*** 1 Progress [(O(1),O(n^1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sel(0(),cons(X,Y)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) Weak DP Rules: Weak TRS Rules: Signature: {activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1} Obligation: Innermost basic terms: {activate,from,s,sel}/{0,cons,n__from,n__s} Applied Processor: InnermostRuleRemoval Proof: Arguments of following rules are not normal-forms. sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) All above mentioned rules can be savely removed. *** 1.1 Progress [(O(1),O(n^1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sel(0(),cons(X,Y)) -> X Weak DP Rules: Weak TRS Rules: Signature: {activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1} Obligation: Innermost basic terms: {activate,from,s,sel}/{0,cons,n__from,n__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(from) = {1}, uargs(s) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [0] p(activate) = [2] x1 + [0] p(cons) = [1] x1 + [1] p(from) = [1] x1 + [0] p(n__from) = [1] x1 + [0] p(n__s) = [1] x1 + [1] p(s) = [1] x1 + [0] p(sel) = [1] x2 + [0] Following rules are strictly oriented: activate(n__s(X)) = [2] X + [2] > [2] X + [0] = s(activate(X)) sel(0(),cons(X,Y)) = [1] X + [1] > [1] X + [0] = X Following rules are (at-least) weakly oriented: activate(X) = [2] X + [0] >= [1] X + [0] = X activate(n__from(X)) = [2] X + [0] >= [2] X + [0] = from(activate(X)) from(X) = [1] X + [0] >= [1] X + [1] = cons(X,n__from(n__s(X))) from(X) = [1] X + [0] >= [1] X + [0] = n__from(X) s(X) = [1] X + [0] >= [1] X + [1] = n__s(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: activate(X) -> X activate(n__from(X)) -> from(activate(X)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) Weak DP Rules: Weak TRS Rules: activate(n__s(X)) -> s(activate(X)) sel(0(),cons(X,Y)) -> X Signature: {activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1} Obligation: Innermost basic terms: {activate,from,s,sel}/{0,cons,n__from,n__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(from) = {1}, uargs(s) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [1] p(activate) = [1] x1 + [0] p(cons) = [1] x1 + [2] p(from) = [1] x1 + [2] p(n__from) = [1] x1 + [0] p(n__s) = [1] x1 + [4] p(s) = [1] x1 + [2] p(sel) = [1] x1 + [9] x2 + [7] Following rules are strictly oriented: from(X) = [1] X + [2] > [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 + [2] = from(activate(X)) activate(n__s(X)) = [1] X + [4] >= [1] X + [2] = s(activate(X)) from(X) = [1] X + [2] >= [1] X + [2] = cons(X,n__from(n__s(X))) s(X) = [1] X + [2] >= [1] X + [4] = n__s(X) sel(0(),cons(X,Y)) = [9] X + [26] >= [1] X + [0] = 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(n^1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: activate(X) -> X activate(n__from(X)) -> from(activate(X)) from(X) -> cons(X,n__from(n__s(X))) s(X) -> n__s(X) Weak DP Rules: Weak TRS Rules: activate(n__s(X)) -> s(activate(X)) from(X) -> n__from(X) sel(0(),cons(X,Y)) -> X Signature: {activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1} Obligation: Innermost basic terms: {activate,from,s,sel}/{0,cons,n__from,n__s} Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy} Proof: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(from) = {1}, uargs(s) = {1} Following symbols are considered usable: {activate,from,s,sel} TcT has computed the following interpretation: p(0) = [3] p(activate) = [2] x1 + [2] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [4] p(n__from) = [1] x1 + [2] p(n__s) = [1] x1 + [1] p(s) = [1] x1 + [2] p(sel) = [8] x1 + [2] x2 + [2] Following rules are strictly oriented: activate(X) = [2] X + [2] > [1] X + [0] = X from(X) = [1] X + [4] > [1] X + [0] = cons(X,n__from(n__s(X))) s(X) = [1] X + [2] > [1] X + [1] = n__s(X) Following rules are (at-least) weakly oriented: activate(n__from(X)) = [2] X + [6] >= [2] X + [6] = from(activate(X)) activate(n__s(X)) = [2] X + [4] >= [2] X + [4] = s(activate(X)) from(X) = [1] X + [4] >= [1] X + [2] = n__from(X) sel(0(),cons(X,Y)) = [2] X + [26] >= [1] X + [0] = X *** 1.1.1.1.1 Progress [(O(1),O(n^1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: activate(n__from(X)) -> from(activate(X)) Weak DP Rules: Weak TRS Rules: activate(X) -> X activate(n__s(X)) -> s(activate(X)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sel(0(),cons(X,Y)) -> X Signature: {activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1} Obligation: Innermost basic terms: {activate,from,s,sel}/{0,cons,n__from,n__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(from) = {1}, uargs(s) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [0] p(activate) = [7] x1 + [0] p(cons) = [1] x1 + [2] p(from) = [1] x1 + [2] p(n__from) = [1] x1 + [1] p(n__s) = [1] x1 + [0] p(s) = [1] x1 + [0] p(sel) = [1] x2 + [4] Following rules are strictly oriented: activate(n__from(X)) = [7] X + [7] > [7] X + [2] = from(activate(X)) Following rules are (at-least) weakly oriented: activate(X) = [7] X + [0] >= [1] X + [0] = X activate(n__s(X)) = [7] X + [0] >= [7] X + [0] = s(activate(X)) from(X) = [1] X + [2] >= [1] X + [2] = cons(X,n__from(n__s(X))) from(X) = [1] X + [2] >= [1] X + [1] = n__from(X) s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) sel(0(),cons(X,Y)) = [1] X + [6] >= [1] X + [0] = X Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.1.1 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(activate(X)) activate(n__s(X)) -> s(activate(X)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sel(0(),cons(X,Y)) -> X Signature: {activate/1,from/1,s/1,sel/2} / {0/0,cons/2,n__from/1,n__s/1} Obligation: Innermost basic terms: {activate,from,s,sel}/{0,cons,n__from,n__s} Applied Processor: EmptyProcessor Proof: The problem is already closed. The intended complexity is O(1).