WORST_CASE(?,O(n^1)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__terms(X)) -> terms(activate(X)) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) s(X) -> n__s(X) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__first/2,n__s/1,n__terms/1,nil/0,recip/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0 ,cons,n__first,n__s,n__terms,nil,recip} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. add(s(X),Y) -> s(add(X,Y)) dbl(s(X)) -> s(s(dbl(X))) first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) sqr(s(X)) -> s(add(sqr(X),dbl(X))) All above mentioned rules can be savely removed. * Step 2: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__terms(X)) -> terms(activate(X)) add(0(),X) -> X dbl(0()) -> 0() first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() s(X) -> n__s(X) sqr(0()) -> 0() terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__first/2,n__s/1,n__terms/1,nil/0,recip/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0 ,cons,n__first,n__s,n__terms,nil,recip} + 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(cons) = {1}, uargs(first) = {1,2}, uargs(recip) = {1}, uargs(s) = {1}, uargs(terms) = {1} Following symbols are considered usable: {activate,add,dbl,first,s,sqr,terms} TcT has computed the following interpretation: p(0) = [0] p(activate) = [4] x1 + [1] p(add) = [4] x2 + [0] p(cons) = [1] x1 + [0] p(dbl) = [1] p(first) = [1] x1 + [1] x2 + [2] p(n__first) = [1] x1 + [1] x2 + [2] p(n__s) = [1] x1 + [1] p(n__terms) = [1] x1 + [4] p(nil) = [0] p(recip) = [1] x1 + [0] p(s) = [1] x1 + [4] p(sqr) = [1] x1 + [0] p(terms) = [1] x1 + [5] Following rules are strictly oriented: activate(X) = [4] X + [1] > [1] X + [0] = X activate(n__first(X1,X2)) = [4] X1 + [4] X2 + [9] > [4] X1 + [4] X2 + [4] = first(activate(X1),activate(X2)) activate(n__terms(X)) = [4] X + [17] > [4] X + [6] = terms(activate(X)) dbl(0()) = [1] > [0] = 0() first(0(),X) = [1] X + [2] > [0] = nil() s(X) = [1] X + [4] > [1] X + [1] = n__s(X) terms(N) = [1] N + [5] > [1] N + [0] = cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) = [1] X + [5] > [1] X + [4] = n__terms(X) Following rules are (at-least) weakly oriented: activate(n__s(X)) = [4] X + [5] >= [4] X + [5] = s(activate(X)) add(0(),X) = [4] X + [0] >= [1] X + [0] = X first(X1,X2) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = n__first(X1,X2) sqr(0()) = [0] >= [0] = 0() * Step 3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__s(X)) -> s(activate(X)) add(0(),X) -> X first(X1,X2) -> n__first(X1,X2) sqr(0()) -> 0() - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__terms(X)) -> terms(activate(X)) dbl(0()) -> 0() first(0(),X) -> nil() s(X) -> n__s(X) terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__first/2,n__s/1,n__terms/1,nil/0,recip/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0 ,cons,n__first,n__s,n__terms,nil,recip} + 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(cons) = {1}, uargs(first) = {1,2}, uargs(recip) = {1}, uargs(s) = {1}, uargs(terms) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(activate) = [5] x1 + [0] p(add) = [5] x1 + [8] x2 + [5] p(cons) = [1] x1 + [0] p(dbl) = [1] x1 + [13] p(first) = [1] x1 + [1] x2 + [5] p(n__first) = [1] x1 + [1] x2 + [1] p(n__s) = [1] x1 + [5] p(n__terms) = [1] x1 + [1] p(nil) = [1] p(recip) = [1] x1 + [2] p(s) = [1] x1 + [6] p(sqr) = [1] x1 + [2] p(terms) = [1] x1 + [5] Following rules are strictly oriented: activate(n__s(X)) = [5] X + [25] > [5] X + [6] = s(activate(X)) add(0(),X) = [8] X + [15] > [1] X + [0] = X first(X1,X2) = [1] X1 + [1] X2 + [5] > [1] X1 + [1] X2 + [1] = n__first(X1,X2) sqr(0()) = [4] > [2] = 0() Following rules are (at-least) weakly oriented: activate(X) = [5] X + [0] >= [1] X + [0] = X activate(n__first(X1,X2)) = [5] X1 + [5] X2 + [5] >= [5] X1 + [5] X2 + [5] = first(activate(X1),activate(X2)) activate(n__terms(X)) = [5] X + [5] >= [5] X + [5] = terms(activate(X)) dbl(0()) = [15] >= [2] = 0() first(0(),X) = [1] X + [7] >= [1] = nil() s(X) = [1] X + [6] >= [1] X + [5] = n__s(X) terms(N) = [1] N + [5] >= [1] N + [4] = cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) = [1] X + [5] >= [1] X + [1] = n__terms(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__terms(X)) -> terms(activate(X)) add(0(),X) -> X dbl(0()) -> 0() first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() s(X) -> n__s(X) sqr(0()) -> 0() terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__first/2,n__s/1,n__terms/1,nil/0,recip/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0 ,cons,n__first,n__s,n__terms,nil,recip} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))