MAYBE * Step 1: WeightGap MAYBE + Considered Problem: - Strict TRS: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,Z)) 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,Z)) 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z)) from(X) -> cons(X,from(s(X))) pi(X) -> 2ndspos(X,from(0())) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) square(X) -> times(X,X) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,from/1,pi/1,plus/2,square/1,times/2} / {0/0,cons/2,cons2/2,negrecip/1,posrecip/1 ,rcons/2,rnil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,from,pi,plus,square ,times} and constructors {0,cons,cons2,negrecip,posrecip,rcons,rnil,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(2ndspos) = {2}, uargs(cons) = {2}, uargs(plus) = {2}, uargs(rcons) = {2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(2ndsneg) = [1] x2 + [5] p(2ndspos) = [1] x2 + [0] p(cons) = [1] x2 + [1] p(cons2) = [1] x2 + [0] p(from) = [2] x1 + [6] p(negrecip) = [2] p(pi) = [4] x1 + [1] p(plus) = [1] x2 + [1] p(posrecip) = [4] p(rcons) = [1] x1 + [1] x2 + [2] p(rnil) = [0] p(s) = [1] x1 + [0] p(square) = [6] x1 + [1] p(times) = [4] x1 + [2] x2 + [0] Following rules are strictly oriented: 2ndsneg(0(),Z) = [1] Z + [5] > [0] = rnil() 2ndsneg(s(N),cons(X,Z)) = [1] Z + [6] > [1] Z + [5] = 2ndsneg(s(N),cons2(X,Z)) 2ndsneg(s(N),cons2(X,cons(Y,Z))) = [1] Z + [6] > [1] Z + [4] = rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(s(N),cons(X,Z)) = [1] Z + [1] > [1] Z + [0] = 2ndspos(s(N),cons2(X,Z)) plus(0(),Y) = [1] Y + [1] > [1] Y + [0] = Y square(X) = [6] X + [1] > [6] X + [0] = times(X,X) times(0(),Y) = [2] Y + [4] > [1] = 0() Following rules are (at-least) weakly oriented: 2ndspos(0(),Z) = [1] Z + [0] >= [0] = rnil() 2ndspos(s(N),cons2(X,cons(Y,Z))) = [1] Z + [1] >= [1] Z + [11] = rcons(posrecip(Y),2ndsneg(N,Z)) from(X) = [2] X + [6] >= [2] X + [7] = cons(X,from(s(X))) pi(X) = [4] X + [1] >= [8] = 2ndspos(X,from(0())) plus(s(X),Y) = [1] Y + [1] >= [1] Y + [1] = s(plus(X,Y)) times(s(X),Y) = [4] X + [2] Y + [0] >= [4] X + [2] Y + [1] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap MAYBE + Considered Problem: - Strict TRS: 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z)) from(X) -> cons(X,from(s(X))) pi(X) -> 2ndspos(X,from(0())) plus(s(X),Y) -> s(plus(X,Y)) times(s(X),Y) -> plus(Y,times(X,Y)) - Weak TRS: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,Z)) 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,Z)) plus(0(),Y) -> Y square(X) -> times(X,X) times(0(),Y) -> 0() - Signature: {2ndsneg/2,2ndspos/2,from/1,pi/1,plus/2,square/1,times/2} / {0/0,cons/2,cons2/2,negrecip/1,posrecip/1 ,rcons/2,rnil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,from,pi,plus,square ,times} and constructors {0,cons,cons2,negrecip,posrecip,rcons,rnil,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(2ndspos) = {2}, uargs(cons) = {2}, uargs(plus) = {2}, uargs(rcons) = {2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [3] p(2ndsneg) = [1] x2 + [6] p(2ndspos) = [1] x2 + [1] p(cons) = [1] x2 + [1] p(cons2) = [1] x2 + [0] p(from) = [5] p(negrecip) = [2] p(pi) = [2] p(plus) = [1] x2 + [0] p(posrecip) = [2] p(rcons) = [1] x1 + [1] x2 + [2] p(rnil) = [6] p(s) = [1] x1 + [6] p(square) = [7] x1 + [0] p(times) = [1] x1 + [6] x2 + [0] Following rules are strictly oriented: times(s(X),Y) = [1] X + [6] Y + [6] > [1] X + [6] Y + [0] = plus(Y,times(X,Y)) Following rules are (at-least) weakly oriented: 2ndsneg(0(),Z) = [1] Z + [6] >= [6] = rnil() 2ndsneg(s(N),cons(X,Z)) = [1] Z + [7] >= [1] Z + [6] = 2ndsneg(s(N),cons2(X,Z)) 2ndsneg(s(N),cons2(X,cons(Y,Z))) = [1] Z + [7] >= [1] Z + [5] = rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(0(),Z) = [1] Z + [1] >= [6] = rnil() 2ndspos(s(N),cons(X,Z)) = [1] Z + [2] >= [1] Z + [1] = 2ndspos(s(N),cons2(X,Z)) 2ndspos(s(N),cons2(X,cons(Y,Z))) = [1] Z + [2] >= [1] Z + [10] = rcons(posrecip(Y),2ndsneg(N,Z)) from(X) = [5] >= [6] = cons(X,from(s(X))) pi(X) = [2] >= [6] = 2ndspos(X,from(0())) plus(0(),Y) = [1] Y + [0] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [0] >= [1] Y + [6] = s(plus(X,Y)) square(X) = [7] X + [0] >= [7] X + [0] = times(X,X) times(0(),Y) = [6] Y + [3] >= [3] = 0() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: NaturalMI MAYBE + Considered Problem: - Strict TRS: 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z)) from(X) -> cons(X,from(s(X))) pi(X) -> 2ndspos(X,from(0())) plus(s(X),Y) -> s(plus(X,Y)) - Weak TRS: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,Z)) 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,Z)) plus(0(),Y) -> Y square(X) -> times(X,X) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,from/1,pi/1,plus/2,square/1,times/2} / {0/0,cons/2,cons2/2,negrecip/1,posrecip/1 ,rcons/2,rnil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,from,pi,plus,square ,times} and constructors {0,cons,cons2,negrecip,posrecip,rcons,rnil,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, 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(2ndspos) = {2}, uargs(cons) = {2}, uargs(plus) = {2}, uargs(rcons) = {2}, uargs(s) = {1} Following symbols are considered usable: {2ndsneg,2ndspos,from,pi,plus,square,times} TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [8] x2 + [4] p(2ndspos) = [8] x2 + [4] p(cons) = [1] x2 + [0] p(cons2) = [1] x2 + [0] p(from) = [0] p(negrecip) = [1] p(pi) = [2] x1 + [8] p(plus) = [8] x2 + [0] p(posrecip) = [8] p(rcons) = [1] x2 + [0] p(rnil) = [4] p(s) = [1] x1 + [0] p(square) = [8] x1 + [9] p(times) = [0] Following rules are strictly oriented: pi(X) = [2] X + [8] > [4] = 2ndspos(X,from(0())) Following rules are (at-least) weakly oriented: 2ndsneg(0(),Z) = [8] Z + [4] >= [4] = rnil() 2ndsneg(s(N),cons(X,Z)) = [8] Z + [4] >= [8] Z + [4] = 2ndsneg(s(N),cons2(X,Z)) 2ndsneg(s(N),cons2(X,cons(Y,Z))) = [8] Z + [4] >= [8] Z + [4] = rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(0(),Z) = [8] Z + [4] >= [4] = rnil() 2ndspos(s(N),cons(X,Z)) = [8] Z + [4] >= [8] Z + [4] = 2ndspos(s(N),cons2(X,Z)) 2ndspos(s(N),cons2(X,cons(Y,Z))) = [8] Z + [4] >= [8] Z + [4] = rcons(posrecip(Y),2ndsneg(N,Z)) from(X) = [0] >= [0] = cons(X,from(s(X))) plus(0(),Y) = [8] Y + [0] >= [1] Y + [0] = Y plus(s(X),Y) = [8] Y + [0] >= [8] Y + [0] = s(plus(X,Y)) square(X) = [8] X + [9] >= [0] = times(X,X) times(0(),Y) = [0] >= [0] = 0() times(s(X),Y) = [0] >= [0] = plus(Y,times(X,Y)) * Step 4: WeightGap MAYBE + Considered Problem: - Strict TRS: 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z)) from(X) -> cons(X,from(s(X))) plus(s(X),Y) -> s(plus(X,Y)) - Weak TRS: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,Z)) 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,Z)) pi(X) -> 2ndspos(X,from(0())) plus(0(),Y) -> Y square(X) -> times(X,X) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,from/1,pi/1,plus/2,square/1,times/2} / {0/0,cons/2,cons2/2,negrecip/1,posrecip/1 ,rcons/2,rnil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,from,pi,plus,square ,times} and constructors {0,cons,cons2,negrecip,posrecip,rcons,rnil,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(2ndspos) = {2}, uargs(cons) = {2}, uargs(plus) = {2}, uargs(rcons) = {2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(2ndsneg) = [1] x2 + [6] p(2ndspos) = [1] x2 + [0] p(cons) = [1] x2 + [5] p(cons2) = [1] x2 + [2] p(from) = [4] x1 + [2] p(negrecip) = [1] x1 + [0] p(pi) = [6] p(plus) = [1] x2 + [0] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [0] p(square) = [7] x1 + [0] p(times) = [2] x1 + [5] x2 + [0] Following rules are strictly