U, U'), false, true) IF_THEN_ELSE_FI(y_2, false, true) -> U61^1(tt, y_2, true) U61^1(tt, y_0, true) -> U62^1(tt, y_0, true) U62^1(tt, y_0, true) -> U63^1(tt, y_0, true) The TRS R consists of the following rules: U42(tt, U', U) -> if_then_else_fi(_isEqualTo_(U, U'), false, true) U32(tt, U', U) -> U33(equal(_isNotEqualTo_(U, U'), true)) U72(tt, U) -> U U41(tt, U', U) -> U42(tt, U', U) U33(tt) -> false U31(tt, U', U) -> U32(tt, U', U) U62(tt, B, U') -> U63(tt, B, U') _isNotEqualTo_(U, U') -> U41(tt, U', U) U63(tt, B, U') -> U64(equal(_isNotEqualTo_(B, true), true), U') equal(X, X) -> tt _isNotEqualTo_(U, U) -> false _isEqualTo_(U, U') -> U31(tt, U', U) if_then_else_fi(true, U, U') -> U71(tt, U) if_then_else_fi(B, U, U') -> U61(tt, B, U') U61(tt, B, U') -> U62(tt, B, U') U71(tt, U) -> U72(tt, U) _isEqualTo_(U, U) -> true U64(tt, U') -> U' Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (28) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U63^1(tt, B, U') -> _ISNOTEQUALTO_(B, true) we obtained the following new rules [LPAR04]: (U63^1(tt, y_0, true) -> _ISNOTEQUALTO_(y_0, true),U63^1(tt, y_0, true) -> _ISNOTEQUALTO_(y_0, true)) ---------------------------------------- (29) Obligation: Q DP problem: The TRS P consists of the following rules: U32^1(tt, U', U) -> _ISNOTEQUALTO_(U, U') U31^1(tt, U', U) -> U32^1(tt, U', U) _ISNOTEQUALTO_(U, U') -> U41^1(tt, U', U) U42^1(tt, U', U) -> _ISEQUALTO_(U, U') _ISEQUALTO_(U, U') -> U31^1(tt, U', U) U41^1(tt, U', U) -> U42^1(tt, U', U) U42^1(tt, U', U) -> IF_THEN_ELSE_FI(_isEqualTo_(U, U'), false, true) IF_THEN_ELSE_FI(y_2, false, true) -> U61^1(tt, y_2, true) U61^1(tt, y_0, true) -> U62^1(tt, y_0, true) U62^1(tt, y_0, true) -> U63^1(tt, y_0, true) U63^1(tt, y_0, true) -> _ISNOTEQUALTO_(y_0, true) The TRS R consists of the following rules: U42(tt, U', U) -> if_then_else_fi(_isEqualTo_(U, U'), false, true) U32(tt, U', U) -> U33(equal(_isNotEqualTo_(U, U'), true)) U72(tt, U) -> U U41(tt, U', U) -> U42(tt, U', U) U33(tt) -> false U31(tt, U', U) -> U32(tt, U', U) U62(tt, B, U') -> U63(tt, B, U') _isNotEqualTo_(U, U') -> U41(tt, U', U) U63(tt, B, U') -> U64(equal(_isNotEqualTo_(B, true), true), U') equal(X, X) -> tt _isNotEqualTo_(U, U) -> false _isEqualTo_(U, U') -> U31(tt, U', U) if_then_else_fi(true, U, U') -> U71(tt, U) if_then_else_fi(B, U, U') -> U61(tt, B, U') U61(tt, B, U') -> U62(tt, B, U') U71(tt, U) -> U72(tt, U) _isEqualTo_(U, U) -> true U64(tt, U') -> U' Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (30) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by narrowing to the right: s = U32^1(tt, U'', U1) evaluates to t =U32^1(tt, U'', U1) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence U32^1(tt, U'', U1) -> _ISNOTEQUALTO_(U1, U'') with rule U32^1(tt, U', U) -> _ISNOTEQUALTO_(U, U') and matcher [U' / U'', U / U1]. _ISNOTEQUALTO_(U1, U'') -> U41^1(tt, U'', U1) with rule _ISNOTEQUALTO_(U''', U'''') -> U41^1(tt, U'''', U''') at position [] and matcher [U''' / U1, U'''' / U''] U41^1(tt, U'', U1) -> U42^1(tt, U'', U1) with rule U41^1(tt, U', U) -> U42^1(tt, U', U) at position [] and matcher [U' / U'', U / U1] U42^1(tt, U'', U1) -> _ISEQUALTO_(U1, U'') with rule U42^1(tt, U''', U1') -> _ISEQUALTO_(U1', U''') at position [] and matcher [U''' / U'', U1' / U1] _ISEQUALTO_(U1, U'') -> U31^1(tt, U'', U1) with rule _ISEQUALTO_(U, U') -> U31^1(tt, U', U) at position [] and matcher [U / U1, U' / U''] U31^1(tt, U'', U1) -> U32^1(tt, U'', U1) with rule U31^1(tt, U''', U1') -> U32^1(tt, U''', U1') at position [] and matcher [U''' / U'', U1' / U1] Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence All these steps are and every following step will be a correct step w.r.t to Q. ---------------------------------------- (31) NO ---------------------------------------- (32) Obligation: The TRS P consists of the following rules: U13^1(tt, A, B, C) -> _AND_(A, B) _AND_(_and_(A, _xor_(B, C)), ext) -> U11^1(tt, A, B, C) _AND_(_and_(A, A), ext) -> _AND_(A, ext) U12^1(tt, A, B, C) -> U13^1(tt, A, B, C) _AND_(A, _xor_(B, C)) -> U11^1(tt, A, B, C) U11^1(tt, A, B, C) -> U12^1(tt, A, B, C) _AND_(_and_(A, _xor_(B, C)), ext) -> _AND_(U11(tt, A, B, C), ext) _AND_(_and_(false, A), ext) -> _AND_(false, ext) U13^1(tt, A, B, C) -> _AND_(A, C) The TRS R consists of the following rules: U11(tt, A, B, C) -> U12(tt, A, B, C) U12(tt, A, B, C) -> U13(tt, A, B, C) U13(tt, A, B, C) -> _xor_(_and_(A, B), _and_(A, C)) U31(tt, U', U) -> U32(tt, U', U) U32(tt, U', U) -> U33(equal(_isNotEqualTo_(U, U'), true)) U33(tt) -> false U41(tt, U', U) -> U42(tt, U', U) U42(tt, U', U) -> if_then_else_fi(_isEqualTo_(U, U'), false, true) U61(tt, B, U') -> U62(tt, B, U') U62(tt, B, U') -> U63(tt, B, U') U63(tt, B, U') -> U64(equal(_isNotEqualTo_(B, true), true), U') U64(tt, U') -> U' U71(tt, U) -> U72(tt, U) U72(tt, U) -> U _and_(A, A) -> A _and_(A, _xor_(B, C)) -> U11(tt, A, B, C) _and_(false, A) -> false _and_(true, A) -> A _isEqualTo_(U, U') -> U31(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> U41(tt, U', U) _isNotEqualTo_(U, U) -> false equal(X, X) -> tt if_then_else_fi(B, U, U') -> U61(tt, B, U') if_then_else_fi(true, U, U') -> U71(tt, U) _and_(_and_(A, A), ext) -> _and_(A, ext) _and_(_and_(A, _xor_(B, C)), ext) -> _and_(U11(tt, A, B, C), ext) _and_(_and_(false, A), ext) -> _and_(false, 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) _AND_(_and_(x, y), z) == _AND_(x, _and_(y, z)) We have to consider all minimal (P,E#,R,E)-chains