MAYBE * Step 1: WeightGap MAYBE + Considered Problem: - Strict TRS: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(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,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,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 + [10] p(2ndspos) = [1] x2 + [0] p(cons) = [1] x2 + [0] p(from) = [7] x1 + [0] p(negrecip) = [11] p(pi) = [8] x1 + [0] p(plus) = [1] x2 + [0] p(posrecip) = [9] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [0] p(square) = [8] x1 + [0] p(times) = [5] x1 + [0] Following rules are strictly oriented: 2ndsneg(0(),Z) = [1] Z + [10] > [0] = rnil() times(0(),Y) = [15] > [3] = 0() Following rules are (at-least) weakly oriented: 2ndsneg(s(N),cons(X,cons(Y,Z))) = [1] Z + [10] >= [1] Z + [11] = rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(0(),Z) = [1] Z + [0] >= [0] = rnil() 2ndspos(s(N),cons(X,cons(Y,Z))) = [1] Z + [0] >= [1] Z + [19] = rcons(posrecip(Y),2ndsneg(N,Z)) from(X) = [7] X + [0] >= [7] X + [0] = cons(X,from(s(X))) pi(X) = [8] X + [0] >= [21] = 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) = [8] X + [0] >= [5] X + [0] = times(X,X) times(s(X),Y) = [5] X + [0] >= [5] X + [0] = 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: 2ndsneg(s(N),cons(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(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(s(X),Y) -> plus(Y,times(X,Y)) - Weak TRS: 2ndsneg(0(),Z) -> rnil() times(0(),Y) -> 0() - Signature: {2ndsneg/2,2ndspos/2,from/1,pi/1,plus/2,square/1,times/2} / {0/0,cons/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,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) = [13] p(2ndsneg) = [1] x2 + [11] p(2ndspos) = [1] x2 + [0] p(cons) = [1] x2 + [0] p(from) = [1] x1 + [0] p(negrecip) = [1] x1 + [0] p(pi) = [1] x1 + [0] p(plus) = [1] x2 + [0] p(posrecip) = [1] x1 + [1] p(rcons) = [1] x2 + [9] p(rnil) = [0] p(s) = [1] x1 + [1] p(square) = [4] x1 + [0] p(times) = [4] x2 + [13] Following rules are strictly oriented: 2ndsneg(s(N),cons(X,cons(Y,Z))) = [1] Z + [11] > [1] Z + [9] = rcons(negrecip(Y),2ndspos(N,Z)) Following rules are (at-least) weakly oriented: 2ndsneg(0(),Z) = [1] Z + [11] >= [0] = rnil() 2ndspos(0(),Z) = [1] Z + [0] >= [0] = rnil() 2ndspos(s(N),cons(X,cons(Y,Z))) = [1] Z + [0] >= [1] Z + [20] = rcons(posrecip(Y),2ndsneg(N,Z)) from(X) = [1] X + [0] >= [1] X + [1] = cons(X,from(s(X))) pi(X) = [1] X + [0] >= [13] = 2ndspos(X,from(0())) plus(0(),Y) = [1] Y + [0] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [0] >= [1] Y + [1] = s(plus(X,Y)) square(X) = [4] X + [0] >= [4] X + [13] = times(X,X) times(0(),Y) = [4] Y + [13] >= [13] = 0() times(s(X),Y) = [4] Y + [13] >= [4] Y + [13] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap MAYBE + Considered Problem: - Strict TRS: 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(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(s(X),Y) -> plus(Y,times(X,Y)) - Weak TRS: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z)) times(0(),Y) -> 0() - Signature: {2ndsneg/2,2ndspos/2,from/1,pi/1,plus/2,square/1,times/2} / {0/0,cons/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,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) = [4] p(2ndsneg) = [1] x2 + [1] p(2ndspos) = [1] x2 + [1] p(cons) = [1] x2 + [0] p(from) = [2] x1 + [0] p(negrecip) = [0] p(pi) = [0] p(plus) = [1] x2 + [0] p(posrecip) = [0] p(rcons) = [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [2] p(square) = [6] x1 + [4] p(times) = [1] x1 + [5] x2 + [0] Following rules are strictly oriented: 2ndspos(0(),Z) = [1] Z + [1] > [0] = rnil() square(X) = [6] X + [4] > [6] X + [0] = times(X,X) times(s(X),Y) = [1] X + [5] Y + [2] > [1] X + [5] Y + [0] = plus(Y,times(X,Y)) Following rules are (at-least) weakly oriented: 2ndsneg(0(),Z) = [1] Z + [1] >= [0] = rnil() 2ndsneg(s(N),cons(X,cons(Y,Z))) = [1] Z + [1] >= [1] Z + [1] = rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(s(N),cons(X,cons(Y,Z))) = [1] Z + [1] >= [1] Z + [1] = rcons(posrecip(Y),2ndsneg(N,Z)) from(X) = [2] X + [0] >= [2] X + [4] = cons(X,from(s(X))) pi(X) = [0] >= [9] = 2ndspos(X,from(0())) plus(0(),Y) = [1] Y + [0] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [0] >= [1] Y + [2] = s(plus(X,Y)) times(0(),Y) = [5] Y + [4] >= [4] = 0() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap MAYBE + Considered Problem: - Strict TRS: 2ndspos(s(N),cons(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)) - Weak