MAYBE * Step 1: WeightGap MAYBE + Considered Problem: - Strict TRS: a__f(X) -> f(X) a__f(0()) -> cons(0(),f(s(0()))) a__f(s(0())) -> a__f(a__p(s(0()))) a__p(X) -> p(X) a__p(s(X)) -> mark(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(f(X)) -> a__f(mark(X)) mark(p(X)) -> a__p(mark(X)) mark(s(X)) -> s(mark(X)) - Signature: {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,a__p,mark} and constructors {0,cons,f,p,s} + 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__f) = {1}, uargs(a__p) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__f) = [1] x1 + [12] p(a__p) = [1] x1 + [0] p(cons) = [1] x1 + [0] p(f) = [0] p(mark) = [0] p(p) = [1] x1 + [0] p(s) = [1] x1 + [0] Following rules are strictly oriented: a__f(X) = [1] X + [12] > [0] = f(X) a__f(0()) = [12] > [0] = cons(0(),f(s(0()))) Following rules are (at-least) weakly oriented: a__f(s(0())) = [12] >= [12] = a__f(a__p(s(0()))) a__p(X) = [1] X + [0] >= [1] X + [0] = p(X) a__p(s(X)) = [1] X + [0] >= [0] = mark(X) mark(0()) = [0] >= [0] = 0() mark(cons(X1,X2)) = [0] >= [0] = cons(mark(X1),X2) mark(f(X)) = [0] >= [12] = a__f(mark(X)) mark(p(X)) = [0] >= [0] = a__p(mark(X)) mark(s(X)) = [0] >= [0] = 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__f(s(0())) -> a__f(a__p(s(0()))) a__p(X) -> p(X) a__p(s(X)) -> mark(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(f(X)) -> a__f(mark(X)) mark(p(X)) -> a__p(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__f(X) -> f(X) a__f(0()) -> cons(0(),f(s(0()))) - Signature: {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,a__p,mark} and constructors {0,cons,f,p,s} + 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__f) = {1}, uargs(a__p) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [10] p(a__f) = [1] x1 + [0] p(a__p) = [1] x1 + [1] p(cons) = [1] x1 + [0] p(f) = [1] x1 + [0] p(mark) = [1] x1 + [0] p(p) = [1] x1 + [4] p(s) = [1] x1 + [6] Following rules are strictly oriented: a__p(s(X)) = [1] X + [7] > [1] X + [0] = mark(X) mark(p(X)) = [1] X + [4] > [1] X + [1] = a__p(mark(X)) Following rules are (at-least) weakly oriented: a__f(X) = [1] X + [0] >= [1] X + [0] = f(X) a__f(0()) = [10] >= [10] = cons(0(),f(s(0()))) a__f(s(0())) = [16] >= [17] = a__f(a__p(s(0()))) a__p(X) = [1] X + [1] >= [1] X + [4] = p(X) mark(0()) = [10] >= [10] = 0() mark(cons(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = cons(mark(X1),X2) mark(f(X)) = [1] X + [0] >= [1] X + [0] = a__f(mark(X)) mark(s(X)) = [1] X + [6] >= [1] X + [6] = s(mark(X)) 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__f(s(0())) -> a__f(a__p(s(0()))) a__p(X) -> p(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(f(X)) -> a__f(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__f(X) -> f(X) a__f(0()) -> cons(0(),f(s(0()))) a__p(s(X)) -> mark(X) mark(p(X)) -> a__p(mark(X)) - Signature: {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,a__p,mark} and constructors {0,cons,f,p,s} + 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__f) = {1}, uargs(a__p) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__f) = [1] x1 + [1] p(a__p) = [1] x1 + [0] p(cons) = [1] x1 + [0] p(f) = [0] p(mark) = [1] p(p) = [1] x1 + [0] p(s) = [1] x1 + [6] Following rules are strictly oriented: mark(0()) = [1] > [0] = 0() Following rules are (at-least) weakly oriented: a__f(X) = [1] X + [1] >= [0] = f(X) a__f(0()) = [1] >= [0] = cons(0(),f(s(0()))) a__f(s(0())) = [7] >= [7] = a__f(a__p(s(0()))) a__p(X) = [1] X + [0] >= [1] X + [0] = p(X) a__p(s(X)) = [1] X + [6] >= [1] = mark(X) mark(cons(X1,X2)) = [1] >= [1] = cons(mark(X1),X2) mark(f(X)) = [1] >= [2] = a__f(mark(X)) mark(p(X)) = [1] >= [1] = a__p(mark(X)) mark(s(X)) = [1] >= [7] = s(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: NaturalMI MAYBE + Considered Problem: - Strict TRS: a__f(s(0())) -> a__f(a__p(s(0()))) a__p(X) -> p(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(f(X)) -> a__f(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__f(X) -> f(X) a__f(0()) -> cons(0(),f(s(0()))) a__p(s(X)) -> mark(X) mark(0()) -> 0() mark(p(X)) -> a__p(mark(X)) - Signature: {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,a__p,mark} and constructors {0,cons,f,p,s} + 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__f) = {1}, uargs(a__p) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__f,a__p,mark} TcT has computed the following interpretation: p(0) = [1] [1] p(a__f) = [1 0] x1 + [0] [0 1] [2] p(a__p) = [1 0] x1 + [0] [0 1] [0] p(cons) = [1 0] x1 + [0] [0 1] [0] p(f) = [1 0] x1 + [0] [0 1] [2] p(mark) = [1 1] x1 + [0] [0 1] [0] p(p) = [1 0] x1 + [0] [0 1] [0] p(s) = [1 5] x1 + [2] [0 1] [6] Following rules are strictly oriented: mark(f(X)) = [1 1] X + [2] [0 1] [2] > [1 1] X + [0] [0 1] [2] = a__f(mark(X)) mark(s(X)) = [1 6] X + [8] [0 1] [6] > [1 6] X + [2] [0 1] [6] = s(mark(X)) Following rules are (at-least) weakly oriented: a__f(X) = [1 0] X + [0] [0 1] [2] >= [1 0] X + [0] [0 1] [2] = f(X) a__f(0()) = [1] [3] >= [1] [1] = cons(0(),f(s(0()))) a__f(s(0())) = [8] [9] >= [8] [9] = a__f(a__p(s(0()))) a__p(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = p(X) a__p(s(X)) = [1 5] X + [2] [0 1] [6] >= [1 1] X + [0] [0 1] [0] = mark(X) mark(0()) = [2] [1] >= [1] [1] = 0() mark(cons(X1,X2)) = [1 1] X1 + [0] [0 1] [0] >= [1 1] X1 + [0] [0 1] [0] = cons(mark(X1),X2) mark(p(X)) = [1 1] X + [0] [0 1] [0] >= [1 1] X + [0] [0 1] [0] = a__p(mark(X)) * Step 5: NaturalMI MAYBE + Considered Problem: - Strict TRS: a__f(s(0())) -> a__f(a__p(s(0()))) a__p(X) -> p(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) - Weak TRS: a__f(X) -> f(X) a__f(0()) -> cons(0(),f(s(0()))) a__p(s(X)) -> mark(X) mark(0()) -> 0() mark(f(X)) -> a__f(mark(X)) mark(p(X)) -> a__p(mark(X)) mark(s(X)) -> s(mark(X)) - Signature: {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,a__p,mark} and constructors {0,cons,f,p,s} + 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__f) = {1}, uargs(a__p) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__f,a__p,mark} TcT has computed the following interpretation: p(0) = [0] [0] p(a__f) = [1 0] x1 + [0] [0 1] [2] p(a__p) = [1 1] x1 + [0] [0 1] [0] p(cons) = [1 0] x1 + [0] [0 1] [2] p(f) = [1 0] x1 + [0] [0 1] [2] p(mark) = [1 2] x1 + [0] [0 1] [0] p(p) = [1 1] x1 + [0] [0 1] [0] p(s) = [1 1] x1 + [0] [0 1] [0] Following rules are strictly oriented: mark(cons(X1,X2)) = [1 2] X1 + [4] [0 1] [2] > [1 2] X1 + [0] [0 1] [2] = cons(mark(X1),X2) Following rules are (at-least) weakly oriented: a__f(X) = [1 0] X + [0] [0 1] [2] >= [1 0] X + [0] [0 1] [2] = f(X) a__f(0()) = [0] [2] >= [0] [2] = cons(0(),f(s(0()))) a__f(s(0())) = [0] [2] >= [0] [2] = a__f(a__p(s(0()))) a__p(X) = [1 1] X + [0] [0 1] [0] >= [1 1] X + [0] [0 1] [0] = p(X) a__p(s(X)) = [1 2] X + [0] [0 1] [0] >= [1 2] X + [0] [0 1] [0] = mark(X) mark(0()) = [0] [0] >= [0] [0] = 0() mark(f(X)) = [1 2] X + [4] [0 1] [2] >= [1 2] X + [0] [0 1] [2] = a__f(mark(X)) mark(p(X)) = [1 3] X + [0] [0 1] [0] >= [1 3] X + [0] [0 1] [0] = a__p(mark(X)) mark(s(X)) = [1 3] X + [0] [0 1] [0] >= [1 3] X + [0] [0 1] [0] = s(mark(X)) * Step 6: NaturalMI MAYBE + Considered Problem: - Strict TRS: a__f(s(0())) -> a__f(a__p(s(0()))) a__p(X) -> p(X) - Weak TRS: a__f(X) -> f(X) a__f(0()) -> cons(0(),f(s(0()))) a__p(s(X)) -> mark(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(f(X)) -> a__f(mark(X)) mark(p(X)) -> a__p(mark(X)) mark(s(X)) -> s(mark(X)) - Signature: {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,a__p,mark} and constructors {0,cons,f,p,s} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 4, 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__f) = {1}, uargs(a__p) = {1}, uargs(cons) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__f,a__p,mark} TcT has computed the following interpretation: p(0) = [0] [0] [1] [0] p(a__f) = [1 0 1 0] [0] [0 1 0 1] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [1] p(a__p) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 1] [0] [0 0 0 1] [0] p(cons) = [1 0 0 0] [0 0 0 0] [0] [0 1 0 0] x1 + [0 0 1 0] x2 + [0] [0 0 0 1] [0 0 0 0] [0] [0 0 0 1] [0 0 0 0] [0] p(f) = [1 0 0 0] [0] [0 1 0 1] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [1] p(mark) = [1 1 0 1] [0] [0 1 0 0] x1 + [0] [0 0 0 1] [1] [0 0 0 1] [0] p(p) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [0] p(s) = [1 1 0 1] [0] [0 1 0 0] x1 + [0] [0 0 1 0] [1] [0 0 0 1] [1] Following rules are strictly oriented: a__f(s(0())) = [2] [1] [0] [2] > [1] [1] [0] [2] = a__f(a__p(s(0()))) Following rules are (at-least) weakly oriented: a__f(X) = [1 0 1 0] [0] [0 1 0 1] X + [0] [0 0 0 0] [0] [0 0 0 1] [1] >= [1 0 0 0] [0] [0 1 0 1] X + [0] [0 0 0 0] [0] [0 0 0 1] [1] = f(X) a__f(0()) = [1] [0] [0] [1] >= [0] [0] [0] [0] = cons(0(),f(s(0()))) a__p(X) = [1 0 0 0] [0] [0 1 0 0] X + [0] [0 0 0 1] [0] [0 0 0 1] [0] >= [1 0 0 0] [0] [0 1 0 0] X + [0] [0 0 0 0] [0] [0 0 0 1] [0] = p(X) a__p(s(X)) = [1 1 0 1] [0] [0 1 0 0] X + [0] [0 0 0 1] [1] [0 0 0 1] [1] >= [1 1 0 1] [0] [0 1 0 0] X + [0] [0 0 0 1] [1] [0 0 0 1] [0] = mark(X) mark(0()) = [0] [0] [1] [0] >= [0] [0] [1] [0] = 0() mark(cons(X1,X2)) = [1 1 0 1] [0 0 1 0] [0] [0 1 0 0] X1 + [0 0 1 0] X2 + [0] [0 0 0 1] [0 0 0 0] [1] [0 0 0 1] [0 0 0 0] [0] >= [1 1 0 1] [0 0 0 0] [0] [0 1 0 0] X1 + [0 0 1 0] X2 + [0] [0 0 0 1] [0 0 0 0] [0] [0 0 0 1] [0 0 0 0] [0] = cons(mark(X1),X2) mark(f(X)) = [1 1 0 2] [1] [0 1 0 1] X + [0] [0 0 0 1] [2] [0 0 0 1] [1] >= [1 1 0 2] [1] [0 1 0 1] X + [0] [0 0 0 0] [0] [0 0 0 1] [1] = a__f(mark(X)) mark(p(X)) = [1 1 0 1] [0] [0 1 0 0] X + [0] [0 0 0 1] [1] [0 0 0 1] [0] >= [1 1 0 1] [0] [0 1 0 0] X + [0] [0 0 0 1] [0] [0 0 0 1] [0] = a__p(mark(X)) mark(s(X)) = [1 2 0 2] [1] [0 1 0 0] X + [0] [0 0 0 1] [2] [0 0 0 1] [1] >= [1 2 0 2] [0] [0 1 0 0] X + [0] [0 0 0 1] [2] [0 0 0 1] [1] = s(mark(X)) * Step 7: Failure MAYBE + Considered Problem: - Strict TRS: a__p(X) -> p(X) - Weak TRS: a__f(X) -> f(X) a__f(0()) -> cons(0(),f(s(0()))) a__f(s(0())) -> a__f(a__p(s(0()))) a__p(s(X)) -> mark(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(f(X)) -> a__f(mark(X)) mark(p(X)) -> a__p(mark(X)) mark(s(X)) -> s(mark(X)) - Signature: {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,a__p,mark} and constructors {0,cons,f,p,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE