WORST_CASE(?,O(n^2)) * Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__sqr(X) -> sqr(X) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) -> terms(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(nil()) -> nil() mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(sqr(X)) -> a__sqr(mark(X)) mark(terms(X)) -> a__terms(mark(X)) - Signature: {a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1 ,s/1,sqr/1,terms/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms ,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms} + 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(a__add) = {1,2}, uargs(a__dbl) = {1}, uargs(a__first) = {1,2}, uargs(a__sqr) = {1}, uargs(a__terms) = {1}, uargs(cons) = {1}, uargs(recip) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(a__add) = [1] x1 + [1] x2 + [0] p(a__dbl) = [1] x1 + [0] p(a__first) = [1] x1 + [1] x2 + [0] p(a__sqr) = [1] x1 + [1] p(a__terms) = [1] x1 + [0] p(add) = [0] p(cons) = [1] x1 + [0] p(dbl) = [4] p(first) = [0] p(mark) = [0] p(nil) = [0] p(recip) = [1] x1 + [0] p(s) = [1] p(sqr) = [0] p(terms) = [1] x1 + [0] Following rules are strictly oriented: a__add(0(),X) = [1] X + [1] > [0] = mark(X) a__first(0(),X) = [1] X + [1] > [0] = nil() a__first(s(X),cons(Y,Z)) = [1] Y + [1] > [0] = cons(mark(Y),first(X,Z)) a__sqr(X) = [1] X + [1] > [0] = sqr(X) a__sqr(0()) = [2] > [1] = 0() a__sqr(s(X)) = [2] > [1] = s(add(sqr(X),dbl(X))) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [1] X2 + [0] >= [0] = add(X1,X2) a__add(s(X),Y) = [1] Y + [1] >= [1] = s(add(X,Y)) a__dbl(X) = [1] X + [0] >= [4] = dbl(X) a__dbl(0()) = [1] >= [1] = 0() a__dbl(s(X)) = [1] >= [1] = s(s(dbl(X))) a__first(X1,X2) = [1] X1 + [1] X2 + [0] >= [0] = first(X1,X2) a__terms(N) = [1] N + [0] >= [1] = cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) = [1] X + [0] >= [1] X + [0] = terms(X) mark(0()) = [0] >= [1] = 0() mark(add(X1,X2)) = [0] >= [0] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [0] >= [0] = cons(mark(X1),X2) mark(dbl(X)) = [0] >= [0] = a__dbl(mark(X)) mark(first(X1,X2)) = [0] >= [0] = a__first(mark(X1),mark(X2)) mark(nil()) = [0] >= [0] = nil() mark(recip(X)) = [0] >= [0] = recip(mark(X)) mark(s(X)) = [0] >= [1] = s(X) mark(sqr(X)) = [0] >= [1] = a__sqr(mark(X)) mark(terms(X)) = [0] >= [0] = a__terms(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__add(s(X),Y) -> s(add(X,Y)) a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__first(X1,X2) -> first(X1,X2) a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) -> terms(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(nil()) -> nil() mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(sqr(X)) -> a__sqr(mark(X)) mark(terms(X)) -> a__terms(mark(X)) - Weak TRS: a__add(0(),X) -> mark(X) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__sqr(X) -> sqr(X) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1 ,s/1,sqr/1,terms/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms ,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms} + 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(a__add) = {1,2}, uargs(a__dbl) = {1}, uargs(a__first) = {1,2}, uargs(a__sqr) = {1}, uargs(a__terms) = {1}, uargs(cons) = {1}, uargs(recip) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [6] p(a__add) = [1] x1 + [1] x2 + [4] p(a__dbl) = [1] x1 + [0] p(a__first) = [1] x1 + [1] x2 + [0] p(a__sqr) = [1] x1 + [0] p(a__terms) = [1] x1 + [0] p(add) = [0] p(cons) = [1] x1 + [5] p(dbl) = [4] p(first) = [1] x1 + [1] x2 + [0] p(mark) = [4] p(nil) = [6] p(recip) = [1] x1 + [5] p(s) = [4] p(sqr) = [0] p(terms) = [1] x1 + [0] Following rules are strictly oriented: a__add(X1,X2) = [1] X1 + [1] X2 + [4] > [0] = add(X1,X2) a__add(s(X),Y) = [1] Y + [8] > [4] = s(add(X,Y)) Following rules are (at-least) weakly oriented: a__add(0(),X) = [1] X + [10] >= [4] = mark(X) a__dbl(X) = [1] X + [0] >= [4] = dbl(X) a__dbl(0()) = [6] >= [6] = 0() a__dbl(s(X)) = [4] >= [4] = s(s(dbl(X))) a__first(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = first(X1,X2) a__first(0(),X) = [1] X + [6] >= [6] = nil() a__first(s(X),cons(Y,Z)) = [1] Y + [9] >= [9] = cons(mark(Y),first(X,Z)) a__sqr(X) = [1] X + [0] >= [0] = sqr(X) a__sqr(0()) = [6] >= [6] = 0() a__sqr(s(X)) = [4] >= [4] = s(add(sqr(X),dbl(X))) a__terms(N) = [1] N + [0] >= [14] = cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) = [1] X + [0] >= [1] X + [0] = terms(X) mark(0()) = [4] >= [6] = 0() mark(add(X1,X2)) = [4] >= [12] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [4] >= [9] = cons(mark(X1),X2) mark(dbl(X)) = [4] >= [4] = a__dbl(mark(X)) mark(first(X1,X2)) = [4] >= [8] = a__first(mark(X1),mark(X2)) mark(nil()) = [4] >= [6] = nil() mark(recip(X)) = [4] >= [9] = recip(mark(X)) mark(s(X)) = [4] >= [4] = s(X) mark(sqr(X)) = [4] >= [4] = a__sqr(mark(X)) mark(terms(X)) = [4] >= [4] = a__terms(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__first(X1,X2) -> first(X1,X2) a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) -> terms(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(nil()) -> nil() mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(sqr(X)) -> a__sqr(mark(X)) mark(terms(X)) -> a__terms(mark(X)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__sqr(X) -> sqr(X) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1 ,s/1,sqr/1,terms/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms ,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms} + 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(a__add) = {1,2}, uargs(a__dbl) = {1}, uargs(a__first) = {1,2}, uargs(a__sqr) = {1}, uargs(a__terms) = {1}, uargs(cons) = {1}, uargs(recip) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__add) = [1] x1 + [1] x2 + [5] p(a__dbl) = [1] x1 + [0] p(a__first) = [1] x1 + [1] x2 + [3] p(a__sqr) = [1] x1 + [0] p(a__terms) = [1] x1 + [0] p(add) = [0] p(cons) = [1] x1 + [0] p(dbl) = [0] p(first) = [1] x1 + [1] x2 + [0] p(mark) = [5] p(nil) = [3] p(recip) = [1] x1 + [2] p(s) = [2] p(sqr) = [0] p(terms) = [1] x1 + [0] Following rules are strictly oriented: a__first(X1,X2) = [1] X1 + [1] X2 + [3] > [1] X1 + [1] X2 + [0] = first(X1,X2) mark(0()) = [5] > [0] = 0() mark(nil()) = [5] > [3] = nil() mark(s(X)) = [5] > [2] = s(X) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [1] X2 + [5] >= [0] = add(X1,X2) a__add(0(),X) = [1] X + [5] >= [5] = mark(X) a__add(s(X),Y) = [1] Y + [7] >= [2] = s(add(X,Y)) a__dbl(X) = [1] X + [0] >= [0] = dbl(X) a__dbl(0()) = [0] >= [0] = 0() a__dbl(s(X)) = [2] >= [2] = s(s(dbl(X))) a__first(0(),X) = [1] X + [3] >= [3] = nil() a__first(s(X),cons(Y,Z)) = [1] Y + [5] >= [5] = cons(mark(Y),first(X,Z)) a__sqr(X) = [1] X + [0] >= [0] = sqr(X) a__sqr(0()) = [0] >= [0] = 0() a__sqr(s(X)) = [2] >= [2] = s(add(sqr(X),dbl(X))) a__terms(N) = [1] N + [0] >= [7] = cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) = [1] X + [0] >= [1] X + [0] = terms(X) mark(add(X1,X2)) = [5] >= [15] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [5] >= [5] = cons(mark(X1),X2) mark(dbl(X)) = [5] >= [5] = a__dbl(mark(X)) mark(first(X1,X2)) = [5] >= [13] = a__first(mark(X1),mark(X2)) mark(recip(X)) = [5] >= [7] = recip(mark(X)) mark(sqr(X)) = [5] >= [5] = a__sqr(mark(X)) mark(terms(X)) = [5] >= [5] = a__terms(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) -> terms(X) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(recip(X)) -> recip(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(terms(X)) -> a__terms(mark(X)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__sqr(X) -> sqr(X) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) mark(0()) -> 0() mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1 ,s/1,sqr/1,terms/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms ,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms} + 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(a__add) = {1,2}, uargs(a__dbl) = {1}, uargs(a__first) = {1,2}, uargs(a__sqr) = {1}, uargs(a__terms) = {1}, uargs(cons) = {1}, uargs(recip) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__add) = [1] x1 + [1] x2 + [1] p(a__dbl) = [1] x1 + [0] p(a__first) = [1] x1 + [1] x2 + [0] p(a__sqr) = [1] x1 + [0] p(a__terms) = [1] x1 + [4] p(add) = [1] x2 + [0] p(cons) = [1] x1 + [2] p(dbl) = [1] x1 + [0] p(first) = [1] x1 + [1] x2 + [0] p(mark) = [0] p(nil) = [0] p(recip) = [1] x1 + [1] p(s) = [0] p(sqr) = [1] x1 + [0] p(terms) = [1] x1 + [0] Following rules are strictly oriented: a__terms(N) = [1] N + [4] > [3] = cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) = [1] X + [4] > [1] X + [0] = terms(X) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [1] X2 + [1] >= [1] X2 + [0] = add(X1,X2) a__add(0(),X) = [1] X + [1] >= [0] = mark(X) a__add(s(X),Y) = [1] Y + [1] >= [0] = s(add(X,Y)) a__dbl(X) = [1] X + [0] >= [1] X + [0] = dbl(X) a__dbl(0()) = [0] >= [0] = 0() a__dbl(s(X)) = [0] >= [0] = s(s(dbl(X))) a__first(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = first(X1,X2) a__first(0(),X) = [1] X + [0] >= [0] = nil() a__first(s(X),cons(Y,Z)) = [1] Y + [2] >= [2] = cons(mark(Y),first(X,Z)) a__sqr(X) = [1] X + [0] >= [1] X + [0] = sqr(X) a__sqr(0()) = [0] >= [0] = 0() a__sqr(s(X)) = [0] >= [0] = s(add(sqr(X),dbl(X))) mark(0()) = [0] >= [0] = 0() mark(add(X1,X2)) = [0] >= [1] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [0] >= [2] = cons(mark(X1),X2) mark(dbl(X)) = [0] >= [0] = a__dbl(mark(X)) mark(first(X1,X2)) = [0] >= [0] = a__first(mark(X1),mark(X2)) mark(nil()) = [0] >= [0] = nil() mark(recip(X)) = [0] >= [1] = recip(mark(X)) mark(s(X)) = [0] >= [0] = s(X) mark(sqr(X)) = [0] >= [0] = a__sqr(mark(X)) mark(terms(X)) = [0] >= [4] = a__terms(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(recip(X)) -> recip(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(terms(X)) -> a__terms(mark(X)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__sqr(X) -> sqr(X) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) -> terms(X) mark(0()) -> 0() mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1 ,s/1,sqr/1,terms/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms ,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms} + 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(a__add) = {1,2}, uargs(a__dbl) = {1}, uargs(a__first) = {1,2}, uargs(a__sqr) = {1}, uargs(a__terms) = {1}, uargs(cons) = {1}, uargs(recip) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__add) = [1] x1 + [1] x2 + [4] p(a__dbl) = [1] x1 + [0] p(a__first) = [1] x1 + [1] x2 + [0] p(a__sqr) = [1] x1 + [1] p(a__terms) = [1] x1 + [3] p(add) = [1] x1 + [1] x2 + [4] p(cons) = [1] x1 + [0] p(dbl) = [1] x1 + [2] p(first) = [1] x1 + [1] x2 + [0] p(mark) = [1] x1 + [0] p(nil) = [0] p(recip) = [1] x1 + [2] p(s) = [1] p(sqr) = [1] x1 + [0] p(terms) = [1] x1 + [0] Following rules are strictly oriented: mark(dbl(X)) = [1] X + [2] > [1] X + [0] = a__dbl(mark(X)) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [4] = add(X1,X2) a__add(0(),X) = [1] X + [4] >= [1] X + [0] = mark(X) a__add(s(X),Y) = [1] Y + [5] >= [1] = s(add(X,Y)) a__dbl(X) = [1] X + [0] >= [1] X + [2] = dbl(X) a__dbl(0()) = [0] >= [0] = 0() a__dbl(s(X)) = [1] >= [1] = s(s(dbl(X))) a__first(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = first(X1,X2) a__first(0(),X) = [1] X + [0] >= [0] = nil() a__first(s(X),cons(Y,Z)) = [1] Y + [1] >= [1] Y + [0] = cons(mark(Y),first(X,Z)) a__sqr(X) = [1] X + [1] >= [1] X + [0] = sqr(X) a__sqr(0()) = [1] >= [0] = 0() a__sqr(s(X)) = [2] >= [1] = s(add(sqr(X),dbl(X))) a__terms(N) = [1] N + [3] >= [1] N + [3] = cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) = [1] X + [3] >= [1] X + [0] = terms(X) mark(0()) = [0] >= [0] = 0() mark(add(X1,X2)) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [4] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = cons(mark(X1),X2) mark(first(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = a__first(mark(X1),mark(X2)) mark(nil()) = [0] >= [0] = nil() mark(recip(X)) = [1] X + [2] >= [1] X + [2] = recip(mark(X)) mark(s(X)) = [1] >= [1] = s(X) mark(sqr(X)) = [1] X + [0] >= [1] X + [1] = a__sqr(mark(X)) mark(terms(X)) = [1] X + [0] >= [1] X + [3] = a__terms(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(recip(X)) -> recip(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(terms(X)) -> a__terms(mark(X)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__sqr(X) -> sqr(X) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) -> terms(X) mark(0()) -> 0() mark(dbl(X)) -> a__dbl(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1 ,s/1,sqr/1,terms/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms ,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms} + 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(a__add) = {1,2}, uargs(a__dbl) = {1}, uargs(a__first) = {1,2}, uargs(a__sqr) = {1}, uargs(a__terms) = {1}, uargs(cons) = {1}, uargs(recip) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(a__add) = [1] x1 + [1] x2 + [5] p(a__dbl) = [1] x1 + [1] p(a__first) = [1] x1 + [1] x2 + [5] p(a__sqr) = [1] x1 + [4] p(a__terms) = [1] x1 + [6] p(add) = [1] x1 + [1] x2 + [0] p(cons) = [1] x1 + [0] p(dbl) = [1] x1 + [4] p(first) = [1] x1 + [1] x2 + [0] p(mark) = [1] x1 + [0] p(nil) = [1] p(recip) = [1] x1 + [1] p(s) = [0] p(sqr) = [1] x1 + [0] p(terms) = [1] x1 + [0] Following rules are strictly oriented: a__dbl(0()) = [2] > [1] = 0() a__dbl(s(X)) = [1] > [0] = s(s(dbl(X))) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [0] = add(X1,X2) a__add(0(),X) = [1] X + [6] >= [1] X + [0] = mark(X) a__add(s(X),Y) = [1] Y + [5] >= [0] = s(add(X,Y)) a__dbl(X) = [1] X + [1] >= [1] X + [4] = dbl(X) a__first(X1,X2) = [1] X1 + [1] X2 + [5] >= [1] X1 + [1] X2 + [0] = first(X1,X2) a__first(0(),X) = [1] X + [6] >= [1] = nil() a__first(s(X),cons(Y,Z)) = [1] Y + [5] >= [1] Y + [0] = cons(mark(Y),first(X,Z)) a__sqr(X) = [1] X + [4] >= [1] X + [0] = sqr(X) a__sqr(0()) = [5] >= [1] = 0() a__sqr(s(X)) = [4] >= [0] = s(add(sqr(X),dbl(X))) a__terms(N) = [1] N + [6] >= [1] N + [5] = cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) = [1] X + [6] >= [1] X + [0] = terms(X) mark(0()) = [1] >= [1] = 0() mark(add(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [5] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = cons(mark(X1),X2) mark(dbl(X)) = [1] X + [4] >= [1] X + [1] = a__dbl(mark(X)) mark(first(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [5] = a__first(mark(X1),mark(X2)) mark(nil()) = [1] >= [1] = nil() mark(recip(X)) = [1] X + [1] >= [1] X + [1] = recip(mark(X)) mark(s(X)) = [0] >= [0] = s(X) mark(sqr(X)) = [1] X + [0] >= [1] X + [4] = a__sqr(mark(X)) mark(terms(X)) = [1] X + [0] >= [1] X + [6] = a__terms(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__dbl(X) -> dbl(X) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(recip(X)) -> recip(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(terms(X)) -> a__terms(mark(X)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__sqr(X) -> sqr(X) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) -> terms(X) mark(0()) -> 0() mark(dbl(X)) -> a__dbl(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1 ,s/1,sqr/1,terms/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms ,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__add) = {1,2}, uargs(a__dbl) = {1}, uargs(a__first) = {1,2}, uargs(a__sqr) = {1}, uargs(a__terms) = {1}, uargs(cons) = {1}, uargs(recip) = {1} Following symbols are considered usable: {a__add,a__dbl,a__first,a__sqr,a__terms,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__add) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__dbl) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 1] [1] p(a__first) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__sqr) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(a__terms) = [1 1 1] [0] [0 1 0] x1 + [0] [0 1 1] [0] p(add) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(cons) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(dbl) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(first) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(mark) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(recip) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(s) = [0 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] p(sqr) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(terms) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] Following rules are strictly oriented: a__dbl(X) = [1 1 0] [1] [0 0 0] X + [0] [0 0 1] [1] > [1 1 0] [0] [0 0 0] X + [0] [0 0 1] [1] = dbl(X) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = add(X1,X2) a__add(0(),X) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(X) a__add(s(X),Y) = [0 1 0] [1 0 1] [1] [0 0 0] X + [0 0 0] Y + [0] [0 0 0] [0 0 1] [0] >= [1] [0] [0] = s(add(X,Y)) a__dbl(0()) = [1] [0] [1] >= [0] [0] [0] = 0() a__dbl(s(X)) = [0 1 0] [2] [0 0 0] X + [0] [0 0 0] [1] >= [1] [0] [0] = s(s(dbl(X))) a__first(X1,X2) = [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = first(X1,X2) a__first(0(),X) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0] [0] [0] = nil() a__first(s(X),cons(Y,Z)) = [0 1 0] [1 1 1] [1] [0 0 0] X + [0 0 0] Y + [0] [0 0 0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] Y + [0] [0 0 1] [0] = cons(mark(Y),first(X,Z)) a__sqr(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = sqr(X) a__sqr(0()) = [0] [0] [0] >= [0] [0] [0] = 0() a__sqr(s(X)) = [0 1 0] [1] [0 0 0] X + [0] [0 0 0] [0] >= [1] [0] [0] = s(add(sqr(X),dbl(X))) a__terms(N) = [1 1 1] [0] [0 1 0] N + [0] [0 1 1] [0] >= [1 0 1] [0] [0 0 0] N + [0] [0 0 1] [0] = cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) = [1 1 1] [0] [0 1 0] X + [0] [0 1 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = terms(X) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(add(X1,X2)) = [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 1 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(dbl(X)) = [1 1 1] [1] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [1] = a__dbl(mark(X)) mark(first(X1,X2)) = [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__first(mark(X1),mark(X2)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(recip(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = recip(mark(X)) mark(s(X)) = [0 1 0] [1] [0 0 0] X + [0] [0 0 0] [0] >= [0 1 0] [1] [0 0 0] X + [0] [0 0 0] [0] = s(X) mark(sqr(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__sqr(mark(X)) mark(terms(X)) = [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] = a__terms(mark(X)) * Step 8: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(recip(X)) -> recip(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(terms(X)) -> a__terms(mark(X)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__sqr(X) -> sqr(X) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) -> terms(X) mark(0()) -> 0() mark(dbl(X)) -> a__dbl(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1 ,s/1,sqr/1,terms/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms ,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__add) = {1,2}, uargs(a__dbl) = {1}, uargs(a__first) = {1,2}, uargs(a__sqr) = {1}, uargs(a__terms) = {1}, uargs(cons) = {1}, uargs(recip) = {1} Following symbols are considered usable: {a__add,a__dbl,a__first,a__sqr,a__terms,mark} TcT has computed the following interpretation: p(0) = [1] [0] [0] p(a__add) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__dbl) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [0] p(a__first) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__sqr) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__terms) = [1 0 1] [1] [0 0 0] x1 + [0] [0 1 1] [1] p(add) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(cons) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(dbl) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [0] p(first) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(mark) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(recip) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 1] [1] p(s) = [0] [0] [0] p(sqr) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(terms) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] Following rules are strictly oriented: mark(recip(X)) = [1 0 1] [2] [0 0 0] X + [0] [0 0 1] [1] > [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [1] = recip(mark(X)) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = add(X1,X2) a__add(0(),X) = [1 0 1] [1] [0 1 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(X) a__add(s(X),Y) = [1 0 1] [0] [0 1 0] Y + [0] [0 0 1] [0] >= [0] [0] [0] = s(add(X,Y)) a__dbl(X) = [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [0] = dbl(X) a__dbl(0()) = [2] [0] [0] >= [1] [0] [0] = 0() a__dbl(s(X)) = [1] [0] [0] >= [0] [0] [0] = s(s(dbl(X))) a__first(X1,X2) = [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = first(X1,X2) a__first(0(),X) = [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [0] >= [0] [0] [0] = nil() a__first(s(X),cons(Y,Z)) = [1 0 1] [0] [0 0 0] Y + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] Y + [0] [0 0 1] [0] = cons(mark(Y),first(X,Z)) a__sqr(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = sqr(X) a__sqr(0()) = [1] [0] [0] >= [1] [0] [0] = 0() a__sqr(s(X)) = [0] [0] [0] >= [0] [0] [0] = s(add(sqr(X),dbl(X))) a__terms(N) = [1 0 1] [1] [0 0 0] N + [0] [0 1 1] [1] >= [1 0 1] [1] [0 0 0] N + [0] [0 0 1] [1] = cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) = [1 0 1] [1] [0 