oriented: 2ndspos(s(N),cons2(X,cons(Y,Z))) = [1] Z + [7] > [1] Z + [6] = rcons(posrecip(Y),2ndsneg(N,Z)) Following rules are (at-least) weakly oriented: 2ndsneg(0(),Z) = [1] Z + [6] >= [0] = rnil() 2ndsneg(s(N),cons(X,Z)) = [1] Z + [11] >= [1] Z + [8] = 2ndsneg(s(N),cons2(X,Z)) 2ndsneg(s(N),cons2(X,cons(Y,Z))) = [1] Z + [13] >= [1] Z + [0] = rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(0(),Z) = [1] Z + [0] >= [0] = rnil() 2ndspos(s(N),cons(X,Z)) = [1] Z + [5] >= [1] Z + [2] = 2ndspos(s(N),cons2(X,Z)) from(X) = [4] X + [2] >= [4] X + [7] = cons(X,from(s(X))) pi(X) = [6] >= [6] = 2ndspos(X,from(0())) plus(0(),Y) = [1] Y + [0] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [0] >= [1] Y + [0] = s(plus(X,Y)) square(X) = [7] X + [0] >= [7] X + [0] = times(X,X) times(0(),Y) = [5] Y + [2] >= [1] = 0() times(s(X),Y) = [2] X + [5] Y + [0] >= [2] X + [5] Y + [0] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap MAYBE + Considered Problem: - Strict TRS: 2ndspos(0(),Z) -> rnil() from(X) -> cons(X,from(s(X))) plus(s(X),Y) -> s(plus(X,Y)) - Weak TRS: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,Z)) 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,Z)) 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z)) pi(X) -> 2ndspos(X,from(0())) plus(0(),Y) -> Y square(X) -> times(X,X) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,from/1,pi/1,plus/2,square/1,times/2} / {0/0,cons/2,cons2/2,negrecip/1,posrecip/1 ,rcons/2,rnil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,from,pi,plus,square ,times} and constructors {0,cons,cons2,negrecip,posrecip,rcons,rnil,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(2ndspos) = {2}, uargs(cons) = {2}, uargs(plus) = {2}, uargs(rcons) = {2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [1] x2 + [2] p(2ndspos) = [1] x2 + [3] p(cons) = [1] x2 + [6] p(cons2) = [1] x2 + [0] p(from) = [4] x1 + [0] p(negrecip) = [4] p(pi) = [1] x1 + [3] p(plus) = [1] x2 + [2] p(posrecip) = [0] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [1] p(s) = [1] x1 + [1] p(square) = [6] x1 + [0] p(times) = [2] x1 + [4] x2 + [0] Following rules are strictly oriented: 2ndspos(0(),Z) = [1] Z + [3] > [1] = rnil() Following rules are (at-least) weakly oriented: 2ndsneg(0(),Z) = [1] Z + [2] >= [1] = rnil() 2ndsneg(s(N),cons(X,Z)) = [1] Z + [8] >= [1] Z + [2] = 2ndsneg(s(N),cons2(X,Z)) 2ndsneg(s(N),cons2(X,cons(Y,Z))) = [1] Z + [8] >= [1] Z + [7] = rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(s(N),cons(X,Z)) = [1] Z + [9] >= [1] Z + [3] = 2ndspos(s(N),cons2(X,Z)) 2ndspos(s(N),cons2(X,cons(Y,Z))) = [1] Z + [9] >= [1] Z + [2] = rcons(posrecip(Y),2ndsneg(N,Z)) from(X) = [4] X + [0] >= [4] X + [10] = cons(X,from(s(X))) pi(X) = [1] X + [3] >= [3] = 2ndspos(X,from(0())) plus(0(),Y) = [1] Y + [2] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [2] >= [1] Y + [3] = s(plus(X,Y)) square(X) = [6] X + [0] >= [6] X + [0] = times(X,X) times(0(),Y) = [4] Y + [0] >= [0] = 0() times(s(X),Y) = [2] X + [4] Y + [2] >= [2] X + [4] Y + [2] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: Failure MAYBE + Considered Problem: - Strict TRS: from(X) -> cons(X,from(s(X))) plus(s(X),Y) -> s(plus(X,Y)) - Weak TRS: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,Z)) 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,Z)) 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z)) pi(X) -> 2ndspos(X,from(0())) plus(0(),Y) -> Y square(X) -> times(X,X) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,from/1,pi/1,plus/2,square/1,times/2} / {0/0,cons/2,cons2/2,negrecip/1,posrecip/1 ,rcons/2,rnil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,from,pi,plus,square ,times} and constructors {0,cons,cons2,negrecip,posrecip,rcons,rnil,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE