MAYBE * Step 1: WeightGap MAYBE + Considered Problem: - Strict TRS: a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U11(tt(),M,N) -> a__U12(tt(),M,N) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> a__U11(tt(),M,N) a__plus(X1,X2) -> plus(X1,X2) mark(0()) -> 0() mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() - Signature: {a__U11/3,a__U12/3,a__plus/2,mark/1} / {0/0,U11/3,U12/3,plus/2,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U12,a__plus,mark} and constructors {0,U11,U12 ,plus,s,tt} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U12) = {1}, uargs(a__plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [1] x1 + [0] p(U12) = [1] x1 + [0] p(a__U11) = [1] x1 + [0] p(a__U12) = [1] x1 + [0] p(a__plus) = [1] x1 + [1] x2 + [0] p(mark) = [7] p(plus) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [0] p(tt) = [0] Following rules are strictly oriented: mark(0()) = [7] > [0] = 0() mark(tt()) = [7] > [0] = tt() Following rules are (at-least) weakly oriented: a__U11(X1,X2,X3) = [1] X1 + [0] >= [1] X1 + [0] = U11(X1,X2,X3) a__U11(tt(),M,N) = [0] >= [0] = a__U12(tt(),M,N) a__U12(X1,X2,X3) = [1] X1 + [0] >= [1] X1 + [0] = U12(X1,X2,X3) a__U12(tt(),M,N) = [0] >= [14] = s(a__plus(mark(N),mark(M))) a__plus(N,0()) = [1] N + [0] >= [7] = mark(N) a__plus(N,s(M)) = [1] M + [1] N + [0] >= [0] = a__U11(tt(),M,N) a__plus(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = plus(X1,X2) mark(U11(X1,X2,X3)) = [7] >= [7] = a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) = [7] >= [7] = a__U12(mark(X1),X2,X3) mark(plus(X1,X2)) = [7] >= [14] = a__plus(mark(X1),mark(X2)) mark(s(X)) = [7] >= [7] = s(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap MAYBE + Considered Problem: - Strict TRS: a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U11(tt(),M,N) -> a__U12(tt(),M,N) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> a__U11(tt(),M,N) a__plus(X1,X2) -> plus(X1,X2) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) - Weak TRS: mark(0()) -> 0() mark(tt()) -> tt() - Signature: {a__U11/3,a__U12/3,a__plus/2,mark/1} / {0/0,U11/3,U12/3,plus/2,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U12,a__plus,mark} and constructors {0,U11,U12 ,plus,s,tt} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U12) = {1}, uargs(a__plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(U11) = [1] x1 + [1] x2 + [1] x3 + [2] p(U12) = [1] x1 + [1] x2 + [1] x3 + [1] p(a__U11) = [1] x1 + [1] x2 + [1] x3 + [0] p(a__U12) = [1] x1 + [1] x2 + [1] x3 + [1] p(a__plus) = [1] x1 + [1] x2 + [1] p(mark) = [1] x1 + [4] p(plus) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [0] p(tt) = [7] Following rules are strictly oriented: a__plus(X1,X2) = [1] X1 + [1] X2 + [1] > [1] X1 + [1] X2 + [0] = plus(X1,X2) mark(U11(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [6] > [1] X1 + [1] X2 + [1] X3 + [4] = a__U11(mark(X1),X2,X3) Following rules are (at-least) weakly oriented: a__U11(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [2] = U11(X1,X2,X3) a__U11(tt(),M,N) = [1] M + [1] N + [7] >= [1] M + [1] N + [8] = a__U12(tt(),M,N) a__U12(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [1] >= [1] X1 + [1] X2 + [1] X3 + [1] = U12(X1,X2,X3) a__U12(tt(),M,N) = [1] M + [1] N + [8] >= [1] M + [1] N + [9] = s(a__plus(mark(N),mark(M))) a__plus(N,0()) = [1] N + [3] >= [1] N + [4] = mark(N) a__plus(N,s(M)) = [1] M + [1] N + [1] >= [1] M + [1] N + [7] = a__U11(tt(),M,N) mark(0()) = [6] >= [2] = 0() mark(U12(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [5] >= [1] X1 + [1] X2 + [1] X3 + [5] = a__U12(mark(X1),X2,X3) mark(plus(X1,X2)) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [9] = a__plus(mark(X1),mark(X2)) mark(s(X)) = [1] X + [4] >= [1] X + [4] = s(mark(X)) mark(tt()) = [11] >= [7] = tt() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap MAYBE + Considered Problem: - Strict TRS: a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U11(tt(),M,N) -> a__U12(tt(),M,N) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> a__U11(tt(),M,N) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__plus(X1,X2) -> plus(X1,X2) mark(0()) -> 0() mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(tt()) -> tt() - Signature: {a__U11/3,a__U12/3,a__plus/2,mark/1} / {0/0,U11/3,U12/3,plus/2,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U12,a__plus,mark} and constructors {0,U11,U12 ,plus,s,tt} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U12) = {1}, uargs(a__plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [4] p(U11) = [1] x1 + [1] x2 + [1] x3 + [2] p(U12) = [1] x1 + [1] x2 + [1] x3 + [4] p(a__U11) = [1] x1 + [1] x2 + [1] x3 + [0] p(a__U12) = [1] x1 + [1] x2 + [1] x3 + [2] p(a__plus) = [1] x1 + [1] x2 + [4] p(mark) = [1] x1 + [0] p(plus) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [5] p(tt) = [0] Following rules are strictly oriented: a__plus(N,0()) = [1] N + [8] > [1] N + [0] = mark(N) a__plus(N,s(M)) = [1] M + [1] N + [9] > [1] M + [1] N + [0] = a__U11(tt(),M,N) mark(U12(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [4] > [1] X1 + [1] X2 + [1] X3 + [2] = a__U12(mark(X1),X2,X3) Following rules are (at-least) weakly oriented: a__U11(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [2] = U11(X1,X2,X3) a__U11(tt(),M,N) = [1] M + [1] N + [0] >= [1] M + [1] N + [2] = a__U12(tt(),M,N) a__U12(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [2] >= [1] X1 + [1] X2 + [1] X3 + [4] = U12(X1,X2,X3) a__U12(tt(),M,N) = [1] M + [1] N + [2] >= [1] M + [1] N + [9] = s(a__plus(mark(N),mark(M))) a__plus(X1,X2) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [0] = plus(X1,X2) mark(0()) = [4] >= [4] = 0() mark(U11(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [2] >= [1] X1 + [1] X2 + [1] X3 + [0] = a__U11(mark(X1),X2,X3) mark(plus(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [4] = a__plus(mark(X1),mark(X2)) mark(s(X)) = [1] X + [5] >= [1] X + [5] = s(mark(X)) mark(tt()) = [0] >= [0] = tt() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap MAYBE + Considered Problem: - Strict TRS: a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U11(tt(),M,N) -> a__U12(tt(),M,N) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M))) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> a__U11(tt(),M,N) a__plus(X1,X2) -> plus(X1,X2) mark(0()) -> 0() mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(tt()) -> tt() - Signature: {a__U11/3,a__U12/3,a__plus/2,mark/1} / {0/0,U11/3,U12/3,plus/2,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U12,a__plus,mark} and constructors {0,U11,U12 ,plus,s,tt} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U12) = {1}, uargs(a__plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [4] p(U11) = [1] x1 + [1] x2 + [1] x3 + [1] p(U12) = [1] x1 + [1] x2 + [1] x3 + [0] p(a__U11) = [1] x1 + [1] x2 + [1] x3 + [1] p(a__U12) = [1] x1 + [1] x2 + [1] x3 + [0] p(a__plus) = [1] x1 + [1] x2 + [1] p(mark) = [1] x1 + [0] p(plus) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [0] p(tt) = [0] Following rules are strictly oriented: a__U11(tt(),M,N) = [1] M + [1] N + [1] > [1] M + [1] N + [0] = a__U12(tt(),M,N) Following rules are (at-least) weakly oriented: a__U11(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [1] >= [1] X1 + [1] X2 + [1] X3 + [1] = U11(X1,X2,X3) a__U12(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = U12(X1,X2,X3) a__U12(tt(),M,N) = [1] M + [1] N + [0] >= [1] M + [1] N + [1] = s(a__plus(mark(N),mark(M))) a__plus(N,0()) = [1] N + [5] >= [1] N + [0] = mark(N) a__plus(N,s(M)) = [1] M + [1] N + [1] >= [1] M + [1] N + [1] = a__U11(tt(),M,N) a__plus(X1,X2) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [0] = plus(X1,X2) mark(0()) = [4] >= [4] = 