0 0] X + [0] [0 1 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = terms(X) mark(0()) = [1] [0] [0] >= [1] [0] [0] = 0() mark(add(X1,X2)) = [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(dbl(X)) = [1 0 2] [1] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 2] [1] [0 0 0] X + [0] [0 0 1] [0] = a__dbl(mark(X)) mark(first(X1,X2)) = [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__first(mark(X1),mark(X2)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(s(X)) = [0] [0] [0] >= [0] [0] [0] = s(X) mark(sqr(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__sqr(mark(X)) mark(terms(X)) = [1 0 2] [1] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 2] [1] [0 0 0] X + [0] [0 0 1] [1] = a__terms(mark(X)) * Step 9: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(sqr(X)) -> a__sqr(mark(X)) mark(terms(X)) -> a__terms(mark(X)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__sqr(X) -> sqr(X) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) -> terms(X) mark(0()) -> 0() mark(dbl(X)) -> a__dbl(mark(X)) mark(nil()) -> nil() mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) - Signature: {a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1 ,s/1,sqr/1,terms/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms ,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__add) = {1,2}, uargs(a__dbl) = {1}, uargs(a__first) = {1,2}, uargs(a__sqr) = {1}, uargs(a__terms) = {1}, uargs(cons) = {1}, uargs(recip) = {1} Following symbols are considered usable: {a__add,a__dbl,a__first,a__sqr,a__terms,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__add) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__dbl) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__first) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__sqr) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__terms) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(add) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(cons) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(dbl) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(first) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(mark) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(recip) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(s) = [0] [0] [0] p(sqr) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(terms) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] Following rules are strictly oriented: mark(add(X1,X2)) = [1 0 1] [1 0 2] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] > [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = a__add(mark(X1),mark(X2)) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = add(X1,X2) a__add(0(),X) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(X) a__add(s(X),Y) = [1 0 1] [0] [0 0 0] Y + [0] [0 0 1] [1] >= [0] [0] [0] = s(add(X,Y)) a__dbl(X) = [1 1 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 1 0] [0] [0 0 0] X + [0] [0 0 1] [0] = dbl(X) a__dbl(0()) = [0] [0] [0] >= [0] [0] [0] = 0() a__dbl(s(X)) = [0] [0] [0] >= [0] [0] [0] = s(s(dbl(X))) a__first(X1,X2) = [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = first(X1,X2) a__first(0(),X) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0] [0] [0] = nil() a__first(s(X),cons(Y,Z)) = [1 0 1] [0] [0 0 0] Y + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] Y + [0] [0 0 1] [0] = cons(mark(Y),first(X,Z)) a__sqr(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = sqr(X) a__sqr(0()) = [0] [0] [0] >= [0] [0] [0] = 0() a__sqr(s(X)) = [0] [0] [0] >= [0] [0] [0] = s(add(sqr(X),dbl(X))) a__terms(N) = [1 0 1] [0] [0 0 0] N + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] N + [0] [0 0 1] [0] = cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = terms(X) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(cons(X1,X2)) = [1 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(dbl(X)) = [1 1 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__dbl(mark(X)) mark(first(X1,X2)) = [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__first(mark(X1),mark(X2)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(recip(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = recip(mark(X)) mark(s(X)) = [0] [0] [0] >= [0] [0] [0] = s(X) mark(sqr(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__sqr(mark(X)) mark(terms(X)) = [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] = a__terms(mark(X)) * Step 10: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(sqr(X)) -> a__sqr(mark(X)) mark(terms(X)) -> a__terms(mark(X)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__sqr(X) -> sqr(X) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) -> terms(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(nil()) -> nil() mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) - Signature: {a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1 ,s/1,sqr/1,terms/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms ,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__add) = {1,2}, uargs(a__dbl) = {1}, uargs(a__first) = {1,2}, uargs(a__sqr) = {1}, uargs(a__terms) = {1}, uargs(cons) = {1}, uargs(recip) = {1} Following symbols are considered usable: {a__add,a__dbl,a__first,a__sqr,a__terms,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__add) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__dbl) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__first) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__sqr) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__terms) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(add) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(cons) = [1 0 0] [0 1 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 0] [0] p(dbl) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(first) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(mark) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(recip) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(s) = [0] [0] [0] p(sqr) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(terms) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] Following rules are strictly oriented: mark(first(X1,X2)) = [1 0 1] [1 0 2] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] > [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = a__first(mark(X1),mark(X2)) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = add(X1,X2) a__add(0(),X) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(X) a__add(s(X),Y) = [1 0 1] [0] [0 0 0] Y + [0] [0 0 1] [1] >= [0] [0] [0] = s(add(X,Y)) a__dbl(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = dbl(X) a__dbl(0()) = [0] [0] [0] >= [0] [0] [0] = 0() a__dbl(s(X)) = [0] [0] [0] >= [0] [0] [0] = s(s(dbl(X))) a__first(X1,X2) = [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = first(X1,X2) a__first(0(),X) = [1 0 1] [0] [0 1 0] X + [0] [0 0 1] [1] >= [0] [0] [0] = nil() a__first(s(X),cons(Y,Z)) = [1 0 1] [0 1 0] [0] [0 0 0] Y + [0 0 0] Z + [0] [0 0 1] [0 0 0] [1] >= [1 0 1] [0] [0 0 0] Y + [0] [0 0 1] [0] = cons(mark(Y),first(X,Z)) a__sqr(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = sqr(X) a__sqr(0()) = [0] [0] [0] >= [0] [0] [0] = 0() a__sqr(s(X)) = [0] [0] [0] >= [0] [0] [0] = s(add(sqr(X),dbl(X))) a__terms(N) = [1 0 1] [0] [0 0 0] N + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] N + [0] [0 0 1] [0] = cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = terms(X) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(add(X1,X2)) = [1 0 1] [1 0 2] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 0 1] [0 1 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 0] [0] >= [1 0 1] [0 1 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 0] [0] = cons(mark(X1),X2) mark(dbl(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__dbl(mark(X)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(recip(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = recip(mark(X)) mark(s(X)) = [0] [0] [0] >= [0] [0] [0] = s(X) mark(sqr(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__sqr(mark(X)) mark(terms(X)) = [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] = a__terms(mark(X)) * Step 11: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(sqr(X)) -> a__sqr(mark(X)) mark(terms(X)) -> a__terms(mark(X)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__sqr(X) -> sqr(X) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) -> terms(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(nil()) -> nil() mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) - Signature: {a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1 ,s/1,sqr/1,terms/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms ,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__add) = {1,2}, uargs(a__dbl) = {1}, uargs(a__first) = {1,2}, uargs(a__sqr) = {1}, uargs(a__terms) = {1}, uargs(cons) = {1}, uargs(recip) = {1} Following symbols are considered usable: {a__add,a__dbl,a__first,a__sqr,a__terms,mark} TcT has computed the following interpretation: p(0) = [1] [0] [0] p(a__add) = [1 0 0] [1 0 1] [1] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__dbl) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [0] p(a__first) = [1 0 1] [1 0 1] [1] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__sqr) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__terms) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(add) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(cons) = [1 0 0] [0 1 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 0] [0] p(dbl) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [0] p(first) = [1 0 1] [1 0 1] [1] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(mark) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(recip) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(s) = [0 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] p(sqr) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(terms) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] Following rules are strictly oriented: mark(terms(X)) = [1 0 2] [1] [0 0 0] X + [0] [0 0 1] [1] > [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [1] = a__terms(mark(X)) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1 0 0] [1 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = add(X1,X2) a__add(0(),X) = [1 0 1] [2] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(X) a__add(s(X),Y) = [0 1 0] [1 0 1] [2] [0 0 0] X + [0 0 0] Y + [0] [0 0 0] [0 0 1] [1] >= [1] [0] [0] = s(add(X,Y)) a__dbl(X) = [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [0] = dbl(X) a__dbl(0()) = [2] [0] [0] >= [1] [0] [0] = 0() a__dbl(s(X)) = [0 1 0] [2] [0 0 0] X + [0] [0 0 0] [0] >= [1] [0] [0] = s(s(dbl(X))) a__first(X1,X2) = [1 0 1] [1 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = first(X1,X2) a__first(0(),X) = [1 0 1] [2] [0 0 0] X + [0] [0 0 1] [0] >= [0] [0] [0] = nil() a__first(s(X),cons(Y,Z)) = [0 1 0] [1 0 1] [0 1 0] [2] [0 0 0] X + [0 0 0] Y + [0 0 0] Z + [0] [0 0 0] [0 0 1] [0 0 0] [0] >= [1 0 1] [0] [0 0 0] Y + [0] [0 0 1] [0] = cons(mark(Y),first(X,Z)) a__sqr(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = sqr(X) a__sqr(0()) = [1] [0] [0] >= [1] [0] [0] = 0() a__sqr(s(X)) = [0 1 0] [1] [0 0 0] X + [0] [0 0 0] [0] >= [1] [0] [0] = s(add(sqr(X),dbl(X))) a__terms(N) = [1 0 1] [0] [0 0 0] N + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] N + [0] [0 0 1] [0] = cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = terms(X) mark(0()) = [1] [0] [0] >= [1] [0] [0] = 0() mark(add(X1,X2)) = [1 0 1] [1 0 2] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 0 1] [1 0 2] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 0 1] [0 1 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 0] [0] >= [1 0 1] [0 1 0] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 0] [0] = cons(mark(X1),X2) mark(dbl(X)) = [1 0 2] [1] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 2] [1] [0 0 0] X + [0] [0 0 1] [0] = a__dbl(mark(X)) mark(first(X1,X2)) = [1 0 2] [1 0 2] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 2] [1 0 2] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__first(mark(X1),mark(X2)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(recip(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = recip(mark(X)) mark(s(X)) = [0 1 0] [1] [0 0 0] X + [0] [0 0 0] [0] >= [0 1 0] [1] [0 0 0] X + [0] [0 0 0] [0] = s(X) mark(sqr(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__sqr(mark(X)) * Step 12: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(sqr(X)) -> a__sqr(mark(X)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__sqr(X) -> sqr(X) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) -> terms(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(nil()) -> nil() mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(terms(X)) -> a__terms(mark(X)) - Signature: {a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1 ,s/1,sqr/1,terms/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms ,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__add) = {1,2}, uargs(a__dbl) = {1}, uargs(a__first) = {1,2}, uargs(a__sqr) = {1}, uargs(a__terms) = {1}, uargs(cons) = {1}, uargs(recip) = {1} Following symbols are considered usable: {a__add,a__dbl,a__first,a__sqr,a__terms,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__add) = [1 1 0] [1 0 1] [1] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__dbl) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(a__first) = [1 1 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__sqr) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(a__terms) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(add) = [1 0 0] [1 0 1] [1] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(cons) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(dbl) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(first) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(mark) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [0] p(recip) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(s) = [0] [0] [0] p(sqr) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(terms) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] Following rules are strictly oriented: mark(sqr(X)) = [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [1] > [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = a__sqr(mark(X)) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1 1 0] [1 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = add(X1,X2) a__add(0(),X) = [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(X) a__add(s(X),Y) = [1 0 1] [1] [0 0 0] Y + [0] [0 0 1] [0] >= [0] [0] [0] = s(add(X,Y)) a__dbl(X) = [1 1 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = dbl(X) a__dbl(0()) = [0] [0] [0] >= [0] [0] [0] = 0() a__dbl(s(X)) = [0] [0] [0] >= [0] [0] [0] = s(s(dbl(X))) a__first(X1,X2) = [1 1 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = first(X1,X2) a__first(0(),X) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0] [0] [0] = nil() a__first(s(X),cons(Y,Z)) = [1 0 1] [0] [0 0 0] Y + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] Y + [0] [0 0 1] [0] = cons(mark(Y),first(X,Z)) a__sqr(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] = sqr(X) a__sqr(0()) = [0] [0] [1] >= [0] [0] [0] = 0() a__sqr(s(X)) = [0] [0] [1] >= [0] [0] [0] = s(add(sqr(X),dbl(X))) a__terms(N) = [1 0 1] [0] [0 0 0] N + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] N + [0] [0 0 1] [1] = cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = terms(X) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(add(X1,X2)) = [1 0 1] [1 0 2] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 0 2] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(dbl(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__dbl(mark(X)) mark(first(X1,X2)) = [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__first(mark(X1),mark(X2)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(recip(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = recip(mark(X)) mark(s(X)) = [0] [0] [0] >= [0] [0] [0] = s(X) mark(terms(X)) = [1 