*** 1 Progress [(O(1),O(n^1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: activate(X) -> X activate(n__from(X)) -> from(X) after(0(),XS) -> XS after(s(N),cons(X,XS)) -> after(N,activate(XS)) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Weak DP Rules: Weak TRS Rules: Signature: {activate/1,after/2,from/1} / {0/0,cons/2,n__from/1,s/1} Obligation: Innermost basic terms: {activate,after,from}/{0,cons,n__from,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: 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(after) = {2} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [0] p(after) = [1] x2 + [11] p(cons) = [1] x2 + [0] p(from) = [1] x1 + [1] p(n__from) = [1] x1 + [0] p(s) = [1] x1 + [0] Following rules are strictly oriented: after(0(),XS) = [1] XS + [11] > [1] XS + [0] = XS from(X) = [1] X + [1] > [1] X + [0] = cons(X,n__from(s(X))) from(X) = [1] X + [1] > [1] X + [0] = n__from(X) Following rules are (at-least) weakly oriented: activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__from(X)) = [1] X + [0] >= [1] X + [1] = from(X) after(s(N),cons(X,XS)) = [1] XS + [11] >= [1] XS + [11] = after(N,activate(XS)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1 Progress [(O(1),O(n^1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: activate(X) -> X activate(n__from(X)) -> from(X) after(s(N),cons(X,XS)) -> after(N,activate(XS)) Weak DP Rules: Weak TRS Rules: after(0(),XS) -> XS from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Signature: {activate/1,after/2,from/1} / {0/0,cons/2,n__from/1,s/1} Obligation: Innermost basic terms: {activate,after,from}/{0,cons,n__from,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: 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(after) = {2} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [13] p(after) = [1] x2 + [0] p(cons) = [1] x2 + [0] p(from) = [1] x1 + [0] p(n__from) = [1] x1 + [0] p(s) = [1] x1 + [0] Following rules are strictly oriented: activate(X) = [1] X + [13] > [1] X + [0] = X activate(n__from(X)) = [1] X + [13] > [1] X + [0] = from(X) Following rules are (at-least) weakly oriented: after(0(),XS) = [1] XS + [0] >= [1] XS + [0] = XS after(s(N),cons(X,XS)) = [1] XS + [0] >= [1] XS + [13] = after(N,activate(XS)) from(X) = [1] X + [0] >= [1] X + [0] = cons(X,n__from(s(X))) from(X) = [1] X + [0] >= [1] X + [0] = n__from(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1 Progress [(O(1),O(n^1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: after(s(N),cons(X,XS)) -> after(N,activate(XS)) Weak DP Rules: Weak TRS Rules: activate(X) -> X activate(n__from(X)) -> from(X) after(0(),XS) -> XS from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Signature: {activate/1,after/2,from/1} / {0/0,cons/2,n__from/1,s/1} Obligation: Innermost basic terms: {activate,after,from}/{0,cons,n__from,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: 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(after) = {2} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [8] p(activate) = [1] x1 + [0] p(after) = [1] x1 + [1] x2 + [2] p(cons) = [1] x2 + [0] p(from) = [0] p(n__from) = [0] p(s) = [1] x1 + [1] Following rules are strictly oriented: after(s(N),cons(X,XS)) = [1] N + [1] XS + [3] > [1] N + [1] XS + [2] = after(N,activate(XS)) Following rules are (at-least) weakly oriented: activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__from(X)) = [0] >= [0] = from(X) after(0(),XS) = [1] XS + [10] >= [1] XS + [0] = XS from(X) = [0] >= [0] = cons(X,n__from(s(X))) from(X) = [0] >= [0] = n__from(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP Rules: Weak TRS Rules: activate(X) -> X activate(n__from(X)) -> from(X) after(0(),XS) -> XS after(s(N),cons(X,XS)) -> after(N,activate(XS)) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) Signature: {activate/1,after/2,from/1} / {0/0,cons/2,n__from/1,s/1} Obligation: Innermost basic terms: {activate,after,from}/{0,cons,n__from,s} Applied Processor: EmptyProcessor Proof: The problem is already closed. The intended complexity is O(1).