YES(O(1),O(n^1)) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { a__and(X1, X2) -> and(X1, X2) , a__and(true(), X) -> mark(X) , a__and(false(), Y) -> false() , mark(true()) -> true() , mark(false()) -> false() , mark(0()) -> 0() , mark(s(X)) -> s(X) , mark(add(X1, X2)) -> a__add(mark(X1), X2) , mark(nil()) -> nil() , mark(cons(X1, X2)) -> cons(X1, X2) , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) , mark(from(X)) -> a__from(X) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) , a__if(X1, X2, X3) -> if(X1, X2, X3) , a__if(true(), X, Y) -> mark(X) , a__if(false(), X, Y) -> mark(Y) , 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(Y, first(X, Z)) , a__from(X) -> cons(X, from(s(X))) , a__from(X) -> from(X) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, Uargs(a__first) = {1, 2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [a__and](x1, x2) = [1] x1 + [0] [true] = [0] [mark](x1) = [0] [false] = [0] [a__if](x1, x2, x3) = [1] x1 + [0] [a__add](x1, x2) = [1] x1 + [0] [0] = [0] [s](x1) = [0] [add](x1, x2) = [1] x1 + [5] [a__first](x1, x2) = [1] x1 + [1] x2 + [0] [nil] = [7] [cons](x1, x2) = [0] [first](x1, x2) = [1] x1 + [1] x2 + [5] [a__from](x1) = [5] [from](x1) = [5] [and](x1, x2) = [1] x1 + [7] [if](x1, x2, x3) = [1] x1 + [7] The following symbols are considered usable {a__and, mark, a__if, a__add, a__first, a__from} The order satisfies the following ordering constraints: [a__and(X1, X2)] = [1] X1 + [0] ? [1] X1 + [7] = [and(X1, X2)] [a__and(true(), X)] = [0] >= [0] = [mark(X)] [a__and(false(), Y)] = [0] >= [0] = [false()] [mark(true())] = [0] >= [0] = [true()] [mark(false())] = [0] >= [0] = [false()] [mark(0())] = [0] >= [0] = [0()] [mark(s(X))] = [0] >= [0] = [s(X)] [mark(add(X1, X2))] = [0] >= [0] = [a__add(mark(X1), X2)] [mark(nil())] = [0] ? [7] = [nil()] [mark(cons(X1, X2))] = [0] >= [0] = [cons(X1, X2)] [mark(first(X1, X2))] = [0] >= [0] = [a__first(mark(X1), mark(X2))] [mark(from(X))] = [0] ? [5] = [a__from(X)] [mark(and(X1, X2))] = [0] >= [0] = [a__and(mark(X1), X2)] [mark(if(X1, X2, X3))] = [0] >= [0] = [a__if(mark(X1), X2, X3)] [a__if(X1, X2, X3)] = [1] X1 + [0] ? [1] X1 + [7] = [if(X1, X2, X3)] [a__if(true(), X, Y)] = [0] >= [0] = [mark(X)] [a__if(false(), X, Y)] = [0] >= [0] = [mark(Y)] [a__add(X1, X2)] = [1] X1 + [0] ? [1] X1 + [5] = [add(X1, X2)] [a__add(0(), X)] = [0] >= [0] = [mark(X)] [a__add(s(X), Y)] = [0] >= [0] = [s(add(X, Y))] [a__first(X1, X2)] = [1] X1 + [1] X2 + [0] ? [1] X1 + [1] X2 + [5] = [first(X1, X2)] [a__first(0(), X)] = [1] X + [0] ? [7] = [nil()] [a__first(s(X), cons(Y, Z))] = [0] >= [0] = [cons(Y, first(X, Z))] [a__from(X)] = [5] > [0] = [cons(X, from(s(X)))] [a__from(X)] = [5] >= [5] = [from(X)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { a__and(X1, X2) -> and(X1, X2) , a__and(true(), X) -> mark(X) , a__and(false(), Y) -> false() , mark(true()) -> true() , mark(false()) -> false() , mark(0()) -> 0() , mark(s(X)) -> s(X) , mark(add(X1, X2)) -> a__add(mark(X1), X2) , mark(nil()) -> nil() , mark(cons(X1, X2)) -> cons(X1, X2) , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) , mark(from(X)) -> a__from(X) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) , a__if(X1, X2, X3) -> if(X1, X2, X3) , a__if(true(), X, Y) -> mark(X) , a__if(false(), X, Y) -> mark(Y) , 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(Y, first(X, Z)) , a__from(X) -> from(X) } Weak Trs: { a__from(X) -> cons(X, from(s(X))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, Uargs(a__first) = {1, 2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [a__and](x1, x2) = [1] x1 + [4] [true] = [0] [mark](x1) = [0] [false] = [0] [a__if](x1, x2, x3) = [1] x1 + [0] [a__add](x1, x2) = [1] x1 + [0] [0] = [0] [s](x1) = [0] [add](x1, x2) = [1] x1 + [5] [a__first](x1, x2) = [1] x1 + [1] x2 + [0] [nil] = [7] [cons](x1, x2) = [0] [first](x1, x2) = [1] x1 + [1] x2 + [5] [a__from](x1) = [5] [from](x1) = [5] [and](x1, x2) = [1] x1 + [3] [if](x1, x2, x3) = [1] x1 + [7] The following symbols are considered usable {a__and, mark, a__if, a__add, a__first, a__from} The order satisfies the following ordering constraints: [a__and(X1, X2)] = [1] X1 + [4] > [1] X1 + [3] = [and(X1, X2)] [a__and(true(), X)] = [4] > [0] = [mark(X)] [a__and(false(), Y)] = [4] > [0] = [false()] [mark(true())] = [0] >= [0] = [true()] [mark(false())] = [0] >= [0] = [false()] [mark(0())] = [0] >= [0] = [0()] [mark(s(X))] = [0] >= [0] = [s(X)] [mark(add(X1, X2))] = [0] >= [0] = [a__add(mark(X1), X2)] [mark(nil())] = [0] ? [7] = [nil()] [mark(cons(X1, X2))] = [0] >= [0] = [cons(X1, X2)] [mark(first(X1, X2))] = [0] >= [0] = [a__first(mark(X1), mark(X2))] [mark(from(X))] = [0] ? [5] = [a__from(X)] [mark(and(X1, X2))] = [0] ? [4] = [a__and(mark(X1), X2)] [mark(if(X1, X2, X3))] = [0] >= [0] = [a__if(mark(X1), X2, X3)] [a__if(X1, X2, X3)] = [1] X1 + [0] ? [1] X1 + [7] = [if(X1, X2, X3)] [a__if(true(), X, Y)] = [0] >= [0] = [mark(X)] [a__if(false(), X, Y)] = [0] >= [0] = [mark(Y)] [a__add(X1, X2)] = [1] X1 + [0] ? [1] X1 + [5] = [add(X1, X2)] [a__add(0(), X)] = [0] >= [0] = [mark(X)] [a__add(s(X), Y)] = [0] >= [0] = [s(add(X, Y))] [a__first(X1, X2)] = [1] X1 + [1] X2 + [0] ? [1] X1 + [1] X2 + [5] = [first(X1, X2)] [a__first(0(), X)] = [1] X + [0] ? [7] = [nil()] [a__first(s(X), cons(Y, Z))] = [0] >= [0] = [cons(Y, first(X, Z))] [a__from(X)] = [5] > [0] = [cons(X, from(s(X)))] [a__from(X)] = [5] >= [5] = [from(X)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { mark(true()) -> true() , mark(false()) -> false() , mark(0()) -> 0() , mark(s(X)) -> s(X) , mark(add(X1, X2)) -> a__add(mark(X1), X2) , mark(nil()) -> nil() , mark(cons(X1, X2)) -> cons(X1, X2) , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) , mark(from(X)) -> a__from(X) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) , a__if(X1, X2, X3) -> if(X1, X2, X3) , a__if(true(), X, Y) -> mark(X) , a__if(false(), X, Y) -> mark(Y) , 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(Y, first(X, Z)) , a__from(X) -> from(X) } Weak Trs: { a__and(X1, X2) -> and(X1, X2) , a__and(true(), X) -> mark(X) , a__and(false(), Y) -> false() , a__from(X) -> cons(X, from(s(X))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, Uargs(a__first) = {1, 2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [a__and](x1, x2) = [1] x1 + [4] [true] = [0] [mark](x1) = [0] [false] = [0] [a__if](x1, x2, x3) = [1] x1 + [4] [a__add](x1, x2) = [1] x1 + [0] [0] = [0] [s](x1) = [0] [add](x1, x2) = [1] x1 + [5] [a__first](x1, x2) = [1] x1 + [1] x2 + [0] [nil] = [7] [cons](x1, x2) = [0] [first](x1, x2) = [1] x1 + [1] x2 + [5] [a__from](x1) = [5] [from](x1) = [5] [and](x1, x2) = [1] x1 + [3] [if](x1, x2, x3) = [1] x1 + [7] The following symbols are considered usable {a__and, mark, a__if, a__add, a__first, a__from} The order satisfies the following ordering constraints: [a__and(X1, X2)] = [1] X1 + [4] > [1] X1 + [3] = [and(X1, X2)] [a__and(true(), X)] = [4] > [0] = [mark(X)] [a__and(false(), Y)] = [4] > [0] = [false()] [mark(true())] = [0] >= [0] = [true()] [mark(false())] = [0] >= [0] = [false()] [mark(0())] = [0] >= [0] = [0()] [mark(s(X))] = [0] >= [0] = [s(X)] [mark(add(X1, X2))] = [0] >= [0] = [a__add(mark(X1), X2)] [mark(nil())] = [0] ? [7] = [nil()] [mark(cons(X1, X2))] = [0] >= [0] = [cons(X1, X2)] [mark(first(X1, X2))] = [0] >= [0] = [a__first(mark(X1), mark(X2))] [mark(from(X))] = [0] ? [5] = [a__from(X)] [mark(and(X1, X2))] = [0] ? [4] = [a__and(mark(X1), X2)] [mark(if(X1, X2, X3))] = [0] ? [4] = [a__if(mark(X1), X2, X3)] [a__if(X1, X2, X3)] = [1] X1 + [4] ? [1] X1 + [7] = [if(X1, X2, X3)] [a__if(true(), X, Y)] = [4] > [0] = [mark(X)] [a__if(false(), X, Y)] = [4] > [0] = [mark(Y)] [a__add(X1, X2)] = [1] X1 + [0] ? [1] X1 + [5] = [add(X1, X2)] [a__add(0(), X)] = [0] >= [0] = [mark(X)] [a__add(s(X), Y)] = [0] >= [0] = [s(add(X, Y))] [a__first(X1, X2)] = [1] X1 + [1] X2 + [0] ? [1] X1 + [1] X2 + [5] = [first(X1, X2)] [a__first(0(), X)] = [1] X + [0] ? [7] = [nil()] [a__first(s(X), cons(Y, Z))] = [0] >= [0] = [cons(Y, first(X, Z))] [a__from(X)] = [5] > [0] = [cons(X, from(s(X)))] [a__from(X)] = [5] >= [5] = [from(X)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { mark(true()) -> true() , mark(false()) -> false() , mark(0()) -> 0() , mark(s(X)) -> s(X) , mark(add(X1, X2)) -> a__add(mark(X1), X2) , mark(nil()) -> nil() , mark(cons(X1, X2)) -> cons(X1, X2) , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) , mark(from(X)) -> a__from(X) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) , a__if(X1, X2, X3) -> if(X1, X2, X3) , 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(Y, first(X, Z)) , a__from(X) -> from(X) } Weak Trs: { a__and(X1, X2) -> and(X1, X2) , a__and(true(), X) -> mark(X) , a__and(false(), Y) -> false() , a__if(true(), X, Y) -> mark(X) , a__if(false(), X, Y) -> mark(Y) , a__from(X) -> cons(X, from(s(X))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, Uargs(a__first) = {1, 2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [a__and](x1, x2) = [1] x1 + [4] [true] = [0] [mark](x1) = [0] [false] = [0] [a__if](x1, x2, x3) = [1] x1 + [4] [a__add](x1, x2) = [1] x1 + [0] [0] = [0] [s](x1) = [0] [add](x1, x2) = [1] x1 + [5] [a__first](x1, x2) = [1] x1 + [1] x2 + [0] [nil] = [7] [cons](x1, x2) = [0] [first](x1, x2) = [1] x1 + [1] x2 + [5] [a__from](x1) = [5] [from](x1) = [5] [and](x1, x2) = [1] x1 + [3] [if](x1, x2, x3) = [1] x1 + [3] The following symbols are considered usable {a__and, mark, a__if, a__add, a__first, a__from} The order satisfies the following ordering constraints: [a__and(X1, X2)] = [1] X1 + [4] > [1] X1 + [3] = [and(X1, X2)] [a__and(true(), X)] = [4] > [0] = [mark(X)] [a__and(false(), Y)] = [4] > [0] = [false()] [mark(true())] = [0] >= [0] = [true()] [mark(false())] = [0] >= [0] = [false()] [mark(0())] = [0] >= [0] = [0()] [mark(s(X))] = [0] >= [0] = [s(X)] [mark(add(X1, X2))] = [0] >= [0] = [a__add(mark(X1), X2)] [mark(nil())] = [0] ? [7] = [nil()] [mark(cons(X1, X2))] = [0] >= [0] = [cons(X1, X2)] [mark(first(X1, X2))] = [0] >= [0] = [a__first(mark(X1), mark(X2))] [mark(from(X))] = [0] ? [5] = [a__from(X)] [mark(and(X1, X2))] = [0] ? [4] = [a__and(mark(X1), X2)] [mark(if(X1, X2, X3))] = [0] ? [4] = [a__if(mark(X1), X2, X3)] [a__if(X1, X2, X3)] = [1] X1 + [4] > [1] X1 + [3] = [if(X1, X2, X3)] [a__if(true(), X, Y)] = [4] > [0] = [mark(X)] [a__if(false(), X, Y)] = [4] > [0] = [mark(Y)] [a__add(X1, X2)] = [1] X1 + [0] ? [1] X1 + [5] = [add(X1, X2)] [a__add(0(), X)] = [0] >= [0] = [mark(X)] [a__add(s(X), Y)] = [0] >= [0] = [s(add(X, Y))] [a__first(X1, X2)] = [1] X1 + [1] X2 + [0] ? [1] X1 + [1] X2 + [5] = [first(X1, X2)] [a__first(0(), X)] = [1] X + [0] ? [7] = [nil()] [a__first(s(X), cons(Y, Z))] = [0] >= [0] = [cons(Y, first(X, Z))] [a__from(X)] = [5] > [0] = [cons(X, from(s(X)))] [a__from(X)] = [5] >= [5] = [from(X)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { mark(true()) -> true() , mark(false()) -> false() , mark(0()) -> 0() , mark(s(X)) -> s(X) , mark(add(X1, X2)) -> a__add(mark(X1), X2) , mark(nil()) -> nil() , mark(cons(X1, X2)) -> cons(X1, X2) , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) , mark(from(X)) -> a__from(X) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) , 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(Y, first(X, Z)) , a__from(X) -> from(X) } Weak Trs: { a__and(X1, X2) -> and(X1, X2) , a__and(true(), X) -> mark(X) , a__and(false(), Y) -> false() , a__if(X1, X2, X3) -> if(X1, X2, X3) , a__if(true(), X, Y) -> mark(X) , a__if(false(), X, Y) -> mark(Y) , a__from(X) -> cons(X, from(s(X))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, Uargs(a__first) = {1, 2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [a__and](x1, x2) = [1] x1 + [0] [true] = [4] [mark](x1) = [0] [false] = [4] [a__if](x1, x2, x3) = [1] x1 + [0] [a__add](x1, x2) = [1] x1 + [1] [0] = [0] [s](x1) = [0] [add](x1, x2) = [1] x1 + [5] [a__first](x1, x2) = [1] x1 + [1] x2 + [0] [nil] = [7] [cons](x1, x2) = [0] [first](x1, x2) = [1] x1 + [1] x2 + [5] [a__from](x1) = [5] [from](x1) = [5] [and](x1, x2) = [1] x1 + [0] [if](x1, x2, x3) = [1] x1 + [0] The following symbols are considered usable {a__and, mark, a__if, a__add, a__first, a__from} The order satisfies the following ordering constraints: [a__and(X1, X2)] = [1] X1 + [0] >= [1] X1 + [0] = [and(X1, X2)] [a__and(true(), X)] = [4] > [0] = [mark(X)] [a__and(false(), Y)] = [4] >= [4] = [false()] [mark(true())] = [0] ? [4] = [true()] [mark(false())] = [0] ? [4] = [false()] [mark(0())] = [0] >= [0] = [0()] [mark(s(X))] = [0] >= [0] = [s(X)] [mark(add(X1, X2))] = [0] ? [1] = [a__add(mark(X1), X2)] [mark(nil())] = [0] ? [7] = [nil()] [mark(cons(X1, X2))] = [0] >= [0] = [cons(X1, X2)] [mark(first(X1, X2))] = [0] >= [0] = [a__first(mark(X1), mark(X2))] [mark(from(X))] = [0] ? [5] = [a__from(X)] [mark(and(X1, X2))] = [0] >= [0] = [a__and(mark(X1), X2)] [mark(if(X1, X2, X3))] = [0] >= [0] = [a__if(mark(X1), X2, X3)] [a__if(X1, X2, X3)] = [1] X1 + [0] >= [1] X1 + [0] = [if(X1, X2, X3)] [a__if(true(), X, Y)] = [4] > [0] = [mark(X)] [a__if(false(), X, Y)] = [4] > [0] = [mark(Y)] [a__add(X1, X2)] = [1] X1 + [1] ? [1] X1 + [5] = [add(X1, X2)] [a__add(0(), X)] = [1] > [0] = [mark(X)] [a__add(s(X), Y)] = [1] > [0] = [s(add(X, Y))] [a__first(X1, X2)] = [1] X1 + [1] X2 + [0] ? [1] X1 + [1] X2 + [5] = [first(X1, X2)] [a__first(0(), X)] = [1] X + [0] ? [7] = [nil()] [a__first(s(X), cons(Y, Z))] = [0] >= [0] = [cons(Y, first(X, Z))] [a__from(X)] = [5] > [0] = [cons(X, from(s(X)))] [a__from(X)] = [5] >= [5] = [from(X)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { mark(true()) -> true() , mark(false()) -> false() , mark(0()) -> 0() , mark(s(X)) -> s(X) , mark(add(X1, X2)) -> a__add(mark(X1), X2) , mark(nil()) -> nil() , mark(cons(X1, X2)) -> cons(X1, X2) , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) , mark(from(X)) -> a__from(X) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) , a__add(X1, X2) -> add(X1, X2) , 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) -> from(X) } Weak Trs: { a__and(X1, X2) -> and(X1, X2) , a__and(true(), X) -> mark(X) , a__and(false(), Y) -> false() , a__if(X1, X2, X3) -> if(X1, X2, X3) , a__if(true(), X, Y) -> mark(X) , a__if(false(), X, Y) -> mark(Y) , a__add(0(), X) -> mark(X) , a__add(s(X), Y) -> s(add(X, Y)) , a__from(X) -> cons(X, from(s(X))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, Uargs(a__first) = {1, 2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [a__and](x1, x2) = [1] x1 + [4] [true] = [4] [mark](x1) = [0] [false] = [4] [a__if](x1, x2, x3) = [1] x1 + [0] [a__add](x1, x2) = [1] x1 + [4] [0] = [0] [s](x1) = [0] [add](x1, x2) = [1] x1 + [1] [a__first](x1, x2) = [1] x1 + [1] x2 + [0] [nil] = [7] [cons](x1, x2) = [0] [first](x1, x2) = [1] x1 + [1] x2 + [5] [a__from](x1) = [5] [from](x1) = [5] [and](x1, x2) = [1] x1 + [3] [if](x1, x2, x3) = [1] x1 + [0] The following symbols are considered usable {a__and, mark, a__if, a__add, a__first, a__from} The order satisfies the following ordering constraints: [a__and(X1, X2)] = [1] X1 + [4] > [1] X1 + [3] = [and(X1, X2)] [a__and(true(), X)] = [8] > [0] = [mark(X)] [a__and(false(), Y)] = [8] > [4] = [false()] [mark(true())] = [0] ? [4] = [true()] [mark(false())] = [0] ? [4] = [false()] [mark(0())] = [0] >= [0] = [0()] [mark(s(X))] = [0] >= [0] = [s(X)] [mark(add(X1, X2))] = [0] ? [4] = [a__add(mark(X1), X2)] [mark(nil())] = [0] ? [7] = [nil()] [mark(cons(X1, X2))] = [0] >= [0] = [cons(X1, X2)] [mark(first(X1, X2))] = [0] >= [0] = [a__first(mark(X1), mark(X2))] [mark(from(X))] = [0] ? [5] = [a__from(X)] [mark(and(X1, X2))] = [0] ? [4] = [a__and(mark(X1), X2)] [mark(if(X1, X2, X3))] = [0] >= [0] = [a__if(mark(X1), X2, X3)] [a__if(X1, X2, X3)] = [1] X1 + [0] >= [1] X1 + [0] = [if(X1, X2, X3)] [a__if(true(), X, Y)] = [4] > [0] = [mark(X)] [a__if(false(), X, Y)] = [4] > [0] = [mark(Y)] [a__add(X1, X2)] = [1] X1 + [4] > [1] X1 + [1] = [add(X1, X2)] [a__add(0(), X)] = [4] > [0] = [mark(X)] [a__add(s(X), Y)] = [4] > [0] = [s(add(X, Y))] [a__first(X1, X2)] = [1] X1 + [1] X2 + [0] ? [1] X1 + [1] X2 + [5] = [first(X1, X2)] [a__first(0(), X)] = [1] X + [0] ? [7] = [nil()] [a__first(s(X), cons(Y, Z))] = [0] >= [0] = [cons(Y, first(X, Z))] [a__from(X)] = [5] > [0] = [cons(X, from(s(X)))] [a__from(X)] = [5] >= [5] = [from(X)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { mark(true()) -> true() , mark(false()) -> false() , mark(0()) -> 0() , mark(s(X)) -> s(X) , mark(add(X1, X2)) -> a__add(mark(X1), X2) , mark(nil()) -> nil() , mark(cons(X1, X2)) -> cons(X1, X2) , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) , mark(from(X)) -> a__from(X) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) , 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) -> from(X) } Weak Trs: { a__and(X1, X2) -> and(X1, X2) , a__and(true(), X) -> mark(X) , a__and(false(), Y) -> false() , a__if(X1, X2, X3) -> if(X1, X2, X3) , a__if(true(), X, Y) -> mark(X) , a__if(false(), X, Y) -> mark(Y) , a__add(X1, X2) -> add(X1, X2) , a__add(0(), X) -> mark(X) , a__add(s(X), Y) -> s(add(X, Y)) , a__from(X) -> cons(X, from(s(X))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, Uargs(a__first) = {1, 2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [a__and](x1, x2) = [1] x1 + [4] [true] = [4] [mark](x1) = [0] [false] = [4] [a__if](x1, x2, x3) = [1] x1 + [0] [a__add](x1, x2) = [1] x1 + [4] [0] = [0] [s](x1) = [0] [add](x1, x2) = [1] x1 + [2] [a__first](x1, x2) = [1] x1 + [1] x2 + [0] [nil] = [7] [cons](x1, x2) = [0] [first](x1, x2) = [1] x1 + [1] x2 + [5] [a__from](x1) = [5] [from](x1) = [1] [and](x1, x2) = [1] x1 + [3] [if](x1, x2, x3) = [1] x1 + [0] The following symbols are considered usable {a__and, mark, a__if, a__add, a__first, a__from} The order satisfies the following ordering constraints: [a__and(X1, X2)] = [1] X1 + [4] > [1] X1 + [3] = [and(X1, X2)] [a__and(true(), X)] = [8] > [0] = [mark(X)] [a__and(false(), Y)] = [8] > [4] = [false()] [mark(true())] = [0] ? [4] = [true()] [mark(false())] = [0] ? [4] = [false()] [mark(0())] = [0] >= [0] = [0()] [mark(s(X))] = [0] >= [0] = [s(X)] [mark(add(X1, X2))] = [0] ? [4] = [a__add(mark(X1), X2)] [mark(nil())] = [0] ? [7] = [nil()] [mark(cons(X1, X2))] = [0] >= [0] = [cons(X1, X2)] [mark(first(X1, X2))] = [0] >= [0] = [a__first(mark(X1), mark(X2))] [mark(from(X))] = [0] ? [5] = [a__from(X)] [mark(and(X1, X2))] = [0] ? [4] = [a__and(mark(X1), X2)] [mark(if(X1, X2, X3))] = [0] >= [0] = [a__if(mark(X1), X2, X3)] [a__if(X1, X2, X3)] = [1] X1 + [0] >= [1] X1 + [0] = [if(X1, X2, X3)] [a__if(true(), X, Y)] = [4] > [0] = [mark(X)] [a__if(false(), X, Y)] = [4] > [0] = [mark(Y)] [a__add(X1, X2)] = [1] X1 + [4] > [1] X1 + [2] = [add(X1, X2)] [a__add(0(), X)] = [4] > [0] = [mark(X)] [a__add(s(X), Y)] = [4] > [0] = [s(add(X, Y))] [a__first(X1, X2)] = [1] X1 + [1] X2 + [0] ? [1] X1 + [1] X2 + [5] = [first(X1, X2)] [a__first(0(), X)] = [1] X + [0] ? [7] = [nil()] [a__first(s(X), cons(Y, Z))] = [0] >= [0] = [cons(Y, first(X, Z))] [a__from(X)] = [5] > [0] = [cons(X, from(s(X)))] [a__from(X)] = [5] > [1] = [from(X)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { mark(true()) -> true() , mark(false()) -> false() , mark(0()) -> 0() , mark(s(X)) -> s(X) , mark(add(X1, X2)) -> a__add(mark(X1), X2) , mark(nil()) -> nil() , mark(cons(X1, X2)) -> cons(X1, X2) , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) , mark(from(X)) -> a__from(X) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) , a__first(X1, X2) -> first(X1, X2) , a__first(0(), X) -> nil() , a__first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) } Weak Trs: { a__and(X1, X2) -> and(X1, X2) , a__and(true(), X) -> mark(X) , a__and(false(), Y) -> false() , a__if(X1, X2, X3) -> if(X1, X2, X3) , a__if(true(), X, Y) -> mark(X) , a__if(false(), X, Y) -> mark(Y) , a__add(X1, X2) -> add(X1, X2) , a__add(0(), X) -> mark(X) , a__add(s(X), Y) -> s(add(X, Y)) , a__from(X) -> cons(X, from(s(X))) , a__from(X) -> from(X) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, Uargs(a__first) = {1, 2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [a__and](x1, x2) = [1] x1 + [0] [true] = [4] [mark](x1) = [0] [false] = [4] [a__if](x1, x2, x3) = [1] x1 + [0] [a__add](x1, x2) = [1] x1 + [4] [0] = [0] [s](x1) = [0] [add](x1, x2) = [1] x1 + [1] [a__first](x1, x2) = [1] x1 + [1] x2 + [1] [nil] = [6] [cons](x1, x2) = [0] [first](x1, x2) = [1] x1 + [1] x2 + [5] [a__from](x1) = [5] [from](x1) = [2] [and](x1, x2) = [1] x1 + [0] [if](x1, x2, x3) = [1] x1 + [0] The following symbols are considered usable {a__and, mark, a__if, a__add, a__first, a__from} The order satisfies the following ordering constraints: [a__and(X1, X2)] = [1] X1 + [0] >= [1] X1 + [0] = [and(X1, X2)] [a__and(true(), X)] = [4] > [0] = [mark(X)] [a__and(false(), Y)] = [4] >= [4] = [false()] [mark(true())] = [0] ? [4] = [true()] [mark(false())] = [0] ? [4] = [false()] [mark(0())] = [0] >= [0] = [0()] [mark(s(X))] = [0] >= [0] = [s(X)] [mark(add(X1, X2))] = [0] ? [4] = [a__add(mark(X1), X2)] [mark(nil())] = [0] ? [6] = [nil()] [mark(cons(X1, X2))] = [0] >= [0] = [cons(X1, X2)] [mark(first(X1, X2))] = [0] ? [1] = [a__first(mark(X1), mark(X2))] [mark(from(X))] = [0] ? [5] = [a__from(X)] [mark(and(X1, X2))] = [0] >= [0] = [a__and(mark(X1), X2)] [mark(if(X1, X2, X3))] = [0] >= [0] = [a__if(mark(X1), X2, X3)] [a__if(X1, X2, X3)] = [1] X1 + [0] >= [1] X1 + [0] = [if(X1, X2, X3)] [a__if(true(), X, Y)] = [4] > [0] = [mark(X)] [a__if(false(), X, Y)] = [4] > [0] = [mark(Y)] [a__add(X1, X2)] = [1] X1 + [4] > [1] X1 + [1] = [add(X1, X2)] [a__add(0(), X)] = [4] > [0] = [mark(X)] [a__add(s(X), Y)] = [4] > [0] = [s(add(X, Y))] [a__first(X1, X2)] = [1] X1 + [1] X2 + [1] ? [1] X1 + [1] X2 + [5] = [first(X1, X2)] [a__first(0(), X)] = [1] X + [1] ? [6] = [nil()] [a__first(s(X), cons(Y, Z))] = [1] > [0] = [cons(Y, first(X, Z))] [a__from(X)] = [5] > [0] = [cons(X, from(s(X)))] [a__from(X)] = [5] > [2] = [from(X)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { mark(true()) -> true() , mark(false()) -> false() , mark(0()) -> 0() , mark(s(X)) -> s(X) , mark(add(X1, X2)) -> a__add(mark(X1), X2) , mark(nil()) -> nil() , mark(cons(X1, X2)) -> cons(X1, X2) , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) , mark(from(X)) -> a__from(X) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) , a__first(X1, X2) -> first(X1, X2) , a__first(0(), X) -> nil() } Weak Trs: { a__and(X1, X2) -> and(X1, X2) , a__and(true(), X) -> mark(X) , a__and(false(), Y) -> false() , a__if(X1, X2, X3) -> if(X1, X2, X3) , a__if(true(), X, Y) -> mark(X) , a__if(false(), X, Y) -> mark(Y) , a__add(X1, X2) -> add(X1, X2) , a__add(0(), X) -> mark(X) , a__add(s(X), Y) -> s(add(X, Y)) , 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) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, Uargs(a__first) = {1, 2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [a__and](x1, x2) = [1] x1 + [0] [true] = [0] [mark](x1) = [0] [false] = [0] [a__if](x1, x2, x3) = [1] x1 + [4] [a__add](x1, x2) = [1] x1 + [4] [0] = [0] [s](x1) = [0] [add](x1, x2) = [1] x1 + [2] [a__first](x1, x2) = [1] x1 + [1] x2 + [1] [nil] = [0] [cons](x1, x2) = [0] [first](x1, x2) = [1] x1 + [1] x2 + [5] [a__from](x1) = [4] [from](x1) = [2] [and](x1, x2) = [1] x1 + [0] [if](x1, x2, x3) = [1] x1 + [3] The following symbols are considered usable {a__and, mark, a__if, a__add, a__first, a__from} The order satisfies the following ordering constraints: [a__and(X1, X2)] = [1] X1 + [0] >= [1] X1 + [0] = [and(X1, X2)] [a__and(true(), X)] = [0] >= [0] = [mark(X)] [a__and(false(), Y)] = [0] >= [0] = [false()] [mark(true())] = [0] >= [0] = [true()] [mark(false())] = [0] >= [0] = [false()] [mark(0())] = [0] >= [0] = [0()] [mark(s(X))] = [0] >= [0] = [s(X)] [mark(add(X1, X2))] = [0] ? [4] = [a__add(mark(X1), X2)] [mark(nil())] = [0] >= [0] = [nil()] [mark(cons(X1, X2))] = [0] >= [0] = [cons(X1, X2)] [mark(first(X1, X2))] = [0] ? [1] = [a__first(mark(X1), mark(X2))] [mark(from(X))] = [0] ? [4] = [a__from(X)] [mark(and(X1, X2))] = [0] >= [0] = [a__and(mark(X1), X2)] [mark(if(X1, X2, X3))] = [0] ? [4] = [a__if(mark(X1), X2, X3)] [a__if(X1, X2, X3)] = [1] X1 + [4] > [1] X1 + [3] = [if(X1, X2, X3)] [a__if(true(), X, Y)] = [4] > [0] = [mark(X)] [a__if(false(), X, Y)] = [4] > [0] = [mark(Y)] [a__add(X1, X2)] = [1] X1 + [4] > [1] X1 + [2] = [add(X1, X2)] [a__add(0(), X)] = [4] > [0] = [mark(X)] [a__add(s(X), Y)] = [4] > [0] = [s(add(X, Y))] [a__first(X1, X2)] = [1] X1 + [1] X2 + [1] ? [1] X1 + [1] X2 + [5] = [first(X1, X2)] [a__first(0(), X)] = [1] X + [1] > [0] = [nil()] [a__first(s(X), cons(Y, Z))] = [1] > [0] = [cons(Y, first(X, Z))] [a__from(X)] = [4] > [0] = [cons(X, from(s(X)))] [a__from(X)] = [4] > [2] = [from(X)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { mark(true()) -> true() , mark(false()) -> false() , mark(0()) -> 0() , mark(s(X)) -> s(X) , mark(add(X1, X2)) -> a__add(mark(X1), X2) , mark(nil()) -> nil() , mark(cons(X1, X2)) -> cons(X1, X2) , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) , mark(from(X)) -> a__from(X) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) , a__first(X1, X2) -> first(X1, X2) } Weak Trs: { a__and(X1, X2) -> and(X1, X2) , a__and(true(), X) -> mark(X) , a__and(false(), Y) -> false() , a__if(X1, X2, X3) -> if(X1, X2, X3) , a__if(true(), X, Y) -> mark(X) , a__if(false(), X, Y) -> mark(Y) , 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(Y, first(X, Z)) , a__from(X) -> cons(X, from(s(X))) , a__from(X) -> from(X) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, Uargs(a__first) = {1, 2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [a__and](x1, x2) = [1] x1 + [0] [true] = [0] [mark](x1) = [0] [false] = [0] [a__if](x1, x2, x3) = [1] x1 + [4] [a__add](x1, x2) = [1] x1 + [4] [0] = [0] [s](x1) = [0] [add](x1, x2) = [1] x1 + [1] [a__first](x1, x2) = [1] x1 + [1] x2 + [1] [nil] = [1] [cons](x1, x2) = [0] [first](x1, x2) = [1] x1 + [1] x2 + [0] [a__from](x1) = [4] [from](x1) = [2] [and](x1, x2) = [1] x1 + [0] [if](x1, x2, x3) = [1] x1 + [3] The following symbols are considered usable {a__and, mark, a__if, a__add, a__first, a__from} The order satisfies the following ordering constraints: [a__and(X1, X2)] = [1] X1 + [0] >= [1] X1 + [0] = [and(X1, X2)] [a__and(true(), X)] = [0] >= [0] = [mark(X)] [a__and(false(), Y)] = [0] >= [0] = [false()] [mark(true())] = [0] >= [0] = [true()] [mark(false())] = [0] >= [0] = [false()] [mark(0())] = [0] >= [0] = [0()] [mark(s(X))] = [0] >= [0] = [s(X)] [mark(add(X1, X2))] = [0] ? [4] = [a__add(mark(X1), X2)] [mark(nil())] = [0] ? [1] = [nil()] [mark(cons(X1, X2))] = [0] >= [0] = [cons(X1, X2)] [mark(first(X1, X2))] = [0] ? [1] = [a__first(mark(X1), mark(X2))] [mark(from(X))] = [0] ? [4] = [a__from(X)] [mark(and(X1, X2))] = [0] >= [0] = [a__and(mark(X1), X2)] [mark(if(X1, X2, X3))] = [0] ? [4] = [a__if(mark(X1), X2, X3)] [a__if(X1, X2, X3)] = [1] X1 + [4] > [1] X1 + [3] = [if(X1, X2, X3)] [a__if(true(), X, Y)] = [4] > [0] = [mark(X)] [a__if(false(), X, Y)] = [4] > [0] = [mark(Y)] [a__add(X1, X2)] = [1] X1 + [4] > [1] X1 + [1] = [add(X1, X2)] [a__add(0(), X)] = [4] > [0] = [mark(X)] [a__add(s(X), Y)] = [4] > [0] = [s(add(X, Y))] [a__first(X1, X2)] = [1] X1 + [1] X2 + [1] > [1] X1 + [1] X2 + [0] = [first(X1, X2)] [a__first(0(), X)] = [1] X + [1] >= [1] = [nil()] [a__first(s(X), cons(Y, Z))] = [1] > [0] = [cons(Y, first(X, Z))] [a__from(X)] = [4] > [0] = [cons(X, from(s(X)))] [a__from(X)] = [4] > [2] = [from(X)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { mark(true()) -> true() , mark(false()) -> false() , mark(0()) -> 0() , mark(s(X)) -> s(X) , mark(add(X1, X2)) -> a__add(mark(X1), X2) , mark(nil()) -> nil() , mark(cons(X1, X2)) -> cons(X1, X2) , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) , mark(from(X)) -> a__from(X) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) } Weak Trs: { a__and(X1, X2) -> and(X1, X2) , a__and(true(), X) -> mark(X) , a__and(false(), Y) -> false() , a__if(X1, X2, X3) -> if(X1, X2, X3) , a__if(true(), X, Y) -> mark(X) , a__if(false(), X, Y) -> mark(Y) , 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(Y, first(X, Z)) , a__from(X) -> cons(X, from(s(X))) , a__from(X) -> from(X) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, Uargs(a__first) = {1, 2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [a__and](x1, x2) = [1] x1 + [4] [true] = [4] [mark](x1) = [1] [false] = [4] [a__if](x1, x2, x3) = [1] x1 + [0] [a__add](x1, x2) = [1] x1 + [4] [0] = [0] [s](x1) = [0] [add](x1, x2) = [1] x1 + [1] [a__first](x1, x2) = [1] x1 + [1] x2 + [6] [nil] = [0] [cons](x1, x2) = [0] [first](x1, x2) = [1] x1 + [1] x2 + [2] [a__from](x1) = [0] [from](x1) = [0] [and](x1, x2) = [1] x1 + [3] [if](x1, x2, x3) = [1] x1 + [0] The following symbols are considered usable {a__and, mark, a__if, a__add, a__first, a__from} The order satisfies the following ordering constraints: [a__and(X1, X2)] = [1] X1 + [4] > [1] X1 + [3] = [and(X1, X2)] [a__and(true(), X)] = [8] > [1] = [mark(X)] [a__and(false(), Y)] = [8] > [4] = [false()] [mark(true())] = [1] ? [4] = [true()] [mark(false())] = [1] ? [4] = [false()] [mark(0())] = [1] > [0] = [0()] [mark(s(X))] = [1] > [0] = [s(X)] [mark(add(X1, X2))] = [1] ? [5] = [a__add(mark(X1), X2)] [mark(nil())] = [1] > [0] = [nil()] [mark(cons(X1, X2))] = [1] > [0] = [cons(X1, X2)] [mark(first(X1, X2))] = [1] ? [8] = [a__first(mark(X1), mark(X2))] [mark(from(X))] = [1] > [0] = [a__from(X)] [mark(and(X1, X2))] = [1] ? [5] = [a__and(mark(X1), X2)] [mark(if(X1, X2, X3))] = [1] >= [1] = [a__if(mark(X1), X2, X3)] [a__if(X1, X2, X3)] = [1] X1 + [0] >= [1] X1 + [0] = [if(X1, X2, X3)] [a__if(true(), X, Y)] = [4] > [1] = [mark(X)] [a__if(false(), X, Y)] = [4] > [1] = [mark(Y)] [a__add(X1, X2)] = [1] X1 + [4] > [1] X1 + [1] = [add(X1, X2)] [a__add(0(), X)] = [4] > [1] = [mark(X)] [a__add(s(X), Y)] = [4] > [0] = [s(add(X, Y))] [a__first(X1, X2)] = [1] X1 + [1] X2 + [6] > [1] X1 + [1] X2 + [2] = [first(X1, X2)] [a__first(0(), X)] = [1] X + [6] > [0] = [nil()] [a__first(s(X), cons(Y, Z))] = [6] > [0] = [cons(Y, first(X, Z))] [a__from(X)] = [0] >= [0] = [cons(X, from(s(X)))] [a__from(X)] = [0] >= [0] = [from(X)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { mark(true()) -> true() , mark(false()) -> false() , mark(add(X1, X2)) -> a__add(mark(X1), X2) , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) } Weak Trs: { a__and(X1, X2) -> and(X1, X2) , a__and(true(), X) -> mark(X) , a__and(false(), Y) -> false() , mark(0()) -> 0() , mark(s(X)) -> s(X) , mark(nil()) -> nil() , mark(cons(X1, X2)) -> cons(X1, X2) , mark(from(X)) -> a__from(X) , a__if(X1, X2, X3) -> if(X1, X2, X3) , a__if(true(), X, Y) -> mark(X) , a__if(false(), X, Y) -> mark(Y) , 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(Y, first(X, Z)) , a__from(X) -> cons(X, from(s(X))) , a__from(X) -> from(X) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, Uargs(a__first) = {1, 2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [a__and](x1, x2) = [1] x1 + [1] x2 + [0] [true] = [4] [mark](x1) = [1] x1 + [1] [false] = [0] [a__if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [4] [a__add](x1, x2) = [1] x1 + [1] x2 + [0] [0] = [4] [s](x1) = [0] [add](x1, x2) = [1] x1 + [1] x2 + [0] [a__first](x1, x2) = [1] x1 + [1] x2 + [6] [nil] = [5] [cons](x1, x2) = [0] [first](x1, x2) = [1] x1 + [1] x2 + [0] [a__from](x1) = [4] [from](x1) = [3] [and](x1, x2) = [1] x1 + [1] x2 + [0] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] The following symbols are considered usable {a__and, mark, a__if, a__add, a__first, a__from} The order satisfies the following ordering constraints: [a__and(X1, X2)] = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = [and(X1, X2)] [a__and(true(), X)] = [1] X + [4] > [1] X + [1] = [mark(X)] [a__and(false(), Y)] = [1] Y + [0] >= [0] = [false()] [mark(true())] = [5] > [4] = [true()] [mark(false())] = [1] > [0] = [false()] [mark(0())] = [5] > [4] = [0()] [mark(s(X))] = [1] > [0] = [s(X)] [mark(add(X1, X2))] = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = [a__add(mark(X1), X2)] [mark(nil())] = [6] > [5] = [nil()] [mark(cons(X1, X2))] = [1] > [0] = [cons(X1, X2)] [mark(first(X1, X2))] = [1] X1 + [1] X2 + [1] ? [1] X1 + [1] X2 + [8] = [a__first(mark(X1), mark(X2))] [mark(from(X))] = [4] >= [4] = [a__from(X)] [mark(and(X1, X2))] = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = [a__and(mark(X1), X2)] [mark(if(X1, X2, X3))] = [1] X1 + [1] X2 + [1] X3 + [1] ? [1] X1 + [1] X2 + [1] X3 + [5] = [a__if(mark(X1), X2, X3)] [a__if(X1, X2, X3)] = [1] X1 + [1] X2 + [1] X3 + [4] > [1] X1 + [1] X2 + [1] X3 + [0] = [if(X1, X2, X3)] [a__if(true(), X, Y)] = [1] X + [1] Y + [8] > [1] X + [1] = [mark(X)] [a__if(false(), X, Y)] = [1] X + [1] Y + [4] > [1] Y + [1] = [mark(Y)] [a__add(X1, X2)] = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = [add(X1, X2)] [a__add(0(), X)] = [1] X + [4] > [1] X + [1] = [mark(X)] [a__add(s(X), Y)] = [1] Y + [0] >= [0] = [s(add(X, Y))] [a__first(X1, X2)] = [1] X1 + [1] X2 + [6] > [1] X1 + [1] X2 + [0] = [first(X1, X2)] [a__first(0(), X)] = [1] X + [10] > [5] = [nil()] [a__first(s(X), cons(Y, Z))] = [6] > [0] = [cons(Y, first(X, Z))] [a__from(X)] = [4] > [0] = [cons(X, from(s(X)))] [a__from(X)] = [4] > [3] = [from(X)] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { mark(add(X1, X2)) -> a__add(mark(X1), X2) , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) } Weak Trs: { a__and(X1, X2) -> and(X1, X2) , a__and(true(), X) -> mark(X) , a__and(false(), Y) -> false() , mark(true()) -> true() , mark(false()) -> false() , mark(0()) -> 0() , mark(s(X)) -> s(X) , mark(nil()) -> nil() , mark(cons(X1, X2)) -> cons(X1, X2) , mark(from(X)) -> a__from(X) , a__if(X1, X2, X3) -> if(X1, X2, X3) , a__if(true(), X, Y) -> mark(X) , a__if(false(), X, Y) -> mark(Y) , 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(Y, first(X, Z)) , a__from(X) -> cons(X, from(s(X))) , a__from(X) -> from(X) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. Trs: { mark(add(X1, X2)) -> a__add(mark(X1), X2) , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^1)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are usable: Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, Uargs(a__first) = {1, 2} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [a__and](x1, x2) = [1] x1 + [2] x2 + [0] [true] = [0] [mark](x1) = [2] x1 + [0] [false] = [0] [a__if](x1, x2, x3) = [1] x1 + [2] x2 + [2] x3 + [0] [a__add](x1, x2) = [1] x1 + [2] x2 + [4] [0] = [4] [s](x1) = [4] [add](x1, x2) = [1] x1 + [1] x2 + [4] [a__first](x1, x2) = [1] x1 + [1] x2 + [4] [nil] = [4] [cons](x1, x2) = [0] [first](x1, x2) = [1] x1 + [1] x2 + [4] [a__from](x1) = [0] [from](x1) = [0] [and](x1, x2) = [1] x1 + [1] x2 + [0] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] The following symbols are considered usable {a__and, mark, a__if, a__add, a__first, a__from} The order satisfies the following ordering constraints: [a__and(X1, X2)] = [1] X1 + [2] X2 + [0] >= [1] X1 + [1] X2 + [0] = [and(X1, X2)] [a__and(true(), X)] = [2] X + [0] >= [2] X + [0] = [mark(X)] [a__and(false(), Y)] = [2] Y + [0] >= [0] = [false()] [mark(true())] = [0] >= [0] = [true()] [mark(false())] = [0] >= [0] = [false()] [mark(0())] = [8] > [4] = [0()] [mark(s(X))] = [8] > [4] = [s(X)] [mark(add(X1, X2))] = [2] X1 + [2] X2 + [8] > [2] X1 + [2] X2 + [4] = [a__add(mark(X1), X2)] [mark(nil())] = [8] > [4] = [nil()] [mark(cons(X1, X2))] = [0] >= [0] = [cons(X1, X2)] [mark(first(X1, X2))] = [2] X1 + [2] X2 + [8] > [2] X1 + [2] X2 + [4] = [a__first(mark(X1), mark(X2))] [mark(from(X))] = [0] >= [0] = [a__from(X)] [mark(and(X1, X2))] = [2] X1 + [2] X2 + [0] >= [2] X1 + [2] X2 + [0] = [a__and(mark(X1), X2)] [mark(if(X1, X2, X3))] = [2] X1 + [2] X2 + [2] X3 + [0] >= [2] X1 + [2] X2 + [2] X3 + [0] = [a__if(mark(X1), X2, X3)] [a__if(X1, X2, X3)] = [1] X1 + [2] X2 + [2] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = [if(X1, X2, X3)] [a__if(true(), X, Y)] = [2] X + [2] Y + [0] >= [2] X + [0] = [mark(X)] [a__if(false(), X, Y)] = [2] X + [2] Y + [0] >= [2] Y + [0] = [mark(Y)] [a__add(X1, X2)] = [1] X1 + [2] X2 + [4] >= [1] X1 + [1] X2 + [4] = [add(X1, X2)] [a__add(0(), X)] = [2] X + [8] > [2] X + [0] = [mark(X)] [a__add(s(X), Y)] = [2] Y + [8] > [4] = [s(add(X, Y))] [a__first(X1, X2)] = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [4] = [first(X1, X2)] [a__first(0(), X)] = [1] X + [8] > [4] = [nil()] [a__first(s(X), cons(Y, Z))] = [8] > [0] = [cons(Y, first(X, Z))] [a__from(X)] = [0] >= [0] = [cons(X, from(s(X)))] [a__from(X)] = [0] >= [0] = [from(X)] We return to the main proof. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) } Weak Trs: { a__and(X1, X2) -> and(X1, X2) , a__and(true(), X) -> mark(X) , a__and(false(), Y) -> false() , mark(true()) -> true() , mark(false()) -> false() , mark(0()) -> 0() , mark(s(X)) -> s(X) , mark(add(X1, X2)) -> a__add(mark(X1), X2) , mark(nil()) -> nil() , mark(cons(X1, X2)) -> cons(X1, X2) , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) , mark(from(X)) -> a__from(X) , a__if(X1, X2, X3) -> if(X1, X2, X3) , a__if(true(), X, Y) -> mark(X) , a__if(false(), X, Y) -> mark(Y) , 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(Y, first(X, Z)) , a__from(X) -> cons(X, from(s(X))) , a__from(X) -> from(X) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. Trs: { mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^1)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are usable: Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, Uargs(a__first) = {1, 2} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [a__and](x1, x2) = [1] x1 + [4] x2 + [0] [true] = [0] [mark](x1) = [4] x1 + [0] [false] = [0] [a__if](x1, x2, x3) = [1] x1 + [4] x2 + [4] x3 + [1] [a__add](x1, x2) = [1] x1 + [4] x2 + [0] [0] = [2] [s](x1) = [0] [add](x1, x2) = [1] x1 + [1] x2 + [0] [a__first](x1, x2) = [1] x1 + [1] x2 + [0] [nil] = [0] [cons](x1, x2) = [0] [first](x1, x2) = [1] x1 + [1] x2 + [0] [a__from](x1) = [0] [from](x1) = [0] [and](x1, x2) = [1] x1 + [1] x2 + [0] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1] The following symbols are considered usable {a__and, mark, a__if, a__add, a__first, a__from} The order satisfies the following ordering constraints: [a__and(X1, X2)] = [1] X1 + [4] X2 + [0] >= [1] X1 + [1] X2 + [0] = [and(X1, X2)] [a__and(true(), X)] = [4] X + [0] >= [4] X + [0] = [mark(X)] [a__and(false(), Y)] = [4] Y + [0] >= [0] = [false()] [mark(true())] = [0] >= [0] = [true()] [mark(false())] = [0] >= [0] = [false()] [mark(0())] = [8] > [2] = [0()] [mark(s(X))] = [0] >= [0] = [s(X)] [mark(add(X1, X2))] = [4] X1 + [4] X2 + [0] >= [4] X1 + [4] X2 + [0] = [a__add(mark(X1), X2)] [mark(nil())] = [0] >= [0] = [nil()] [mark(cons(X1, X2))] = [0] >= [0] = [cons(X1, X2)] [mark(first(X1, X2))] = [4] X1 + [4] X2 + [0] >= [4] X1 + [4] X2 + [0] = [a__first(mark(X1), mark(X2))] [mark(from(X))] = [0] >= [0] = [a__from(X)] [mark(and(X1, X2))] = [4] X1 + [4] X2 + [0] >= [4] X1 + [4] X2 + [0] = [a__and(mark(X1), X2)] [mark(if(X1, X2, X3))] = [4] X1 + [4] X2 + [4] X3 + [4] > [4] X1 + [4] X2 + [4] X3 + [1] = [a__if(mark(X1), X2, X3)] [a__if(X1, X2, X3)] = [1] X1 + [4] X2 + [4] X3 + [1] >= [1] X1 + [1] X2 + [1] X3 + [1] = [if(X1, X2, X3)] [a__if(true(), X, Y)] = [4] X + [4] Y + [1] > [4] X + [0] = [mark(X)] [a__if(false(), X, Y)] = [4] X + [4] Y + [1] > [4] Y + [0] = [mark(Y)] [a__add(X1, X2)] = [1] X1 + [4] X2 + [0] >= [1] X1 + [1] X2 + [0] = [add(X1, X2)] [a__add(0(), X)] = [4] X + [2] > [4] X + [0] = [mark(X)] [a__add(s(X), Y)] = [4] Y + [0] >= [0] = [s(add(X, Y))] [a__first(X1, X2)] = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = [first(X1, X2)] [a__first(0(), X)] = [1] X + [2] > [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)] We return to the main proof. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { mark(and(X1, X2)) -> a__and(mark(X1), X2) } Weak Trs: { a__and(X1, X2) -> and(X1, X2) , a__and(true(), X) -> mark(X) , a__and(false(), Y) -> false() , mark(true()) -> true() , mark(false()) -> false() , mark(0()) -> 0() , mark(s(X)) -> s(X) , mark(add(X1, X2)) -> a__add(mark(X1), X2) , mark(nil()) -> nil() , mark(cons(X1, X2)) -> cons(X1, X2) , 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) , a__if(X1, X2, X3) -> if(X1, X2, X3) , a__if(true(), X, Y) -> mark(X) , a__if(false(), X, Y) -> mark(Y) , 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(Y, first(X, Z)) , a__from(X) -> cons(X, from(s(X))) , a__from(X) -> from(X) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. Trs: { mark(and(X1, X2)) -> a__and(mark(X1), X2) } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^1)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are usable: Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, Uargs(a__first) = {1, 2} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [a__and](x1, x2) = [1] x1 + [2] x2 + [4] [true] = [4] [mark](x1) = [2] x1 + [0] [false] = [4] [a__if](x1, x2, x3) = [1] x1 + [2] x2 + [2] x3 + [0] [a__add](x1, x2) = [1] x1 + [2] x2 + [4] [0] = [4] [s](x1) = [4] [add](x1, x2) = [1] x1 + [1] x2 + [4] [a__first](x1, x2) = [1] x1 + [1] x2 + [0] [nil] = [4] [cons](x1, x2) = [4] [first](x1, x2) = [1] x1 + [1] x2 + [0] [a__from](x1) = [4] [from](x1) = [4] [and](x1, x2) = [1] x1 + [1] x2 + [4] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] The following symbols are considered usable {a__and, mark, a__if, a__add, a__first, a__from} The order satisfies the following ordering constraints: [a__and(X1, X2)] = [1] X1 + [2] X2 + [4] >= [1] X1 + [1] X2 + [4] = [and(X1, X2)] [a__and(true(), X)] = [2] X + [8] > [2] X + [0] = [mark(X)] [a__and(false(), Y)] = [2] Y + [8] > [4] = [false()] [mark(true())] = [8] > [4] = [true()] [mark(false())] = [8] > [4] = [false()] [mark(0())] = [8] > [4] = [0()] [mark(s(X))] = [8] > [4] = [s(X)] [mark(add(X1, X2))] = [2] X1 + [2] X2 + [8] > [2] X1 + [2] X2 + [4] = [a__add(mark(X1), X2)] [mark(nil())] = [8] > [4] = [nil()] [mark(cons(X1, X2))] = [8] > [4] = [cons(X1, X2)] [mark(first(X1, X2))] = [2] X1 + [2] X2 + [0] >= [2] X1 + [2] X2 + [0] = [a__first(mark(X1), mark(X2))] [mark(from(X))] = [8] > [4] = [a__from(X)] [mark(and(X1, X2))] = [2] X1 + [2] X2 + [8] > [2] X1 + [2] X2 + [4] = [a__and(mark(X1), X2)] [mark(if(X1, X2, X3))] = [2] X1 + [2] X2 + [2] X3 + [0] >= [2] X1 + [2] X2 + [2] X3 + [0] = [a__if(mark(X1), X2, X3)] [a__if(X1, X2, X3)] = [1] X1 + [2] X2 + [2] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = [if(X1, X2, X3)] [a__if(true(), X, Y)] = [2] X + [2] Y + [4] > [2] X + [0] = [mark(X)] [a__if(false(), X, Y)] = [2] X + [2] Y + [4] > [2] Y + [0] = [mark(Y)] [a__add(X1, X2)] = [1] X1 + [2] X2 + [4] >= [1] X1 + [1] X2 + [4] = [add(X1, X2)] [a__add(0(), X)] = [2] X + [8] > [2] X + [0] = [mark(X)] [a__add(s(X), Y)] = [2] Y + [8] > [4] = [s(add(X, Y))] [a__first(X1, X2)] = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = [first(X1, X2)] [a__first(0(), X)] = [1] X + [4] >= [4] = [nil()] [a__first(s(X), cons(Y, Z))] = [8] > [4] = [cons(Y, first(X, Z))] [a__from(X)] = [4] >= [4] = [cons(X, from(s(X)))] [a__from(X)] = [4] >= [4] = [from(X)] We return to the main proof. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak Trs: { a__and(X1, X2) -> and(X1, X2) , a__and(true(), X) -> mark(X) , a__and(false(), Y) -> false() , mark(true()) -> true() , mark(false()) -> false() , mark(0()) -> 0() , mark(s(X)) -> s(X) , mark(add(X1, X2)) -> a__add(mark(X1), X2) , mark(nil()) -> nil() , mark(cons(X1, X2)) -> cons(X1, X2) , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) , mark(from(X)) -> a__from(X) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) , a__if(X1, X2, X3) -> if(X1, X2, X3) , a__if(true(), X, Y) -> mark(X) , a__if(false(), X, Y) -> mark(Y) , 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(Y, first(X, Z)) , a__from(X) -> cons(X, from(s(X))) , a__from(X) -> from(X) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Hurray, we answered YES(O(1),O(n^1))