0 2] [1] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [1] = a__terms(mark(X)) * Step 13: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(cons(X1,X2)) -> cons(mark(X1),X2) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__sqr(X) -> sqr(X) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) -> terms(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(nil()) -> nil() mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(sqr(X)) -> a__sqr(mark(X)) mark(terms(X)) -> a__terms(mark(X)) - Signature: {a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1 ,s/1,sqr/1,terms/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms ,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__add) = {1,2}, uargs(a__dbl) = {1}, uargs(a__first) = {1,2}, uargs(a__sqr) = {1}, uargs(a__terms) = {1}, uargs(cons) = {1}, uargs(recip) = {1} Following symbols are considered usable: {a__add,a__dbl,a__first,a__sqr,a__terms,mark} TcT has computed the following interpretation: p(0) = [1] [1] [0] p(a__add) = [1 0 0] [1 1 0] [1] [0 1 0] x1 + [0 1 0] x2 + [1] [0 0 0] [0 0 1] [0] p(a__dbl) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(a__first) = [1 0 1] [1 1 1] [1] [0 1 1] x1 + [0 1 0] x2 + [1] [0 0 0] [0 0 0] [0] p(a__sqr) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(a__terms) = [1 1 1] [0] [0 1 0] x1 + [1] [0 0 1] [0] p(add) = [1 0 0] [1 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [1] [0 0 0] [0 0 0] [0] p(cons) = [1 0 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(dbl) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(first) = [1 0 1] [1 1 0] [1] [0 1 1] x1 + [0 1 0] x2 + [1] [0 0 0] [0 0 0] [0] p(mark) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(nil) = [0] [0] [0] p(recip) = [1 0 0] [0] [0 1 1] x1 + [0] [0 0 0] [0] p(s) = [0] [0] [0] p(sqr) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(terms) = [1 1 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] Following rules are strictly oriented: mark(cons(X1,X2)) = [1 1 0] [1] [0 1 0] X1 + [1] [0 0 0] [0] > [1 1 0] [0] [0 1 0] X1 + [1] [0 0 0] [0] = cons(mark(X1),X2) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1 0 0] [1 1 0] [1] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 1] [0] >= [1 0 0] [1 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] = add(X1,X2) a__add(0(),X) = [1 1 0] [2] [0 1 0] X + [2] [0 0 1] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = mark(X) a__add(s(X),Y) = [1 1 0] [1] [0 1 0] Y + [1] [0 0 1] [0] >= [0] [0] [0] = s(add(X,Y)) a__dbl(X) = [1 1 0] [0] [0 1 0] X + [0] [0 0 1] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = dbl(X) a__dbl(0()) = [2] [1] [0] >= [1] [1] [0] = 0() a__dbl(s(X)) = [0] [0] [0] >= [0] [0] [0] = s(s(dbl(X))) a__first(X1,X2) = [1 0 1] [1 1 1] [1] [0 1 1] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] >= [1 0 1] [1 1 0] [1] [0 1 1] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] = first(X1,X2) a__first(0(),X) = [1 1 1] [2] [0 1 0] X + [2] [0 0 0] [0] >= [0] [0] [0] = nil() a__first(s(X),cons(Y,Z)) = [1 1 0] [2] [0 1 0] Y + [2] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] Y + [1] [0 0 0] [0] = cons(mark(Y),first(X,Z)) a__sqr(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = sqr(X) a__sqr(0()) = [1] [1] [0] >= [1] [1] [0] = 0() a__sqr(s(X)) = [0] [0] [0] >= [0] [0] [0] = s(add(sqr(X),dbl(X))) a__terms(N) = [1 1 1] [0] [0 1 0] N + [1] [0 0 1] [0] >= [1 1 0] [0] [0 1 0] N + [1] [0 0 0] [0] = cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) = [1 1 1] [0] [0 1 0] X + [1] [0 0 1] [0] >= [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] = terms(X) mark(0()) = [2] [1] [0] >= [1] [1] [0] = 0() mark(add(X1,X2)) = [1 1 0] [1 2 0] [1] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] >= [1 1 0] [1 2 0] [1] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] = a__add(mark(X1),mark(X2)) mark(dbl(X)) = [1 2 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 2 0] [0] [0 1 0] X + [0] [0 0 0] [0] = a__dbl(mark(X)) mark(first(X1,X2)) = [1 1 2] [1 2 0] [2] [0 1 1] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] >= [1 1 0] [1 2 0] [1] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 0] [0] = a__first(mark(X1),mark(X2)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(recip(X)) = [1 1 1] [0] [0 1 1] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = recip(mark(X)) mark(s(X)) = [0] [0] [0] >= [0] [0] [0] = s(X) mark(sqr(X)) = [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = a__sqr(mark(X)) mark(terms(X)) = [1 2 0] [1] [0 1 0] X + [1] [0 0 0] [0] >= [1 2 0] [0] [0 1 0] X + [1] [0 0 0] [0] = a__terms(mark(X)) * Step 14: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__sqr(X) -> sqr(X) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(add(sqr(X),dbl(X))) a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) -> terms(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(nil()) -> nil() mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(X) mark(sqr(X)) -> a__sqr(mark(X)) mark(terms(X)) -> a__terms(mark(X)) - Signature: {a__add/2,a__dbl/1,a__first/2,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,nil/0,recip/1 ,s/1,sqr/1,terms/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__sqr,a__terms ,mark} and constructors {0,add,cons,dbl,first,nil,recip,s,sqr,terms} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))