0() mark(U11(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [1] >= [1] X1 + [1] X2 + [1] X3 + [1] = a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = a__U12(mark(X1),X2,X3) mark(plus(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [1] = a__plus(mark(X1),mark(X2)) mark(s(X)) = [1] X + [0] >= [1] X + [0] = s(mark(X)) mark(tt()) = [0] >= [0] = tt() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: NaturalMI MAYBE + Considered Problem: - Strict TRS: a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M))) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__U11(tt(),M,N) -> a__U12(tt(),M,N) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> a__U11(tt(),M,N) a__plus(X1,X2) -> plus(X1,X2) mark(0()) -> 0() mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(tt()) -> tt() - Signature: {a__U11/3,a__U12/3,a__plus/2,mark/1} / {0/0,U11/3,U12/3,plus/2,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U12,a__plus,mark} and constructors {0,U11,U12 ,plus,s,tt} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__U11) = {1}, uargs(a__U12) = {1}, uargs(a__plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {a__U11,a__U12,a__plus,mark} TcT has computed the following interpretation: p(0) = [0] [2] p(U11) = [1 6] x1 + [1 4] x2 + [1 0] x3 + [3] [0 1] [0 1] [0 1] [4] p(U12) = [1 4] x1 + [1 4] x2 + [1 0] x3 + [0] [0 0] [0 1] [0 1] [6] p(a__U11) = [1 6] x1 + [1 4] x2 + [1 0] x3 + [3] [0 1] [0 1] [0 1] [4] p(a__U12) = [1 4] x1 + [1 4] x2 + [1 0] x3 + [0] [0 0] [0 1] [0 1] [6] p(a__plus) = [1 0] x1 + [1 4] x2 + [0] [0 1] [0 1] [4] p(mark) = [1 0] x1 + [0] [0 1] [0] p(plus) = [1 0] x1 + [1 4] x2 + [0] [0 1] [0 1] [4] p(s) = [1 0] x1 + [7] [0 1] [2] p(tt) = [0] [2] Following rules are strictly oriented: a__U12(tt(),M,N) = [1 4] M + [1 0] N + [8] [0 1] [0 1] [6] > [1 4] M + [1 0] N + [7] [0 1] [0 1] [6] = s(a__plus(mark(N),mark(M))) Following rules are (at-least) weakly oriented: a__U11(X1,X2,X3) = [1 6] X1 + [1 4] X2 + [1 0] X3 + [3] [0 1] [0 1] [0 1] [4] >= [1 6] X1 + [1 4] X2 + [1 0] X3 + [3] [0 1] [0 1] [0 1] [4] = U11(X1,X2,X3) a__U11(tt(),M,N) = [1 4] M + [1 0] N + [15] [0 1] [0 1] [6] >= [1 4] M + [1 0] N + [8] [0 1] [0 1] [6] = a__U12(tt(),M,N) a__U12(X1,X2,X3) = [1 4] X1 + [1 4] X2 + [1 0] X3 + [0] [0 0] [0 1] [0 1] [6] >= [1 4] X1 + [1 4] X2 + [1 0] X3 + [0] [0 0] [0 1] [0 1] [6] = U12(X1,X2,X3) a__plus(N,0()) = [1 0] N + [8] [0 1] [6] >= [1 0] N + [0] [0 1] [0] = mark(N) a__plus(N,s(M)) = [1 4] M + [1 0] N + [15] [0 1] [0 1] [6] >= [1 4] M + [1 0] N + [15] [0 1] [0 1] [6] = a__U11(tt(),M,N) a__plus(X1,X2) = [1 0] X1 + [1 4] X2 + [0] [0 1] [0 1] [4] >= [1 0] X1 + [1 4] X2 + [0] [0 1] [0 1] [4] = plus(X1,X2) mark(0()) = [0] [2] >= [0] [2] = 0() mark(U11(X1,X2,X3)) = [1 6] X1 + [1 4] X2 + [1 0] X3 + [3] [0 1] [0 1] [0 1] [4] >= [1 6] X1 + [1 4] X2 + [1 0] X3 + [3] [0 1] [0 1] [0 1] [4] = a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) = [1 4] X1 + [1 4] X2 + [1 0] X3 + [0] [0 0] [0 1] [0 1] [6] >= [1 4] X1 + [1 4] X2 + [1 0] X3 + [0] [0 0] [0 1] [0 1] [6] = a__U12(mark(X1),X2,X3) mark(plus(X1,X2)) = [1 0] X1 + [1 4] X2 + [0] [0 1] [0 1] [4] >= [1 0] X1 + [1 4] X2 + [0] [0 1] [0 1] [4] = a__plus(mark(X1),mark(X2)) mark(s(X)) = [1 0] X + [7] [0 1] [2] >= [1 0] X + [7] [0 1] [2] = s(mark(X)) mark(tt()) = [0] [2] >= [0] [2] = tt() * Step 6: Failure MAYBE + Considered Problem: - Strict TRS: a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__U11(tt(),M,N) -> a__U12(tt(),M,N) a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> a__U11(tt(),M,N) a__plus(X1,X2) -> plus(X1,X2) mark(0()) -> 0() mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(tt()) -> tt() - Signature: {a__U11/3,a__U12/3,a__plus/2,mark/1} / {0/0,U11/3,U12/3,plus/2,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U12,a__plus,mark} and constructors {0,U11,U12 ,plus,s,tt} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE