WORST_CASE(?,O(n^1)) * Step 1: WeightGap WORST_CASE(?,O(n^1)) + 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__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if ,mark} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + 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}, uargs(a__and) = {1}, uargs(a__first) = {1,2}, uargs(a__if) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [4] p(a__add) = [1] x1 + [0] p(a__and) = [1] x1 + [3] p(a__first) = [1] x1 + [1] x2 + [0] p(a__from) = [0] p(a__if) = [1] x1 + [0] p(add) = [0] p(and) = [0] p(cons) = [0] p(false) = [0] p(first) = [0] p(from) = [0] p(if) = [0] p(mark) = [4] p(nil) = [0] p(s) = [2] p(true) = [5] Following rules are strictly oriented: a__and(X1,X2) = [1] X1 + [3] > [0] = and(X1,X2) a__and(false(),Y) = [3] > [0] = false() a__and(true(),X) = [8] > [4] = mark(X) a__first(0(),X) = [1] X + [4] > [0] = nil() a__first(s(X),cons(Y,Z)) = [2] > [0] = cons(Y,first(X,Z)) a__if(true(),X,Y) = [5] > [4] = mark(X) mark(cons(X1,X2)) = [4] > [0] = cons(X1,X2) mark(false()) = [4] > [0] = false() mark(from(X)) = [4] > [0] = a__from(X) mark(nil()) = [4] > [0] = nil() mark(s(X)) = [4] > [2] = s(X) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [0] >= [0] = add(X1,X2) a__add(0(),X) = [4] >= [4] = mark(X) a__add(s(X),Y) = [2] >= [2] = s(add(X,Y)) a__first(X1,X2) = [1] X1 + [1] X2 + [0] >= [0] = first(X1,X2) a__from(X) = [0] >= [0] = cons(X,from(s(X))) a__from(X) = [0] >= [0] = from(X) a__if(X1,X2,X3) = [1] X1 + [0] >= [0] = if(X1,X2,X3) a__if(false(),X,Y) = [0] >= [4] = mark(Y) mark(0()) = [4] >= [4] = 0() mark(add(X1,X2)) = [4] >= [4] = a__add(mark(X1),X2) mark(and(X1,X2)) = [4] >= [7] = a__and(mark(X1),X2) mark(first(X1,X2)) = [4] >= [8] = a__first(mark(X1),mark(X2)) mark(if(X1,X2,X3)) = [4] >= [4] = a__if(mark(X1),X2,X3) mark(true()) = [4] >= [5] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap WORST_CASE(?,O(n^1)) + 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__first(X1,X2) -> first(X1,X2) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(true()) -> true() - Weak TRS: a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__if(true(),X,Y) -> mark(X) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(from(X)) -> a__from(X) mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if ,mark} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + 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}, uargs(a__and) = {1}, uargs(a__first) = {1,2}, uargs(a__if) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [5] p(a__add) = [1] x1 + [1] x2 + [0] p(a__and) = [1] x1 + [1] x2 + [4] p(a__first) = [1] x1 + [1] x2 + [0] p(a__from) = [0] p(a__if) = [1] x1 + [1] x2 + [1] x3 + [5] p(add) = [1] x1 + [1] x2 + [0] p(and) = [1] x1 + [1] x2 + [0] p(cons) = [1] p(false) = [2] p(first) = [1] x1 + [1] x2 + [2] p(from) = [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [3] p(mark) = [1] x1 + [1] p(nil) = [4] p(s) = [1] x1 + [0] p(true) = [2] Following rules are strictly oriented: a__add(0(),X) = [1] X + [5] > [1] X + [1] = mark(X) a__if(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [5] > [1] X1 + [1] X2 + [1] X3 + [3] = if(X1,X2,X3) a__if(false(),X,Y) = [1] X + [1] Y + [7] > [1] Y + [1] = mark(Y) mark(0()) = [6] > [5] = 0() mark(first(X1,X2)) = [1] X1 + [1] X2 + [3] > [1] X1 + [1] X2 + [2] = a__first(mark(X1),mark(X2)) mark(true()) = [3] > [2] = true() Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = add(X1,X2) a__add(s(X),Y) = [1] X + [1] Y + [0] >= [1] X + [1] Y + [0] = s(add(X,Y)) a__and(X1,X2) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [0] = and(X1,X2) a__and(false(),Y) = [1] Y + [6] >= [2] = false() a__and(true(),X) = [1] X + [6] >= [1] X + [1] = mark(X) a__first(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [2] = first(X1,X2) a__first(0(),X) = [1] X + [5] >= [4] = nil() a__first(s(X),cons(Y,Z)) = [1] X + [1] >= [1] = cons(Y,first(X,Z)) a__from(X) = [0] >= [1] = cons(X,from(s(X))) a__from(X) = [0] >= [0] = from(X) a__if(true(),X,Y) = [1] X + [1] Y + [7] >= [1] X + [1] = mark(X) mark(add(X1,X2)) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = a__add(mark(X1),X2) mark(and(X1,X2)) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [5] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [2] >= [1] = cons(X1,X2) mark(false()) = [3] >= [2] = false() mark(from(X)) = [1] >= [0] = a__from(X) mark(if(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [4] >= [1] X1 + [1] X2 + [1] X3 + [6] = a__if(mark(X1),X2,X3) mark(nil()) = [5] >= [4] = nil() mark(s(X)) = [1] X + [1] >= [1] X + [0] = s(X) 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: a__add(X1,X2) -> add(X1,X2) a__add(s(X),Y) -> s(add(X,Y)) a__first(X1,X2) -> first(X1,X2) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) - Weak TRS: a__add(0(),X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if ,mark} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + 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}, uargs(a__and) = {1}, uargs(a__first) = {1,2}, uargs(a__if) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__add) = [1] x1 + [1] p(a__and) = [1] x1 + [6] p(a__first) = [1] x1 + [1] x2 + [0] p(a__from) = [0] p(a__if) = [1] x1 + [4] p(add) = [0] p(and) = [0] p(cons) = [0] p(false) = [0] p(first) = [1] x1 + [0] p(from) = [0] p(if) = [1] x1 + [0] p(mark) = [0] p(nil) = [0] p(s) = [0] p(true) = [0] Following rules are strictly oriented: a__add(X1,X2) = [1] X1 + [1] > [0] = add(X1,X2) a__add(s(X),Y) = [1] > [0] = s(add(X,Y)) Following rules are (at-least) weakly oriented: a__add(0(),X) = [1] >= [0] = mark(X) a__and(X1,X2) = [1] X1 + [6] >= [0] = and(X1,X2) a__and(false(),Y) = [6] >= [0] = false() a__and(true(),X) = [6] >= [0] = mark(X) a__first(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [0] = first(X1,X2) a__first(0(),X) = [1] X + [0] >= [0] = nil() a__first(s(X),cons(Y,Z)) = [0] >= [0] = cons(Y,first(X,Z)) a__from(X) = [0] >= [0] = cons(X,from(s(X))) a__from(X) = [0] >= [0] = from(X) a__if(X1,X2,X3) = [1] X1 + [4] >= [1] X1 + [0] = if(X1,X2,X3) a__if(false(),X,Y) = [4] >= [0] = mark(Y) a__if(true(),X,Y) = [4] >= [0] = mark(X) mark(0()) = [0] >= [0] = 0() mark(add(X1,X2)) = [0] >= [1] = a__add(mark(X1),X2) mark(and(X1,X2)) = [0] >= [6] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [0] = cons(X1,X2) mark(false()) = [0] >= [0] = false() mark(first(X1,X2)) = [0] >= [0] = a__first(mark(X1),mark(X2)) mark(from(X)) = [0] >= [0] = a__from(X) mark(if(X1,X2,X3)) = [0] >= [4] = a__if(mark(X1),X2,X3) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [0] = s(X) mark(true()) = [0] >= [0] = true() 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: a__first(X1,X2) -> first(X1,X2) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) - 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__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if ,mark} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + 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}, uargs(a__and) = {1}, uargs(a__first) = {1,2}, uargs(a__if) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(a__add) = [1] x1 + [2] x2 + [4] p(a__and) = [1] x1 + [2] x2 + [0] p(a__first) = [1] x1 + [1] x2 + [0] p(a__from) = [2] p(a__if) = [1] x1 + [2] x2 + [2] x3 + [4] p(add) = [1] x1 + [1] x2 + [0] p(and) = [1] x1 + [1] x2 + [0] p(cons) = [0] p(false) = [4] p(first) = [1] x1 + [1] x2 + [0] p(from) = [6] p(if) = [1] x1 + [1] x2 + [1] x3 + [1] p(mark) = [2] x1 + [0] p(nil) = [0] p(s) = [4] p(true) = [4] Following rules are strictly oriented: a__from(X) = [2] > [0] = cons(X,from(s(X))) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [2] X2 + [4] >= [1] X1 + [1] X2 + [0] = add(X1,X2) a__add(0(),X) = [2] X + [5] >= [2] X + [0] = mark(X) a__add(s(X),Y) = [2] Y + [8] >= [4] = s(add(X,Y)) a__and(X1,X2) = [1] X1 + [2] X2 + [0] >= [1] X1 + [1] X2 + [0] = and(X1,X2) a__and(false(),Y) = [2] Y + [4] >= [4] = false() a__and(true(),X) = [2] X + [4] >= [2] X + [0] = mark(X) a__first(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = first(X1,X2) a__first(0(),X) = [1] X + [1] >= [0] = nil() a__first(s(X),cons(Y,Z)) = [4] >= [0] = cons(Y,first(X,Z)) a__from(X) = [2] >= [6] = from(X) a__if(X1,X2,X3) = [1] X1 + [2] X2 + [2] X3 + [4] >= [1] X1 + [1] X2 + [1] X3 + [1] = if(X1,X2,X3) a__if(false(),X,Y) = [2] X + [2] Y + [8] >= [2] Y + [0] = mark(Y) a__if(true(),X,Y) = [2] X + [2] Y + [8] >= [2] X + [0] = mark(X) mark(0()) = [2] >= [1] = 0() mark(add(X1,X2)) = [2] X1 + [2] X2 + [0] >= [2] X1 + [2] X2 + [4] = a__add(mark(X1),X2) mark(and(X1,X2)) = [2] X1 + [2] X2 + [0] >= [2] X1 + [2] X2 + [0] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [0] = cons(X1,X2) mark(false()) = [8] >= [4] = false() mark(first(X1,X2)) = [2] X1 + [2] X2 + [0] >= [2] X1 + [2] X2 + [0] = a__first(mark(X1),mark(X2)) mark(from(X)) = [12] >= [2] = a__from(X) mark(if(X1,X2,X3)) = [2] X1 + [2] X2 + [2] X3 + [2] >= [2] X1 + [2] X2 + [2] X3 + [4] = a__if(mark(X1),X2,X3) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [8] >= [4] = s(X) mark(true()) = [8] >= [4] = true() 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: a__first(X1,X2) -> first(X1,X2) a__from(X) -> from(X) mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) - 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__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if ,mark} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + 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}, uargs(a__and) = {1}, uargs(a__first) = {1,2}, uargs(a__if) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__add) = [1] x1 + [2] x2 + [1] p(a__and) = [1] x1 + [2] x2 + [1] p(a__first) = [1] x1 + [1] x2 + [0] p(a__from) = [0] p(a__if) = [1] x1 + [2] x2 + [2] x3 + [1] p(add) = [1] x1 + [1] x2 + [0] p(and) = [1] x1 + [1] x2 + [1] p(cons) = [0] p(false) = [5] p(first) = [1] x1 + [1] x2 + [0] p(from) = [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [2] x1 + [0] p(nil) = [0] p(s) = [0] p(true) = [4] Following rules are strictly oriented: mark(and(X1,X2)) = [2] X1 + [2] X2 + [2] > [2] X1 + [2] X2 + [1] = a__and(mark(X1),X2) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [2] X2 + [1] >= [1] X1 + [1] X2 + [0] = add(X1,X2) a__add(0(),X) = [2] X + [1] >= [2] X + [0] = mark(X) a__add(s(X),Y) = [2] Y + [1] >= [0] = s(add(X,Y)) a__and(X1,X2) = [1] X1 + [2] X2 + [1] >= [1] X1 + [1] X2 + [1] = and(X1,X2) a__and(false(),Y) = [2] Y + [6] >= [5] = false() a__and(true(),X) = [2] X + [5] >= [2] X + [0] = mark(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)) = [0] >= [0] = cons(Y,first(X,Z)) a__from(X) = [0] >= [0] = cons(X,from(s(X))) a__from(X) = [0] >= [0] = from(X) a__if(X1,X2,X3) = [1] X1 + [2] X2 + [2] X3 + [1] >= [1] X1 + [1] X2 + [1] X3 + [0] = if(X1,X2,X3) a__if(false(),X,Y) = [2] X + [2] Y + [6] >= [2] Y + [0] = mark(Y) a__if(true(),X,Y) = [2] X + [2] Y + [5] >= [2] X + [0] = mark(X) mark(0()) = [0] >= [0] = 0() mark(add(X1,X2)) = [2] X1 + [2] X2 + [0] >= [2] X1 + [2] X2 + [1] = a__add(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [0] = cons(X1,X2) mark(false()) = [10] >= [5] = false() mark(first(X1,X2)) = [2] X1 + [2] X2 + [0] >= [2] X1 + [2] X2 + [0] = a__first(mark(X1),mark(X2)) mark(from(X)) = [0] >= [0] = a__from(X) mark(if(X1,X2,X3)) = [2] X1 + [2] X2 + [2] X3 + [0] >= [2] X1 + [2] X2 + [2] X3 + [1] = a__if(mark(X1),X2,X3) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [0] = s(X) mark(true()) = [8] >= [4] = true() 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: a__first(X1,X2) -> first(X1,X2) a__from(X) -> from(X) mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) - 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__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if ,mark} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + 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}, uargs(a__and) = {1}, uargs(a__first) = {1,2}, uargs(a__if) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(a__add) = [1] x1 + [2] x2 + [0] p(a__and) = [1] x1 + [2] x2 + [1] p(a__first) = [1] x1 + [1] x2 + [0] p(a__from) = [2] x1 + [0] p(a__if) = [1] x1 + [2] x2 + [2] x3 + [4] p(add) = [1] x1 + [1] x2 + [0] p(and) = [1] x1 + [1] x2 + [1] p(cons) = [0] p(false) = [0] p(first) = [1] x1 + [1] x2 + [0] p(from) = [1] x1 + [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [4] p(mark) = [2] x1 + [0] p(nil) = [1] p(s) = [0] p(true) = [5] Following rules are strictly oriented: mark(if(X1,X2,X3)) = [2] X1 + [2] X2 + [2] X3 + [8] > [2] X1 + [2] X2 + [2] X3 + [4] = a__if(mark(X1),X2,X3) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [2] X2 + [0] >= [1] X1 + [1] X2 + [0] = add(X1,X2) a__add(0(),X) = [2] X + [1] >= [2] X + [0] = mark(X) a__add(s(X),Y) = [2] Y + [0] >= [0] = s(add(X,Y)) a__and(X1,X2) = [1] X1 + [2] X2 + [1] >= [1] X1 + [1] X2 + [1] = and(X1,X2) a__and(false(),Y) = [2] Y + [1] >= [0] = false() a__and(true(),X) = [2] X + [6] >= [2] X + [0] = mark(X) a__first(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = first(X1,X2) a__first(0(),X) = [1] X + [1] >= [1] = nil() a__first(s(X),cons(Y,Z)) = [0] >= [0] = cons(Y,first(X,Z)) a__from(X) = [2] X + [0] >= [0] = cons(X,from(s(X))) a__from(X) = [2] X + [0] >= [1] X + [0] = from(X) a__if(X1,X2,X3) = [1] X1 + [2] X2 + [2] X3 + [4] >= [1] X1 + [1] X2 + [1] X3 + [4] = if(X1,X2,X3) a__if(false(),X,Y) = [2] X + [2] Y + [4] >= [2] Y + [0] = mark(Y) a__if(true(),X,Y) = [2] X + [2] Y + [9] >= [2] X + [0] = mark(X) mark(0()) = [2] >= [1] = 0() mark(add(X1,X2)) = [2] X1 + [2] X2 + [0] >= [2] X1 + [2] X2 + [0] = a__add(mark(X1),X2) mark(and(X1,X2)) = [2] X1 + [2] X2 + [2] >= [2] X1 + [2] X2 + [1] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [0] = cons(X1,X2) mark(false()) = [0] >= [0] = false() mark(first(X1,X2)) = [2] X1 + [2] X2 + [0] >= [2] X1 + [2] X2 + [0] = a__first(mark(X1),mark(X2)) mark(from(X)) = [2] X + [0] >= [2] X + [0] = a__from(X) mark(nil()) = [2] >= [1] = nil() mark(s(X)) = [0] >= [0] = s(X) mark(true()) = [10] >= [5] = true() 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: a__first(X1,X2) -> first(X1,X2) a__from(X) -> from(X) mark(add(X1,X2)) -> a__add(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__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if ,mark} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + 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}, uargs(a__and) = {1}, uargs(a__first) = {1,2}, uargs(a__if) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(a__add) = [1] x1 + [4] x2 + [2] p(a__and) = [1] x1 + [4] x2 + [1] p(a__first) = [1] x1 + [1] x2 + [4] p(a__from) = [1] x1 + [2] p(a__if) = [1] x1 + [4] x2 + [4] x3 + [0] p(add) = [1] x1 + [1] x2 + [0] p(and) = [1] x1 + [1] x2 + [1] p(cons) = [1] x1 + [2] p(false) = [3] p(first) = [1] x1 + [1] x2 + [1] p(from) = [1] x1 + [1] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [4] x1 + [0] p(nil) = [2] p(s) = [1] p(true) = [0] Following rules are strictly oriented: a__first(X1,X2) = [1] X1 + [1] X2 + [4] > [1] X1 + [1] X2 + [1] = first(X1,X2) a__from(X) = [1] X + [2] > [1] X + [1] = from(X) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [4] X2 + [2] >= [1] X1 + [1] X2 + [0] = add(X1,X2) a__add(0(),X) = [4] X + [3] >= [4] X + [0] = mark(X) a__add(s(X),Y) = [4] Y + [3] >= [1] = s(add(X,Y)) a__and(X1,X2) = [1] X1 + [4] X2 + [1] >= [1] X1 + [1] X2 + [1] = and(X1,X2) a__and(false(),Y) = [4] Y + [4] >= [3] = false() a__and(true(),X) = [4] X + [1] >= [4] X + [0] = mark(X) a__first(0(),X) = [1] X + [5] >= [2] = nil() a__first(s(X),cons(Y,Z)) = [1] Y + [7] >= [1] Y + [2] = cons(Y,first(X,Z)) a__from(X) = [1] X + [2] >= [1] X + [2] = cons(X,from(s(X))) a__if(X1,X2,X3) = [1] X1 + [4] X2 + [4] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = if(X1,X2,X3) a__if(false(),X,Y) = [4] X + [4] Y + [3] >= [4] Y + [0] = mark(Y) a__if(true(),X,Y) = [4] X + [4] Y + [0] >= [4] X + [0] = mark(X) mark(0()) = [4] >= [1] = 0() mark(add(X1,X2)) = [4] X1 + [4] X2 + [0] >= [4] X1 + [4] X2 + [2] = a__add(mark(X1),X2) mark(and(X1,X2)) = [4] X1 + [4] X2 + [4] >= [4] X1 + [4] X2 + [1] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [4] X1 + [8] >= [1] X1 + [2] = cons(X1,X2) mark(false()) = [12] >= [3] = false() mark(first(X1,X2)) = [4] X1 + [4] X2 + [4] >= [4] X1 + [4] X2 + [4] = a__first(mark(X1),mark(X2)) mark(from(X)) = [4] X + [4] >= [1] X + [2] = a__from(X) mark(if(X1,X2,X3)) = [4] X1 + [4] X2 + [4] X3 + [0] >= [4] X1 + [4] X2 + [4] X3 + [0] = a__if(mark(X1),X2,X3) mark(nil()) = [8] >= [2] = nil() mark(s(X)) = [4] >= [1] = s(X) mark(true()) = [0] >= [0] = true() 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: mark(add(X1,X2)) -> a__add(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__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if ,mark} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + 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}, uargs(a__and) = {1}, uargs(a__first) = {1,2}, uargs(a__if) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__add) = [1] x1 + [5] x2 + [4] p(a__and) = [1] x1 + [5] x2 + [4] p(a__first) = [1] x1 + [1] x2 + [0] p(a__from) = [4] p(a__if) = [1] x1 + [5] x2 + [5] x3 + [0] p(add) = [1] x1 + [1] x2 + [1] p(and) = [1] x1 + [1] x2 + [1] p(cons) = [0] p(false) = [2] p(first) = [1] x1 + [1] x2 + [0] p(from) = [2] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [5] x1 + [0] p(nil) = [0] p(s) = [0] p(true) = [0] Following rules are strictly oriented: mark(add(X1,X2)) = [5] X1 + [5] X2 + [5] > [5] X1 + [5] X2 + [4] = a__add(mark(X1),X2) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [5] X2 + [4] >= [1] X1 + [1] X2 + [1] = add(X1,X2) a__add(0(),X) = [5] X + [4] >= [5] X + [0] = mark(X) a__add(s(X),Y) = [5] Y + [4] >= [0] = s(add(X,Y)) a__and(X1,X2) = [1] X1 + [5] X2 + [4] >= [1] X1 + [1] X2 + [1] = and(X1,X2) a__and(false(),Y) = [5] Y + [6] >= [2] = false() a__and(true(),X) = [5] X + [4] >= [5] X + [0] = mark(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)) = [0] >= [0] = cons(Y,first(X,Z)) a__from(X) = [4] >= [0] = cons(X,from(s(X))) a__from(X) = [4] >= [2] = from(X) a__if(X1,X2,X3) = [1] X1 + [5] X2 + [5] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = if(X1,X2,X3) a__if(false(),X,Y) = [5] X + [5] Y + [2] >= [5] Y + [0] = mark(Y) a__if(true(),X,Y) = [5] X + [5] Y + [0] >= [5] X + [0] = mark(X) mark(0()) = [0] >= [0] = 0() mark(and(X1,X2)) = [5] X1 + [5] X2 + [5] >= [5] X1 + [5] X2 + [4] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [0] = cons(X1,X2) mark(false()) = [10] >= [2] = false() mark(first(X1,X2)) = [5] X1 + [5] X2 + [0] >= [5] X1 + [5] X2 + [0] = a__first(mark(X1),mark(X2)) mark(from(X)) = [10] >= [4] = a__from(X) mark(if(X1,X2,X3)) = [5] X1 + [5] X2 + [5] X3 + [0] >= [5] X1 + [5] X2 + [5] X3 + [0] = a__if(mark(X1),X2,X3) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [0] = s(X) mark(true()) = [0] >= [0] = true() 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: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if ,mark} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))