TRS: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(0(),Z) -> rnil() 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,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,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 + [1] p(2ndspos) = [1] x2 + [5] p(cons) = [1] x2 + [2] p(from) = [1] x1 + [0] p(negrecip) = [1] x1 + [0] p(pi) = [4] x1 + [0] p(plus) = [1] x2 + [0] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [1] p(square) = [7] x1 + [2] p(times) = [1] x1 + [4] x2 + [2] Following rules are strictly oriented: 2ndspos(s(N),cons(X,cons(Y,Z))) = [1] Z + [9] > [1] Z + [1] = rcons(posrecip(Y),2ndsneg(N,Z)) Following rules are (at-least) weakly oriented: 2ndsneg(0(),Z) = [1] Z + [1] >= [0] = rnil() 2ndsneg(s(N),cons(X,cons(Y,Z))) = [1] Z + [5] >= [1] Z + [5] = rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(0(),Z) = [1] Z + [5] >= [0] = rnil() from(X) = [1] X + [0] >= [1] X + [3] = cons(X,from(s(X))) pi(X) = [4] X + [0] >= [5] = 2ndspos(X,from(0())) plus(0(),Y) = [1] Y + [0] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [0] >= [1] Y + [1] = s(plus(X,Y)) square(X) = [7] X + [2] >= [5] X + [2] = times(X,X) times(0(),Y) = [4] Y + [2] >= [0] = 0() times(s(X),Y) = [1] X + [4] Y + [3] >= [1] 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 5: WeightGap MAYBE + Considered Problem: - Strict TRS: 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)) - Weak TRS: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z)) 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,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,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] x1 + [1] x2 + [4] p(2ndspos) = [1] x1 + [1] x2 + [0] p(cons) = [1] x2 + [2] p(from) = [2] p(negrecip) = [0] p(pi) = [4] x1 + [3] p(plus) = [1] x2 + [0] p(posrecip) = [0] p(rcons) = [1] x1 + [1] x2 + [4] p(rnil) = [0] p(s) = [1] x1 + [4] p(square) = [1] x1 + [4] p(times) = [1] x1 + [0] Following rules are strictly oriented: pi(X) = [4] X + [3] > [1] X + [2] = 2ndspos(X,from(0())) Following rules are (at-least) weakly oriented: 2ndsneg(0(),Z) = [1] Z + [5] >= [0] = rnil() 2ndsneg(s(N),cons(X,cons(Y,Z))) = [1] N + [1] Z + [12] >= [1] N + [1] Z + [4] = rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(0(),Z) = [1] Z + [1] >= [0] = rnil() 2ndspos(s(N),cons(X,cons(Y,Z))) = [1] N + [1] Z + [8] >= [1] N + [1] Z + [8] = rcons(posrecip(Y),2ndsneg(N,Z)) from(X) = [2] >= [4] = cons(X,from(s(X))) plus(0(),Y) = [1] Y + [0] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [0] >= [1] Y + [4] = s(plus(X,Y)) square(X) = [1] X + [4] >= [1] X + [0] = times(X,X) times(0(),Y) = [1] >= [1] = 0() times(s(X),Y) = [1] X + [4] >= [1] X + [0] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: WeightGap MAYBE + Considered Problem: - Strict TRS: from(X) -> cons(X,from(s(X))) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) - Weak TRS: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z)) pi(X) -> 2ndspos(X,from(0())) 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,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,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 + [6] p(2ndspos) = [1] x2 + [5] p(cons) = [1] x2 + [2] p(from) = [2] x1 + [0] p(negrecip) = [0] p(pi) = [5] p(plus) = [1] x2 + [2] p(posrecip) = [0] p(rcons) = [1] x2 + [2] p(rnil) = [5] p(s) = [1] x1 + [2] p(square) = [4] x1 + [0] p(times) = [2] x1 + [2] x2 + [0] Following rules are strictly oriented: plus(0(),Y) = [1] Y + [2] > [1] Y + [0] = Y Following rules are (at-least) weakly oriented: 2ndsneg(0(),Z) = [1] Z + [6] >= [5] = rnil() 2ndsneg(s(N),cons(X,cons(Y,Z))) = [1] Z + [10] >= [1] Z + [7] = rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(0(),Z) = [1] Z + [5] >= [5] = rnil() 2ndspos(s(N),cons(X,cons(Y,Z))) = [1] Z + [9] >= [1] Z + [8] = rcons(posrecip(Y),2ndsneg(N,Z)) from(X) = [2] X + [0] >= [2] X + [6] = cons(X,from(s(X))) pi(X) = [5] >= [5] = 2ndspos(X,from(0())) plus(s(X),Y) = [1] Y + [2] >= [1] Y + [4] = s(plus(X,Y)) square(X) = [4] X + [0] >= [4] X + [0] = times(X,X) times(0(),Y) = [2] Y + [0] >= [0] = 0() times(s(X),Y) = [2] X + [2] Y + [4] >= [2] X + [2] Y + [2] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: 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,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(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,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,negrecip,posrecip,rcons,rnil,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE