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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ETRS could be disproven: (0) ETRS (1) RRRPoloETRSProof [EQUIVALENT, 195 ms] (2) ETRS (3) RRRPoloETRSProof [EQUIVALENT, 66 ms] (4) ETRS (5) RRRPoloETRSProof [EQUIVALENT, 58 ms] (6) ETRS (7) RRRPoloETRSProof [EQUIVALENT, 48 ms] (8) ETRS (9) RRRPoloETRSProof [EQUIVALENT, 63 ms] (10) ETRS (11) RRRPoloETRSProof [EQUIVALENT, 31 ms] (12) ETRS (13) EquationalDependencyPairsProof [EQUIVALENT, 10 ms] (14) EDP (15) EDependencyGraphProof [EQUIVALENT, 0 ms] (16) EDP (17) EUsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (18) EDP (19) EDPProblemToQDPProblemProof [EQUIVALENT, 0 ms] (20) QDP (21) TransformationProof [EQUIVALENT, 0 ms] (22) QDP (23) TransformationProof [EQUIVALENT, 0 ms] (24) QDP (25) NonTerminationLoopProof [COMPLETE, 5 ms] (26) NO ---------------------------------------- (0) Obligation: Equational rewrite system: The TRS R consists of the following rules: U11(tt, U', U) -> U12(equal(_isNotEqualTo_(U, U'), true)) U12(tt) -> false U21(tt, B, U') -> U22(equal(_isNotEqualTo_(B, true), true), U') U22(tt, U') -> U' _and_(A, A) -> A _and_(A, _xor_(B, C)) -> _xor_(_and_(A, B), _and_(A, C)) _and_(false, A) -> false _and_(true, A) -> A _implies_(A, B) -> not_(_xor_(A, _and_(A, B))) _isEqualTo_(U, U') -> U11(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> if_then_else_fi(_isEqualTo_(U, U'), false, true) _isNotEqualTo_(U, U) -> false _or_(A, B) -> _xor_(_and_(A, B), _xor_(A, B)) _xor_(A, A) -> false _xor_(false, A) -> A and(tt, X) -> X equal(X, X) -> tt if_then_else_fi(B, U, U') -> U21(tt, B, U') if_then_else_fi(true, U, U') -> 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, U', U) -> U12(equal(_isNotEqualTo_(U, U'), true)) U12(tt) -> false U21(tt, B, U') -> U22(equal(_isNotEqualTo_(B, true), true), U') U22(tt, U') -> U' _and_(A, A) -> A _and_(A, _xor_(B, C)) -> _xor_(_and_(A, B), _and_(A, C)) _and_(false, A) -> false _and_(true, A) -> A _implies_(A, B) -> not_(_xor_(A, _and_(A, B))) _isEqualTo_(U, U') -> U11(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> if_then_else_fi(_isEqualTo_(U, U'), false, true) _isNotEqualTo_(U, U) -> false _or_(A, B) -> _xor_(_and_(A, B), _xor_(A, B)) _xor_(A, A) -> false _xor_(false, A) -> A and(tt, X) -> X equal(X, X) -> tt if_then_else_fi(B, U, U') -> U21(tt, B, U') if_then_else_fi(true, U, U') -> 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: _and_(A, A) -> A _and_(false, A) -> false _and_(true, A) -> A _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)) = 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(U12(x_1)) = x_1 POL(U21(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(U22(x_1, x_2)) = x_1 + 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)) = 3 + 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)) = 3 + 3*x_1 + 2*x_1*x_2 + 3*x_2 POL(_xor_(x_1, x_2)) = 1 + x_1 + x_2 POL(and(x_1, x_2)) = 3*x_1 + 3*x_1*x_2 + 3*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 + 3*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, U', U) -> U12(equal(_isNotEqualTo_(U, U'), true)) U12(tt) -> false U21(tt, B, U') -> U22(equal(_isNotEqualTo_(B, true), true), U') U22(tt, U') -> U' _and_(A, _xor_(B, C)) -> _xor_(_and_(A, B), _and_(A, C)) _implies_(A, B) -> not_(_xor_(A, _and_(A, B))) _isEqualTo_(U, U') -> U11(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> if_then_else_fi(_isEqualTo_(U, U'), false, true) _isNotEqualTo_(U, U) -> false _or_(A, B) -> _xor_(_and_(A, B), _xor_(A, B)) and(tt, X) -> X equal(X, X) -> tt if_then_else_fi(B, U, U') -> U21(tt, B, U') if_then_else_fi(true, U, U') -> 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, U', U) -> U12(equal(_isNotEqualTo_(U, U'), true)) U12(tt) -> false U21(tt, B, U') -> U22(equal(_isNotEqualTo_(B, true), true), U') U22(tt, U') -> U' _and_(A, _xor_(B, C)) -> _xor_(_and_(A, B), _and_(A, C)) _implies_(A, B) -> not_(_xor_(A, _and_(A, B))) _isEqualTo_(U, U') -> U11(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> if_then_else_fi(_isEqualTo_(U, U'), false, true) _isNotEqualTo_(U, U) -> false _or_(A, B) -> _xor_(_and_(A, B), _xor_(A, B)) and(tt, X) -> X equal(X, X) -> tt if_then_else_fi(B, U, U') -> U21(tt, B, U') if_then_else_fi(true, U, U') -> 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: not_(A) -> _xor_(A, true) Used ordering: Polynomial interpretation [POLO]: POL(U11(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(U12(x_1)) = x_1 POL(U21(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(U22(x_1, x_2)) = x_1 + x_2 POL(_and_(x_1, x_2)) = x_1 + x_1*x_2 + x_2 POL(_implies_(x_1, x_2)) = 3 + 3*x_1 + 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)) = 2 + 2*x_1 + x_1*x_2 + 2*x_2 POL(_xor_(x_1, x_2)) = 1 + x_1 + x_2 POL(and(x_1, x_2)) = 3*x_1 + 3*x_1*x_2 + 3*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(not_(x_1)) = 2 + x_1 POL(true) = 0 POL(tt) = 0 ---------------------------------------- (4) Obligation: Equational rewrite system: The TRS R consists of the following rules: U11(tt, U', U) -> U12(equal(_isNotEqualTo_(U, U'), true)) U12(tt) -> false U21(tt, B, U') -> U22(equal(_isNotEqualTo_(B, true), true), U') U22(tt, U') -> U' _and_(A, _xor_(B, C)) -> _xor_(_and_(A, B), _and_(A, C)) _implies_(A, B) -> not_(_xor_(A, _and_(A, B))) _isEqualTo_(U, U') -> U11(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> if_then_else_fi(_isEqualTo_(U, U'), false, true) _isNotEqualTo_(U, U) -> false _or_(A, B) -> _xor_(_and_(A, B), _xor_(A, B)) and(tt, X) -> X equal(X, X) -> tt if_then_else_fi(B, U, U') -> U21(tt, B, U') if_then_else_fi(true, U, U') -> 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)) ---------------------------------------- (5) RRRPoloETRSProof (EQUIVALENT) The following E TRS is given: Equational rewrite system: The TRS R consists of the following rules: U11(tt, U', U) -> U12(equal(_isNotEqualTo_(U, U'), true)) U12(tt) -> false U21(tt, B, U') -> U22(equal(_isNotEqualTo_(B, true), true), U') U22(tt, U') -> U' _and_(A, _xor_(B, C)) -> _xor_(_and_(A, B), _and_(A, C)) _implies_(A, B) -> not_(_xor_(A, _and_(A, B))) _isEqualTo_(U, U') -> U11(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> if_then_else_fi(_isEqualTo_(U, U'), false, true) _isNotEqualTo_(U, U) -> false _or_(A, B) -> _xor_(_and_(A, B), _xor_(A, B)) and(tt, X) -> X equal(X, X) -> tt if_then_else_fi(B, U, U') -> U21(tt, B, U') if_then_else_fi(true, U, U') -> 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: _implies_(A, B) -> not_(_xor_(A, _and_(A, B))) Used ordering: Polynomial interpretation [POLO]: POL(U11(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(U12(x_1)) = x_1 POL(U21(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(U22(x_1, x_2)) = x_1 + x_2 POL(_and_(x_1, x_2)) = x_1 + x_1*x_2 + x_2 POL(_implies_(x_1, x_2)) = 3 + 3*x_1 + 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)) = 2 + 2*x_1 + x_1*x_2 + 2*x_2 POL(_xor_(x_1, x_2)) = 1 + x_1 + x_2 POL(and(x_1, x_2)) = 3*x_1 + 2*x_1*x_2 + 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(not_(x_1)) = 1 + x_1 POL(true) = 0 POL(tt) = 0 ---------------------------------------- (6) Obligation: Equational rewrite system: The TRS R consists of the following rules: U11(tt, U', U) -> U12(equal(_isNotEqualTo_(U, U'), true)) U12(tt) -> false U21(tt, B, U') -> U22(equal(_isNotEqualTo_(B, true), true), U') U22(tt, U') -> U' _and_(A, _xor_(B, C)) -> _xor_(_and_(A, B), _and_(A, C)) _isEqualTo_(U, U') -> U11(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> if_then_else_fi(_isEqualTo_(U, U'), false, true) _isNotEqualTo_(U, U) -> false _or_(A, B) -> _xor_(_and_(A, B), _xor_(A, B)) and(tt, X) -> X equal(X, X) -> tt if_then_else_fi(B, U, U') -> U21(tt, B, U') if_then_else_fi(true, U, U') -> 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, U', U) -> U12(equal(_isNotEqualTo_(U, U'), true)) U12(tt) -> false U21(tt, B, U') -> U22(equal(_isNotEqualTo_(B, true), true), U') U22(tt, U') -> U' _and_(A, _xor_(B, C)) -> _xor_(_and_(A, B), _and_(A, C)) _isEqualTo_(U, U') -> U11(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> if_then_else_fi(_isEqualTo_(U, U'), false, true) _isNotEqualTo_(U, U) -> false _or_(A, B) -> _xor_(_and_(A, B), _xor_(A, B)) and(tt, X) -> X equal(X, X) -> tt if_then_else_fi(B, U, U') -> U21(tt, B, U') if_then_else_fi(true, U, U') -> 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: and(tt, X) -> X Used ordering: Polynomial interpretation [POLO]: POL(U11(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(U12(x_1)) = x_1 POL(U21(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(U22(x_1, x_2)) = x_1 + x_2 POL(_and_(x_1, x_2)) = x_1 + 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)) = 2 + 2*x_1 + x_1*x_2 + 2*x_2 POL(_xor_(x_1, x_2)) = 1 + x_1 + x_2 POL(and(x_1, x_2)) = 2 + 3*x_1 + 3*x_1*x_2 + 3*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 + 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, U', U) -> U12(equal(_isNotEqualTo_(U, U'), true)) U12(tt) -> false U21(tt, B, U') -> U22(equal(_isNotEqualTo_(B, true), true), U') U22(tt, U') -> U' _and_(A, _xor_(B, C)) -> _xor_(_and_(A, B), _and_(A, C)) _isEqualTo_(U, U') -> U11(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> if_then_else_fi(_isEqualTo_(U, U'), false, true) _isNotEqualTo_(U, U) -> false _or_(A, B) -> _xor_(_and_(A, B), _xor_(A, B)) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U21(tt, B, U') if_then_else_fi(true, U, U') -> 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, U', U) -> U12(equal(_isNotEqualTo_(U, U'), true)) U12(tt) -> false U21(tt, B, U') -> U22(equal(_isNotEqualTo_(B, true), true), U') U22(tt, U') -> U' _and_(A, _xor_(B, C)) -> _xor_(_and_(A, B), _and_(A, C)) _isEqualTo_(U, U') -> U11(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> if_then_else_fi(_isEqualTo_(U, U'), false, true) _isNotEqualTo_(U, U) -> false _or_(A, B) -> _xor_(_and_(A, B), _xor_(A, B)) equal(X, X) -> tt if_then_else_fi(B, U, U') -> U21(tt, B, U') if_then_else_fi(true, U, U') -> 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) -> _xor_(_and_(A, B), _xor_(A, B)) Used ordering: Polynomial interpretation [POLO]: POL(U11(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(U12(x_1)) = x_1 POL(U21(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(U22(x_1, x_2)) = x_1 + x_2 POL(_and_(x_1, x_2)) = x_1 + 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 + 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, U', U) -> U12(equal(_isNotEqualTo_(U, U'), true)) U12(tt) -> false U21(tt, B, U') -> U22(equal(_isNotEqualTo_(B, true), true), U') U22(tt, U') -> U' _and_(A, _xor_(B, C)) -> _xor_(_and_(A, B), _and_(A, C)) _isEqualTo_(U, U') -> U11(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> if_then_else_fi(_isEqualTo_(U, U'), false, true) _isNotEqualTo_(U, U) -> false equal(X, X) -> tt if_then_else_fi(B, U, U') -> U21(tt, B, U') if_then_else_fi(true, U, U') -> 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) RRRPoloETRSProof (EQUIVALENT) The following E TRS is given: Equational rewrite system: The TRS R consists of the following rules: U11(tt, U', U) -> U12(equal(_isNotEqualTo_(U, U'), true)) U12(tt) -> false U21(tt, B, U') -> U22(equal(_isNotEqualTo_(B, true), true), U') U22(tt, U') -> U' _and_(A, _xor_(B, C)) -> _xor_(_and_(A, B), _and_(A, C)) _isEqualTo_(U, U') -> U11(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> if_then_else_fi(_isEqualTo_(U, U'), false, true) _isNotEqualTo_(U, U) -> false equal(X, X) -> tt if_then_else_fi(B, U, U') -> U21(tt, B, U') if_then_else_fi(true, U, U') -> 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: _and_(A, _xor_(B, C)) -> _xor_(_and_(A, B), _and_(A, C)) Used ordering: Polynomial interpretation [POLO]: POL(U11(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(U12(x_1)) = x_1 POL(U21(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(U22(x_1, x_2)) = x_1 + x_2 POL(_and_(x_1, x_2)) = 2 + 2*x_1 + x_1*x_2 + 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 + 2*x_1 + 2*x_1*x_2 + 2*x_2 POL(_xor_(x_1, x_2)) = 3 + 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(true) = 0 POL(tt) = 0 ---------------------------------------- (12) Obligation: Equational rewrite system: The TRS R consists of the following rules: U11(tt, U', U) -> U12(equal(_isNotEqualTo_(U, U'), true)) U12(tt) -> false U21(tt, B, U') -> U22(equal(_isNotEqualTo_(B, true), true), U') U22(tt, U') -> U' _isEqualTo_(U, U') -> U11(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> if_then_else_fi(_isEqualTo_(U, U'), false, true) _isNotEqualTo_(U, U) -> false equal(X, X) -> tt if_then_else_fi(B, U, U') -> U21(tt, B, U') if_then_else_fi(true, U, U') -> 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)) ---------------------------------------- (13) 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, U', U) -> U12^1(equal(_isNotEqualTo_(U, U'), true)) U11^1(tt, U', U) -> EQUAL(_isNotEqualTo_(U, U'), true) U11^1(tt, U', U) -> _ISNOTEQUALTO_(U, U') U21^1(tt, B, U') -> U22^1(equal(_isNotEqualTo_(B, true), true), U') U21^1(tt, B, U') -> EQUAL(_isNotEqualTo_(B, true), true) U21^1(tt, B, U') -> _ISNOTEQUALTO_(B, true) _ISEQUALTO_(U, U') -> U11^1(tt, U', U) _ISNOTEQUALTO_(U, U') -> IF_THEN_ELSE_FI(_isEqualTo_(U, U'), false, true) _ISNOTEQUALTO_(U, U') -> _ISEQUALTO_(U, U') IF_THEN_ELSE_FI(B, U, U') -> U21^1(tt, B, U') The TRS R consists of the following rules: U11(tt, U', U) -> U12(equal(_isNotEqualTo_(U, U'), true)) U12(tt) -> false U21(tt, B, U') -> U22(equal(_isNotEqualTo_(B, true), true), U') U22(tt, U') -> U' _isEqualTo_(U, U') -> U11(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> if_then_else_fi(_isEqualTo_(U, U'), false, true) _isNotEqualTo_(U, U) -> false equal(X, X) -> tt if_then_else_fi(B, U, U') -> U21(tt, B, U') if_then_else_fi(true, U, U') -> 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)) E# is empty. We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (14) Obligation: The TRS P consists of the following rules: U11^1(tt, U', U) -> U12^1(equal(_isNotEqualTo_(U, U'), true)) U11^1(tt, U', U) -> EQUAL(_isNotEqualTo_(U, U'), true) U11^1(tt, U', U) -> _ISNOTEQUALTO_(U, U') U21^1(tt, B, U') -> U22^1(equal(_isNotEqualTo_(B, true), true), U') U21^1(tt, B, U') -> EQUAL(_isNotEqualTo_(B, true), true) U21^1(tt, B, U') -> _ISNOTEQUALTO_(B, true) _ISEQUALTO_(U, U') -> U11^1(tt, U', U) _ISNOTEQUALTO_(U, U') -> IF_THEN_ELSE_FI(_isEqualTo_(U, U'), false, true) _ISNOTEQUALTO_(U, U') -> _ISEQUALTO_(U, U') IF_THEN_ELSE_FI(B, U, U') -> U21^1(tt, B, U') The TRS R consists of the following rules: U11(tt, U', U) -> U12(equal(_isNotEqualTo_(U, U'), true)) U12(tt) -> false U21(tt, B, U') -> U22(equal(_isNotEqualTo_(B, true), true), U') U22(tt, U') -> U' _isEqualTo_(U, U') -> U11(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> if_then_else_fi(_isEqualTo_(U, U'), false, true) _isNotEqualTo_(U, U) -> false equal(X, X) -> tt if_then_else_fi(B, U, U') -> U21(tt, B, U') if_then_else_fi(true, U, U') -> 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)) E# is empty. We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (15) EDependencyGraphProof (EQUIVALENT) The approximation of the Equational Dependency Graph [DA_STEIN] contains 1 SCC with 4 less nodes. ---------------------------------------- (16) Obligation: The TRS P consists of the following rules: IF_THEN_ELSE_FI(B, U, U') -> U21^1(tt, B, U') _ISEQUALTO_(U, U') -> U11^1(tt, U', U) _ISNOTEQUALTO_(U, U') -> IF_THEN_ELSE_FI(_isEqualTo_(U, U'), false, true) U21^1(tt, B, U') -> _ISNOTEQUALTO_(B, true) U11^1(tt, U', U) -> _ISNOTEQUALTO_(U, U') _ISNOTEQUALTO_(U, U') -> _ISEQUALTO_(U, U') The TRS R consists of the following rules: U11(tt, U', U) -> U12(equal(_isNotEqualTo_(U, U'), true)) U12(tt) -> false U21(tt, B, U') -> U22(equal(_isNotEqualTo_(B, true), true), U') U22(tt, U') -> U' _isEqualTo_(U, U') -> U11(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> if_then_else_fi(_isEqualTo_(U, U'), false, true) _isNotEqualTo_(U, U) -> false equal(X, X) -> tt if_then_else_fi(B, U, U') -> U21(tt, B, U') if_then_else_fi(true, U, U') -> 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)) E# is empty. We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (17) 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. No rules are removed from R. 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(U11(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3 POL(U11^1(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U12(x_1)) = x_1 POL(U21(x_1, x_2, x_3)) = 2*x_1 + x_2 + 2*x_3 POL(U21^1(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U22(x_1, x_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 ---------------------------------------- (18) Obligation: The TRS P consists of the following rules: IF_THEN_ELSE_FI(B, U, U') -> U21^1(tt, B, U') _ISEQUALTO_(U, U') -> U11^1(tt, U', U) _ISNOTEQUALTO_(U, U') -> IF_THEN_ELSE_FI(_isEqualTo_(U, U'), false, true) U21^1(tt, B, U') -> _ISNOTEQUALTO_(B, true) U11^1(tt, U', U) -> _ISNOTEQUALTO_(U, U') _ISNOTEQUALTO_(U, U') -> _ISEQUALTO_(U, U') The TRS R consists of the following rules: U11(tt, U', U) -> U12(equal(_isNotEqualTo_(U, U'), true)) U12(tt) -> false U21(tt, B, U') -> U22(equal(_isNotEqualTo_(B, true), true), U') U22(tt, U') -> U' _isEqualTo_(U, U') -> U11(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> if_then_else_fi(_isEqualTo_(U, U'), false, true) _isNotEqualTo_(U, U) -> false equal(X, X) -> tt if_then_else_fi(B, U, U') -> U21(tt, B, U') if_then_else_fi(true, U, U') -> U E is empty. E# is empty. We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (19) 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. ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: IF_THEN_ELSE_FI(B, U, U') -> U21^1(tt, B, U') _ISEQUALTO_(U, U') -> U11^1(tt, U', U) _ISNOTEQUALTO_(U, U') -> IF_THEN_ELSE_FI(_isEqualTo_(U, U'), false, true) U21^1(tt, B, U') -> _ISNOTEQUALTO_(B, true) U11^1(tt, U', U) -> _ISNOTEQUALTO_(U, U') _ISNOTEQUALTO_(U, U') -> _ISEQUALTO_(U, U') The TRS R consists of the following rules: U11(tt, U', U) -> U12(equal(_isNotEqualTo_(U, U'), true)) U12(tt) -> false U21(tt, B, U') -> U22(equal(_isNotEqualTo_(B, true), true), U') U22(tt, U') -> U' _isEqualTo_(U, U') -> U11(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> if_then_else_fi(_isEqualTo_(U, U'), false, true) _isNotEqualTo_(U, U) -> false equal(X, X) -> tt if_then_else_fi(B, U, U') -> U21(tt, B, U') if_then_else_fi(true, U, U') -> U Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF_THEN_ELSE_FI(B, U, U') -> U21^1(tt, B, U') we obtained the following new rules [LPAR04]: (IF_THEN_ELSE_FI(y_2, false, true) -> U21^1(tt, y_2, true),IF_THEN_ELSE_FI(y_2, false, true) -> U21^1(tt, y_2, true)) ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: _ISEQUALTO_(U, U') -> U11^1(tt, U', U) _ISNOTEQUALTO_(U, U') -> IF_THEN_ELSE_FI(_isEqualTo_(U, U'), false, true) U21^1(tt, B, U') -> _ISNOTEQUALTO_(B, true) U11^1(tt, U', U) -> _ISNOTEQUALTO_(U, U') _ISNOTEQUALTO_(U, U') -> _ISEQUALTO_(U, U') IF_THEN_ELSE_FI(y_2, false, true) -> U21^1(tt, y_2, true) The TRS R consists of the following rules: U11(tt, U', U) -> U12(equal(_isNotEqualTo_(U, U'), true)) U12(tt) -> false U21(tt, B, U') -> U22(equal(_isNotEqualTo_(B, true), true), U') U22(tt, U') -> U' _isEqualTo_(U, U') -> U11(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> if_then_else_fi(_isEqualTo_(U, U'), false, true) _isNotEqualTo_(U, U) -> false equal(X, X) -> tt if_then_else_fi(B, U, U') -> U21(tt, B, U') if_then_else_fi(true, U, U') -> U Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U21^1(tt, B, U') -> _ISNOTEQUALTO_(B, true) we obtained the following new rules [LPAR04]: (U21^1(tt, y_0, true) -> _ISNOTEQUALTO_(y_0, true),U21^1(tt, y_0, true) -> _ISNOTEQUALTO_(y_0, true)) ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: _ISEQUALTO_(U, U') -> U11^1(tt, U', U) _ISNOTEQUALTO_(U, U') -> IF_THEN_ELSE_FI(_isEqualTo_(U, U'), false, true) U11^1(tt, U', U) -> _ISNOTEQUALTO_(U, U') _ISNOTEQUALTO_(U, U') -> _ISEQUALTO_(U, U') IF_THEN_ELSE_FI(y_2, false, true) -> U21^1(tt, y_2, true) U21^1(tt, y_0, true) -> _ISNOTEQUALTO_(y_0, true) The TRS R consists of the following rules: U11(tt, U', U) -> U12(equal(_isNotEqualTo_(U, U'), true)) U12(tt) -> false U21(tt, B, U') -> U22(equal(_isNotEqualTo_(B, true), true), U') U22(tt, U') -> U' _isEqualTo_(U, U') -> U11(tt, U', U) _isEqualTo_(U, U) -> true _isNotEqualTo_(U, U') -> if_then_else_fi(_isEqualTo_(U, U'), false, true) _isNotEqualTo_(U, U) -> false equal(X, X) -> tt if_then_else_fi(B, U, U') -> U21(tt, B, U') if_then_else_fi(true, U, U') -> U Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) 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 = _ISEQUALTO_(U, U') evaluates to t =_ISEQUALTO_(U, U') Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence _ISEQUALTO_(U, U') -> U11^1(tt, U', U) with rule _ISEQUALTO_(U, U') -> U11^1(tt, U', U) and matcher [ ]. U11^1(tt, U', U) -> _ISNOTEQUALTO_(U, U') with rule U11^1(tt, U'', U1) -> _ISNOTEQUALTO_(U1, U'') at position [] and matcher [U'' / U', U1 / U] _ISNOTEQUALTO_(U, U') -> _ISEQUALTO_(U, U') with rule _ISNOTEQUALTO_(U'', U''') -> _ISEQUALTO_(U'', U''') at position [] and matcher [U'' / U, U''' / U'] 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. ---------------------------------------- (26) NO