WORST_CASE(?,O(n^1)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(X)) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) dbl(X) -> n__dbl(X) dbl(0()) -> 0() dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z))) s(X) -> n__s(X) sqr(X) -> n__sqr(X) sqr(0()) -> 0() sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(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__add/2,n__dbl/1,n__first/2,n__s/1 ,n__sqr/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__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. add(s(X),Y) -> s(n__add(activate(X),Y)) dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z))) sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X)))) All above mentioned rules can be savely removed. * Step 2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(X)) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X dbl(X) -> n__dbl(X) dbl(0()) -> 0() first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() s(X) -> n__s(X) sqr(X) -> n__sqr(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__add/2,n__dbl/1,n__first/2,n__s/1 ,n__sqr/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__add,n__dbl,n__first,n__s,n__sqr,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(add) = {1,2}, uargs(cons) = {1}, uargs(dbl) = {1}, uargs(first) = {1,2}, uargs(recip) = {1}, uargs(sqr) = {1}, uargs(terms) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [2] x1 + [0] p(add) = [1] x1 + [1] x2 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(dbl) = [1] x1 + [0] p(first) = [1] x1 + [1] x2 + [3] p(n__add) = [1] x1 + [1] x2 + [0] p(n__dbl) = [1] x1 + [0] p(n__first) = [1] x1 + [1] x2 + [0] p(n__s) = [0] p(n__sqr) = [1] x1 + [0] p(n__terms) = [1] x1 + [0] p(nil) = [0] p(recip) = [1] x1 + [0] p(s) = [0] p(sqr) = [1] x1 + [0] p(terms) = [1] x1 + [1] Following rules are strictly oriented: first(X1,X2) = [1] X1 + [1] X2 + [3] > [1] X1 + [1] X2 + [0] = n__first(X1,X2) first(0(),X) = [1] X + [3] > [0] = nil() terms(N) = [1] N + [1] > [1] N + [0] = cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) = [1] X + [1] > [1] X + [0] = n__terms(X) Following rules are (at-least) weakly oriented: activate(X) = [2] X + [0] >= [1] X + [0] = X activate(n__add(X1,X2)) = [2] X1 + [2] X2 + [0] >= [2] X1 + [2] X2 + [0] = add(activate(X1),activate(X2)) activate(n__dbl(X)) = [2] X + [0] >= [2] X + [0] = dbl(activate(X)) activate(n__first(X1,X2)) = [2] X1 + [2] X2 + [0] >= [2] X1 + [2] X2 + [3] = first(activate(X1),activate(X2)) activate(n__s(X)) = [0] >= [0] = s(X) activate(n__sqr(X)) = [2] X + [0] >= [2] X + [0] = sqr(activate(X)) activate(n__terms(X)) = [2] X + [0] >= [2] X + [1] = terms(activate(X)) add(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__add(X1,X2) add(0(),X) = [1] X + [0] >= [1] X + [0] = X dbl(X) = [1] X + [0] >= [1] X + [0] = n__dbl(X) dbl(0()) = [0] >= [0] = 0() s(X) = [0] >= [0] = n__s(X) sqr(X) = [1] X + [0] >= [1] X + [0] = n__sqr(X) sqr(0()) = [0] >= [0] = 0() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(X)) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X dbl(X) -> n__dbl(X) dbl(0()) -> 0() s(X) -> n__s(X) sqr(X) -> n__sqr(X) sqr(0()) -> 0() - Weak TRS: first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() 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__add/2,n__dbl/1,n__first/2,n__s/1 ,n__sqr/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__add,n__dbl,n__first,n__s,n__sqr,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(add) = {1,2}, uargs(cons) = {1}, uargs(dbl) = {1}, uargs(first) = {1,2}, uargs(recip) = {1}, uargs(sqr) = {1}, uargs(terms) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(activate) = [2] x1 + [0] p(add) = [1] x1 + [1] x2 + [0] p(cons) = [1] x1 + [3] p(dbl) = [1] x1 + [2] p(first) = [1] x1 + [1] x2 + [2] p(n__add) = [1] x1 + [1] x2 + [2] p(n__dbl) = [1] x1 + [0] p(n__first) = [1] x1 + [1] x2 + [1] p(n__s) = [4] p(n__sqr) = [1] x1 + [1] p(n__terms) = [1] x1 + [0] p(nil) = [4] p(recip) = [1] x1 + [1] p(s) = [0] p(sqr) = [1] x1 + [1] p(terms) = [1] x1 + [6] Following rules are strictly oriented: activate(n__add(X1,X2)) = [2] X1 + [2] X2 + [4] > [2] X1 + [2] X2 + [0] = add(activate(X1),activate(X2)) activate(n__s(X)) = [8] > [0] = s(X) activate(n__sqr(X)) = [2] X + [2] > [2] X + [1] = sqr(activate(X)) add(0(),X) = [1] X + [2] > [1] X + [0] = X dbl(X) = [1] X + [2] > [1] X + [0] = n__dbl(X) dbl(0()) = [4] > [2] = 0() sqr(0()) = [3] > [2] = 0() Following rules are (at-least) weakly oriented: activate(X) = [2] X + [0] >= [1] X + [0] = X activate(n__dbl(X)) = [2] X + [0] >= [2] X + [2] = dbl(activate(X)) activate(n__first(X1,X2)) = [2] X1 + [2] X2 + [2] >= [2] X1 + [2] X2 + [2] = first(activate(X1),activate(X2)) activate(n__terms(X)) = [2] X + [0] >= [2] X + [6] = terms(activate(X)) add(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [2] = n__add(X1,X2) first(X1,X2) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [1] = n__first(X1,X2) first(0(),X) = [1] X + [4] >= [4] = nil() s(X) = [0] >= [4] = n__s(X) sqr(X) = [1] X + [1] >= [1] X + [1] = n__sqr(X) terms(N) = [1] N + [6] >= [1] N + [5] = cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) = [1] X + [6] >= [1] X + [0] = n__terms(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__terms(X)) -> terms(activate(X)) add(X1,X2) -> n__add(X1,X2) s(X) -> n__s(X) sqr(X) -> n__sqr(X) - Weak TRS: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) add(0(),X) -> X dbl(X) -> n__dbl(X) dbl(0()) -> 0() first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() 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__add/2,n__dbl/1,n__first/2,n__s/1 ,n__sqr/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__add,n__dbl,n__first,n__s,n__sqr,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(add) = {1,2}, uargs(cons) = {1}, uargs(dbl) = {1}, uargs(first) = {1,2}, uargs(recip) = {1}, uargs(sqr) = {1}, uargs(terms) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [2] p(add) = [1] x1 + [1] x2 + [0] p(cons) = [1] x1 + [0] p(dbl) = [1] x1 + [1] p(first) = [1] x1 + [1] x2 + [2] p(n__add) = [1] x1 + [1] x2 + [5] p(n__dbl) = [1] x1 + [0] p(n__first) = [1] x1 + [1] x2 + [0] p(n__s) = [5] p(n__sqr) = [1] x1 + [0] p(n__terms) = [1] x1 + [0] p(nil) = [1] p(recip) = [1] x1 + [0] p(s) = [7] p(sqr) = [1] x1 + [0] p(terms) = [1] x1 + [0] Following rules are strictly oriented: activate(X) = [1] X + [2] > [1] X + [0] = X s(X) = [7] > [5] = n__s(X) Following rules are (at-least) weakly oriented: activate(n__add(X1,X2)) = [1] X1 + [1] X2 + [7] >= [1] X1 + [1] X2 + [4] = add(activate(X1),activate(X2)) activate(n__dbl(X)) = [1] X + [2] >= [1] X + [3] = dbl(activate(X)) activate(n__first(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [6] = first(activate(X1),activate(X2)) activate(n__s(X)) = [7] >= [7] = s(X) activate(n__sqr(X)) = [1] X + [2] >= [1] X + [2] = sqr(activate(X)) activate(n__terms(X)) = [1] X + [2] >= [1] X + [2] = terms(activate(X)) add(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [5] = n__add(X1,X2) add(0(),X) = [1] X + [0] >= [1] X + [0] = X dbl(X) = [1] X + [1] >= [1] X + [0] = n__dbl(X) dbl(0()) = [1] >= [0] = 0() first(X1,X2) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [0] = n__first(X1,X2) first(0(),X) = [1] X + [2] >= [1] = nil() sqr(X) = [1] X + [0] >= [1] X + [0] = n__sqr(X) sqr(0()) = [0] >= [0] = 0() terms(N) = [1] N + [0] >= [1] N + [0] = cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) = [1] X + [0] >= [1] X + [0] = n__terms(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__terms(X)) -> terms(activate(X)) add(X1,X2) -> n__add(X1,X2) sqr(X) -> n__sqr(X) - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) add(0(),X) -> X dbl(X) -> n__dbl(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__add/2,n__dbl/1,n__first/2,n__s/1 ,n__sqr/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__add,n__dbl,n__first,n__s,n__sqr,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(add) = {1,2}, uargs(cons) = {1}, uargs(dbl) = {1}, uargs(first) = {1,2}, uargs(recip) = {1}, uargs(sqr) = {1}, uargs(terms) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [4] x1 + [0] p(add) = [1] x1 + [1] x2 + [4] p(cons) = [1] x1 + [0] p(dbl) = [1] x1 + [0] p(first) = [1] x1 + [1] x2 + [4] p(n__add) = [1] x1 + [1] x2 + [2] p(n__dbl) = [1] x1 + [0] p(n__first) = [1] x1 + [1] x2 + [0] p(n__s) = [1] p(n__sqr) = [1] x1 + [2] p(n__terms) = [1] x1 + [0] p(nil) = [1] p(recip) = [1] x1 + [0] p(s) = [4] p(sqr) = [1] x1 + [1] p(terms) = [1] x1 + [4] Following rules are strictly oriented: add(X1,X2) = [1] X1 + [1] X2 + [4] > [1] X1 + [1] X2 + [2] = n__add(X1,X2) Following rules are (at-least) weakly oriented: activate(X) = [4] X + [0] >= [1] X + [0] = X activate(n__add(X1,X2)) = [4] X1 + [4] X2 + [8] >= [4] X1 + [4] X2 + [4] = add(activate(X1),activate(X2)) activate(n__dbl(X)) = [4] X + [0] >= [4] X + [0] = dbl(activate(X)) activate(n__first(X1,X2)) = [4] X1 + [4] X2 + [0] >= [4] X1 + [4] X2 + [4] = first(activate(X1),activate(X2)) activate(n__s(X)) = [4] >= [4] = s(X) activate(n__sqr(X)) = [4] X + [8] >= [4] X + [1] = sqr(activate(X)) activate(n__terms(X)) = [4] X + [0] >= [4] X + [4] = terms(activate(X)) add(0(),X) = [1] X + [4] >= [1] X + [0] = X dbl(X) = [1] X + [0] >= [1] X + [0] = n__dbl(X) dbl(0()) = [0] >= [0] = 0() first(X1,X2) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [0] = n__first(X1,X2) first(0(),X) = [1] X + [4] >= [1] = nil() s(X) = [4] >= [1] = n__s(X) sqr(X) = [1] X + [1] >= [1] X + [2] = n__sqr(X) sqr(0()) = [1] >= [0] = 0() terms(N) = [1] N + [4] >= [1] N + [1] = cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) = [1] X + [4] >= [1] X + [0] = n__terms(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__terms(X)) -> terms(activate(X)) sqr(X) -> n__sqr(X) - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X dbl(X) -> n__dbl(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__add/2,n__dbl/1,n__first/2,n__s/1 ,n__sqr/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__add,n__dbl,n__first,n__s,n__sqr,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(add) = {1,2}, uargs(cons) = {1}, uargs(dbl) = {1}, uargs(first) = {1,2}, uargs(recip) = {1}, uargs(sqr) = {1}, uargs(terms) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [3] p(activate) = [4] x1 + [0] p(add) = [1] x1 + [1] x2 + [0] p(cons) = [1] x1 + [3] p(dbl) = [1] x1 + [0] p(first) = [1] x1 + [1] x2 + [0] p(n__add) = [1] x1 + [1] x2 + [0] p(n__dbl) = [1] x1 + [0] p(n__first) = [1] x1 + [1] x2 + [0] p(n__s) = [1] x1 + [0] p(n__sqr) = [1] x1 + [1] p(n__terms) = [1] x1 + [0] p(nil) = [3] p(recip) = [1] x1 + [0] p(s) = [4] x1 + [0] p(sqr) = [1] x1 + [4] p(terms) = [1] x1 + [7] Following rules are strictly oriented: sqr(X) = [1] X + [4] > [1] X + [1] = n__sqr(X) Following rules are (at-least) weakly oriented: activate(X) = [4] X + [0] >= [1] X + [0] = X activate(n__add(X1,X2)) = [4] X1 + [4] X2 + [0] >= [4] X1 + [4] X2 + [0] = add(activate(X1),activate(X2)) activate(n__dbl(X)) = [4] X + [0] >= [4] X + [0] = dbl(activate(X)) activate(n__first(X1,X2)) = [4] X1 + [4] X2 + [0] >= [4] X1 + [4] X2 + [0] = first(activate(X1),activate(X2)) activate(n__s(X)) = [4] X + [0] >= [4] X + [0] = s(X) activate(n__sqr(X)) = [4] X + [4] >= [4] X + [4] = sqr(activate(X)) activate(n__terms(X)) = [4] X + [0] >= [4] X + [7] = terms(activate(X)) add(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__add(X1,X2) add(0(),X) = [1] X + [3] >= [1] X + [0] = X dbl(X) = [1] X + [0] >= [1] X + [0] = n__dbl(X) dbl(0()) = [3] >= [3] = 0() first(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__first(X1,X2) first(0(),X) = [1] X + [3] >= [3] = nil() s(X) = [4] X + [0] >= [1] X + [0] = n__s(X) sqr(0()) = [7] >= [3] = 0() terms(N) = [1] N + [7] >= [1] N + [7] = cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) = [1] X + [7] >= [1] X + [0] = n__terms(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__terms(X)) -> terms(activate(X)) - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X dbl(X) -> n__dbl(X) dbl(0()) -> 0() first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() s(X) -> n__s(X) sqr(X) -> n__sqr(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__add/2,n__dbl/1,n__first/2,n__s/1 ,n__sqr/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__add,n__dbl,n__first,n__s,n__sqr,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(add) = {1,2}, uargs(cons) = {1}, uargs(dbl) = {1}, uargs(first) = {1,2}, uargs(recip) = {1}, uargs(sqr) = {1}, uargs(terms) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(activate) = [4] x1 + [1] p(add) = [1] x1 + [1] x2 + [4] p(cons) = [1] x1 + [1] x2 + [0] p(dbl) = [1] x1 + [3] p(first) = [1] x1 + [1] x2 + [0] p(n__add) = [1] x1 + [1] x2 + [2] p(n__dbl) = [1] x1 + [3] p(n__first) = [1] x1 + [1] x2 + [0] p(n__s) = [2] p(n__sqr) = [1] x1 + [0] p(n__terms) = [1] x1 + [3] p(nil) = [2] p(recip) = [1] x1 + [0] p(s) = [2] p(sqr) = [1] x1 + [0] p(terms) = [1] x1 + [5] Following rules are strictly oriented: activate(n__dbl(X)) = [4] X + [13] > [4] X + [4] = dbl(activate(X)) activate(n__terms(X)) = [4] X + [13] > [4] X + [6] = terms(activate(X)) Following rules are (at-least) weakly oriented: activate(X) = [4] X + [1] >= [1] X + [0] = X activate(n__add(X1,X2)) = [4] X1 + [4] X2 + [9] >= [4] X1 + [4] X2 + [6] = add(activate(X1),activate(X2)) activate(n__first(X1,X2)) = [4] X1 + [4] X2 + [1] >= [4] X1 + [4] X2 + [2] = first(activate(X1),activate(X2)) activate(n__s(X)) = [9] >= [2] = s(X) activate(n__sqr(X)) = [4] X + [1] >= [4] X + [1] = sqr(activate(X)) add(X1,X2) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [2] = n__add(X1,X2) add(0(),X) = [1] X + [6] >= [1] X + [0] = X dbl(X) = [1] X + [3] >= [1] X + [3] = n__dbl(X) dbl(0()) = [5] >= [2] = 0() first(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__first(X1,X2) first(0(),X) = [1] X + [2] >= [2] = nil() s(X) = [2] >= [2] = n__s(X) sqr(X) = [1] X + [0] >= [1] X + [0] = n__sqr(X) sqr(0()) = [2] >= [2] = 0() terms(N) = [1] N + [5] >= [1] N + [5] = cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) = [1] X + [5] >= [1] X + [3] = n__terms(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 8: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(X)) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X dbl(X) -> n__dbl(X) dbl(0()) -> 0() first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() s(X) -> n__s(X) sqr(X) -> n__sqr(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__add/2,n__dbl/1,n__first/2,n__s/1 ,n__sqr/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__add,n__dbl,n__first,n__s,n__sqr,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(add) = {1,2}, uargs(cons) = {1}, uargs(dbl) = {1}, uargs(first) = {1,2}, uargs(recip) = {1}, uargs(sqr) = {1}, uargs(terms) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [2] x1 + [0] p(add) = [1] x1 + [1] x2 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(dbl) = [1] x1 + [6] p(first) = [1] x1 + [1] x2 + [7] p(n__add) = [1] x1 + [1] x2 + [0] p(n__dbl) = [1] x1 + [3] p(n__first) = [1] x1 + [1] x2 + [5] p(n__s) = [0] p(n__sqr) = [1] x1 + [0] p(n__terms) = [1] x1 + [0] p(nil) = [7] p(recip) = [1] x1 + [0] p(s) = [0] p(sqr) = [1] x1 + [0] p(terms) = [1] x1 + [0] Following rules are strictly oriented: activate(n__first(X1,X2)) = [2] X1 + [2] X2 + [10] > [2] X1 + [2] X2 + [7] = first(activate(X1),activate(X2)) Following rules are (at-least) weakly oriented: activate(X) = [2] X + [0] >= [1] X + [0] = X activate(n__add(X1,X2)) = [2] X1 + [2] X2 + [0] >= [2] X1 + [2] X2 + [0] = add(activate(X1),activate(X2)) activate(n__dbl(X)) = [2] X + [6] >= [2] X + [6] = dbl(activate(X)) activate(n__s(X)) = [0] >= [0] = s(X) activate(n__sqr(X)) = [2] X + [0] >= [2] X + [0] = sqr(activate(X)) activate(n__terms(X)) = [2] X + [0] >= [2] X + [0] = terms(activate(X)) add(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__add(X1,X2) add(0(),X) = [1] X + [0] >= [1] X + [0] = X dbl(X) = [1] X + [6] >= [1] X + [3] = n__dbl(X) dbl(0()) = [6] >= [0] = 0() first(X1,X2) = [1] X1 + [1] X2 + [7] >= [1] X1 + [1] X2 + [5] = n__first(X1,X2) first(0(),X) = [1] X + [7] >= [7] = nil() s(X) = [0] >= [0] = n__s(X) sqr(X) = [1] X + [0] >= [1] X + [0] = n__sqr(X) sqr(0()) = [0] >= [0] = 0() terms(N) = [1] N + [0] >= [1] N + [0] = cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) = [1] X + [0] >= [1] X + [0] = n__terms(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 9: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(X)) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X dbl(X) -> n__dbl(X) dbl(0()) -> 0() first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() s(X) -> n__s(X) sqr(X) -> n__sqr(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__add/2,n__dbl/1,n__first/2,n__s/1 ,n__sqr/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__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))