YES(O(1),O(n^2)) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict Trs: { minus(X1, X2) -> n__minus(X1, X2) , minus(n__0(), Y) -> 0() , minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) , 0() -> n__0() , activate(X) -> X , activate(n__0()) -> 0() , activate(n__s(X)) -> s(activate(X)) , activate(n__div(X1, X2)) -> div(activate(X1), X2) , activate(n__minus(X1, X2)) -> minus(X1, X2) , geq(X, n__0()) -> true() , geq(n__0(), n__s(Y)) -> false() , geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) , div(X1, X2) -> n__div(X1, X2) , div(0(), n__s(Y)) -> 0() , div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0()) , s(X) -> n__s(X) , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) Arguments of following rules are not normal-forms: { div(0(), n__s(Y)) -> 0() , div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0()) } All above mentioned rules can be savely removed. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict Trs: { minus(X1, X2) -> n__minus(X1, X2) , minus(n__0(), Y) -> 0() , minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) , 0() -> n__0() , activate(X) -> X , activate(n__0()) -> 0() , activate(n__s(X)) -> s(activate(X)) , activate(n__div(X1, X2)) -> div(activate(X1), X2) , activate(n__minus(X1, X2)) -> minus(X1, X2) , geq(X, n__0()) -> true() , geq(n__0(), n__s(Y)) -> false() , geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) , div(X1, X2) -> n__div(X1, X2) , s(X) -> n__s(X) , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(minus) = {1, 2}, Uargs(geq) = {1, 2}, Uargs(div) = {1}, Uargs(s) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [minus](x1, x2) = [1] x1 + [1] x2 + [1] [n__0] = [7] [0] = [7] [n__s](x1) = [1] x1 + [7] [activate](x1) = [1] x1 + [7] [geq](x1, x2) = [1] x1 + [1] x2 + [1] [true] = [7] [false] = [7] [div](x1, x2) = [1] x1 + [1] x2 + [7] [s](x1) = [1] x1 + [7] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [7] [n__div](x1, x2) = [1] x1 + [1] x2 + [7] [n__minus](x1, x2) = [1] x1 + [1] x2 + [7] The following symbols are considered usable {minus, 0, activate, geq, div, s, if} The order satisfies the following ordering constraints: [minus(X1, X2)] = [1] X1 + [1] X2 + [1] ? [1] X1 + [1] X2 + [7] = [n__minus(X1, X2)] [minus(n__0(), Y)] = [1] Y + [8] > [7] = [0()] [minus(n__s(X), n__s(Y))] = [1] Y + [1] X + [15] >= [1] Y + [1] X + [15] = [minus(activate(X), activate(Y))] [0()] = [7] >= [7] = [n__0()] [activate(X)] = [1] X + [7] > [1] X + [0] = [X] [activate(n__0())] = [14] > [7] = [0()] [activate(n__s(X))] = [1] X + [14] >= [1] X + [14] = [s(activate(X))] [activate(n__div(X1, X2))] = [1] X1 + [1] X2 + [14] >= [1] X1 + [1] X2 + [14] = [div(activate(X1), X2)] [activate(n__minus(X1, X2))] = [1] X1 + [1] X2 + [14] > [1] X1 + [1] X2 + [1] = [minus(X1, X2)] [geq(X, n__0())] = [1] X + [8] > [7] = [true()] [geq(n__0(), n__s(Y))] = [1] Y + [15] > [7] = [false()] [geq(n__s(X), n__s(Y))] = [1] Y + [1] X + [15] >= [1] Y + [1] X + [15] = [geq(activate(X), activate(Y))] [div(X1, X2)] = [1] X1 + [1] X2 + [7] >= [1] X1 + [1] X2 + [7] = [n__div(X1, X2)] [s(X)] = [1] X + [7] >= [1] X + [7] = [n__s(X)] [if(true(), X, Y)] = [1] Y + [1] X + [14] > [1] X + [7] = [activate(X)] [if(false(), X, Y)] = [1] Y + [1] X + [14] > [1] Y + [7] = [activate(Y)] 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^2)). Strict Trs: { minus(X1, X2) -> n__minus(X1, X2) , minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) , 0() -> n__0() , activate(n__s(X)) -> s(activate(X)) , activate(n__div(X1, X2)) -> div(activate(X1), X2) , geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) , div(X1, X2) -> n__div(X1, X2) , s(X) -> n__s(X) } Weak Trs: { minus(n__0(), Y) -> 0() , activate(X) -> X , activate(n__0()) -> 0() , activate(n__minus(X1, X2)) -> minus(X1, X2) , geq(X, n__0()) -> true() , geq(n__0(), n__s(Y)) -> false() , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(minus) = {1, 2}, Uargs(geq) = {1, 2}, Uargs(div) = {1}, Uargs(s) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [minus](x1, x2) = [1] x1 + [1] x2 + [0] [n__0] = [4] [0] = [0] [n__s](x1) = [1] x1 + [0] [activate](x1) = [1] x1 + [0] [geq](x1, x2) = [1] x1 + [1] x2 + [0] [true] = [0] [false] = [0] [div](x1, x2) = [1] x1 + [1] x2 + [1] [s](x1) = [1] x1 + [0] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [4] [n__div](x1, x2) = [1] x1 + [1] x2 + [0] [n__minus](x1, x2) = [1] x1 + [1] x2 + [5] The following symbols are considered usable {minus, 0, activate, geq, div, s, if} The order satisfies the following ordering constraints: [minus(X1, X2)] = [1] X1 + [1] X2 + [0] ? [1] X1 + [1] X2 + [5] = [n__minus(X1, X2)] [minus(n__0(), Y)] = [1] Y + [4] > [0] = [0()] [minus(n__s(X), n__s(Y))] = [1] Y + [1] X + [0] >= [1] Y + [1] X + [0] = [minus(activate(X), activate(Y))] [0()] = [0] ? [4] = [n__0()] [activate(X)] = [1] X + [0] >= [1] X + [0] = [X] [activate(n__0())] = [4] > [0] = [0()] [activate(n__s(X))] = [1] X + [0] >= [1] X + [0] = [s(activate(X))] [activate(n__div(X1, X2))] = [1] X1 + [1] X2 + [0] ? [1] X1 + [1] X2 + [1] = [div(activate(X1), X2)] [activate(n__minus(X1, X2))] = [1] X1 + [1] X2 + [5] > [1] X1 + [1] X2 + [0] = [minus(X1, X2)] [geq(X, n__0())] = [1] X + [4] > [0] = [true()] [geq(n__0(), n__s(Y))] = [1] Y + [4] > [0] = [false()] [geq(n__s(X), n__s(Y))] = [1] Y + [1] X + [0] >= [1] Y + [1] X + [0] = [geq(activate(X), activate(Y))] [div(X1, X2)] = [1] X1 + [1] X2 + [1] > [1] X1 + [1] X2 + [0] = [n__div(X1, X2)] [s(X)] = [1] X + [0] >= [1] X + [0] = [n__s(X)] [if(true(), X, Y)] = [1] Y + [1] X + [4] > [1] X + [0] = [activate(X)] [if(false(), X, Y)] = [1] Y + [1] X + [4] > [1] Y + [0] = [activate(Y)] 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^2)). Strict Trs: { minus(X1, X2) -> n__minus(X1, X2) , minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) , 0() -> n__0() , activate(n__s(X)) -> s(activate(X)) , activate(n__div(X1, X2)) -> div(activate(X1), X2) , geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) , s(X) -> n__s(X) } Weak Trs: { minus(n__0(), Y) -> 0() , activate(X) -> X , activate(n__0()) -> 0() , activate(n__minus(X1, X2)) -> minus(X1, X2) , geq(X, n__0()) -> true() , geq(n__0(), n__s(Y)) -> false() , div(X1, X2) -> n__div(X1, X2) , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(minus) = {1, 2}, Uargs(geq) = {1, 2}, Uargs(div) = {1}, Uargs(s) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [minus](x1, x2) = [1] x1 + [1] x2 + [0] [n__0] = [4] [0] = [3] [n__s](x1) = [1] x1 + [0] [activate](x1) = [1] x1 + [0] [geq](x1, x2) = [1] x1 + [1] x2 + [0] [true] = [0] [false] = [0] [div](x1, x2) = [1] x1 + [1] x2 + [0] [s](x1) = [1] x1 + [4] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [4] [n__div](x1, x2) = [1] x1 + [1] x2 + [0] [n__minus](x1, x2) = [1] x1 + [1] x2 + [5] The following symbols are considered usable {minus, 0, activate, geq, div, s, if} The order satisfies the following ordering constraints: [minus(X1, X2)] = [1] X1 + [1] X2 + [0] ? [1] X1 + [1] X2 + [5] = [n__minus(X1, X2)] [minus(n__0(), Y)] = [1] Y + [4] > [3] = [0()] [minus(n__s(X), n__s(Y))] = [1] Y + [1] X + [0] >= [1] Y + [1] X + [0] = [minus(activate(X), activate(Y))] [0()] = [3] ? [4] = [n__0()] [activate(X)] = [1] X + [0] >= [1] X + [0] = [X] [activate(n__0())] = [4] > [3] = [0()] [activate(n__s(X))] = [1] X + [0] ? [1] X + [4] = [s(activate(X))] [activate(n__div(X1, X2))] = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = [div(activate(X1), X2)] [activate(n__minus(X1, X2))] = [1] X1 + [1] X2 + [5] > [1] X1 + [1] X2 + [0] = [minus(X1, X2)] [geq(X, n__0())] = [1] X + [4] > [0] = [true()] [geq(n__0(), n__s(Y))] = [1] Y + [4] > [0] = [false()] [geq(n__s(X), n__s(Y))] = [1] Y + [1] X + [0] >= [1] Y + [1] X + [0] = [geq(activate(X), activate(Y))] [div(X1, X2)] = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = [n__div(X1, X2)] [s(X)] = [1] X + [4] > [1] X + [0] = [n__s(X)] [if(true(), X, Y)] = [1] Y + [1] X + [4] > [1] X + [0] = [activate(X)] [if(false(), X, Y)] = [1] Y + [1] X + [4] > [1] Y + [0] = [activate(Y)] 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^2)). Strict Trs: { minus(X1, X2) -> n__minus(X1, X2) , minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) , 0() -> n__0() , activate(n__s(X)) -> s(activate(X)) , activate(n__div(X1, X2)) -> div(activate(X1), X2) , geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) } Weak Trs: { minus(n__0(), Y) -> 0() , activate(X) -> X , activate(n__0()) -> 0() , activate(n__minus(X1, X2)) -> minus(X1, X2) , geq(X, n__0()) -> true() , geq(n__0(), n__s(Y)) -> false() , div(X1, X2) -> n__div(X1, X2) , s(X) -> n__s(X) , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(minus) = {1, 2}, Uargs(geq) = {1, 2}, Uargs(div) = {1}, Uargs(s) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [minus](x1, x2) = [1] x1 + [1] x2 + [4] [n__0] = [4] [0] = [6] [n__s](x1) = [1] x1 + [0] [activate](x1) = [1] x1 + [4] [geq](x1, x2) = [1] x1 + [1] x2 + [0] [true] = [0] [false] = [4] [div](x1, x2) = [1] x1 + [1] x2 + [4] [s](x1) = [1] x1 + [4] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [4] [n__div](x1, x2) = [1] x1 + [1] x2 + [4] [n__minus](x1, x2) = [1] x1 + [1] x2 + [5] The following symbols are considered usable {minus, 0, activate, geq, div, s, if} The order satisfies the following ordering constraints: [minus(X1, X2)] = [1] X1 + [1] X2 + [4] ? [1] X1 + [1] X2 + [5] = [n__minus(X1, X2)] [minus(n__0(), Y)] = [1] Y + [8] > [6] = [0()] [minus(n__s(X), n__s(Y))] = [1] Y + [1] X + [4] ? [1] Y + [1] X + [12] = [minus(activate(X), activate(Y))] [0()] = [6] > [4] = [n__0()] [activate(X)] = [1] X + [4] > [1] X + [0] = [X] [activate(n__0())] = [8] > [6] = [0()] [activate(n__s(X))] = [1] X + [4] ? [1] X + [8] = [s(activate(X))] [activate(n__div(X1, X2))] = [1] X1 + [1] X2 + [8] >= [1] X1 + [1] X2 + [8] = [div(activate(X1), X2)] [activate(n__minus(X1, X2))] = [1] X1 + [1] X2 + [9] > [1] X1 + [1] X2 + [4] = [minus(X1, X2)] [geq(X, n__0())] = [1] X + [4] > [0] = [true()] [geq(n__0(), n__s(Y))] = [1] Y + [4] >= [4] = [false()] [geq(n__s(X), n__s(Y))] = [1] Y + [1] X + [0] ? [1] Y + [1] X + [8] = [geq(activate(X), activate(Y))] [div(X1, X2)] = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [4] = [n__div(X1, X2)] [s(X)] = [1] X + [4] > [1] X + [0] = [n__s(X)] [if(true(), X, Y)] = [1] Y + [1] X + [4] >= [1] X + [4] = [activate(X)] [if(false(), X, Y)] = [1] Y + [1] X + [8] > [1] Y + [4] = [activate(Y)] 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^2)). Strict Trs: { minus(X1, X2) -> n__minus(X1, X2) , minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) , activate(n__s(X)) -> s(activate(X)) , activate(n__div(X1, X2)) -> div(activate(X1), X2) , geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) } Weak Trs: { minus(n__0(), Y) -> 0() , 0() -> n__0() , activate(X) -> X , activate(n__0()) -> 0() , activate(n__minus(X1, X2)) -> minus(X1, X2) , geq(X, n__0()) -> true() , geq(n__0(), n__s(Y)) -> false() , div(X1, X2) -> n__div(X1, X2) , s(X) -> n__s(X) , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(minus) = {1, 2}, Uargs(geq) = {1, 2}, Uargs(div) = {1}, Uargs(s) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [minus](x1, x2) = [1] x1 + [1] x2 + [4] [n__0] = [0] [0] = [3] [n__s](x1) = [1] x1 + [4] [activate](x1) = [1] x1 + [4] [geq](x1, x2) = [1] x1 + [1] x2 + [0] [true] = [0] [false] = [0] [div](x1, x2) = [1] x1 + [1] x2 + [4] [s](x1) = [1] x1 + [4] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [4] [n__div](x1, x2) = [1] x1 + [1] x2 + [1] [n__minus](x1, x2) = [1] x1 + [1] x2 + [1] The following symbols are considered usable {minus, 0, activate, geq, div, s, if} The order satisfies the following ordering constraints: [minus(X1, X2)] = [1] X1 + [1] X2 + [4] > [1] X1 + [1] X2 + [1] = [n__minus(X1, X2)] [minus(n__0(), Y)] = [1] Y + [4] > [3] = [0()] [minus(n__s(X), n__s(Y))] = [1] Y + [1] X + [12] >= [1] Y + [1] X + [12] = [minus(activate(X), activate(Y))] [0()] = [3] > [0] = [n__0()] [activate(X)] = [1] X + [4] > [1] X + [0] = [X] [activate(n__0())] = [4] > [3] = [0()] [activate(n__s(X))] = [1] X + [8] >= [1] X + [8] = [s(activate(X))] [activate(n__div(X1, X2))] = [1] X1 + [1] X2 + [5] ? [1] X1 + [1] X2 + [8] = [div(activate(X1), X2)] [activate(n__minus(X1, X2))] = [1] X1 + [1] X2 + [5] > [1] X1 + [1] X2 + [4] = [minus(X1, X2)] [geq(X, n__0())] = [1] X + [0] >= [0] = [true()] [geq(n__0(), n__s(Y))] = [1] Y + [4] > [0] = [false()] [geq(n__s(X), n__s(Y))] = [1] Y + [1] X + [8] >= [1] Y + [1] X + [8] = [geq(activate(X), activate(Y))] [div(X1, X2)] = [1] X1 + [1] X2 + [4] > [1] X1 + [1] X2 + [1] = [n__div(X1, X2)] [s(X)] = [1] X + [4] >= [1] X + [4] = [n__s(X)] [if(true(), X, Y)] = [1] Y + [1] X + [4] >= [1] X + [4] = [activate(X)] [if(false(), X, Y)] = [1] Y + [1] X + [4] >= [1] Y + [4] = [activate(Y)] 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^2)). Strict Trs: { minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) , activate(n__s(X)) -> s(activate(X)) , activate(n__div(X1, X2)) -> div(activate(X1), X2) , geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) } Weak Trs: { minus(X1, X2) -> n__minus(X1, X2) , minus(n__0(), Y) -> 0() , 0() -> n__0() , activate(X) -> X , activate(n__0()) -> 0() , activate(n__minus(X1, X2)) -> minus(X1, X2) , geq(X, n__0()) -> true() , geq(n__0(), n__s(Y)) -> false() , div(X1, X2) -> n__div(X1, X2) , s(X) -> n__s(X) , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(minus) = {1, 2}, Uargs(geq) = {1, 2}, Uargs(div) = {1}, Uargs(s) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [minus](x1, x2) = [1] x1 + [1] x2 + [0] [n__0] = [4] [0] = [4] [n__s](x1) = [1] x1 + [4] [activate](x1) = [1] x1 + [0] [geq](x1, x2) = [1] x1 + [1] x2 + [0] [true] = [0] [false] = [4] [div](x1, x2) = [1] x1 + [1] x2 + [4] [s](x1) = [1] x1 + [4] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [4] [n__div](x1, x2) = [1] x1 + [1] x2 + [1] [n__minus](x1, x2) = [1] x1 + [1] x2 + [0] The following symbols are considered usable {minus, 0, activate, geq, div, s, if} The order satisfies the following ordering constraints: [minus(X1, X2)] = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = [n__minus(X1, X2)] [minus(n__0(), Y)] = [1] Y + [4] >= [4] = [0()] [minus(n__s(X), n__s(Y))] = [1] Y + [1] X + [8] > [1] Y + [1] X + [0] = [minus(activate(X), activate(Y))] [0()] = [4] >= [4] = [n__0()] [activate(X)] = [1] X + [0] >= [1] X + [0] = [X] [activate(n__0())] = [4] >= [4] = [0()] [activate(n__s(X))] = [1] X + [4] >= [1] X + [4] = [s(activate(X))] [activate(n__div(X1, X2))] = [1] X1 + [1] X2 + [1] ? [1] X1 + [1] X2 + [4] = [div(activate(X1), X2)] [activate(n__minus(X1, X2))] = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = [minus(X1, X2)] [geq(X, n__0())] = [1] X + [4] > [0] = [true()] [geq(n__0(), n__s(Y))] = [1] Y + [8] > [4] = [false()] [geq(n__s(X), n__s(Y))] = [1] Y + [1] X + [8] > [1] Y + [1] X + [0] = [geq(activate(X), activate(Y))] [div(X1, X2)] = [1] X1 + [1] X2 + [4] > [1] X1 + [1] X2 + [1] = [n__div(X1, X2)] [s(X)] = [1] X + [4] >= [1] X + [4] = [n__s(X)] [if(true(), X, Y)] = [1] Y + [1] X + [4] > [1] X + [0] = [activate(X)] [if(false(), X, Y)] = [1] Y + [1] X + [8] > [1] Y + [0] = [activate(Y)] 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^2)). Strict Trs: { activate(n__s(X)) -> s(activate(X)) , activate(n__div(X1, X2)) -> div(activate(X1), X2) } Weak Trs: { minus(X1, X2) -> n__minus(X1, X2) , minus(n__0(), Y) -> 0() , minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) , 0() -> n__0() , activate(X) -> X , activate(n__0()) -> 0() , activate(n__minus(X1, X2)) -> minus(X1, X2) , geq(X, n__0()) -> true() , geq(n__0(), n__s(Y)) -> false() , geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) , div(X1, X2) -> n__div(X1, X2) , s(X) -> n__s(X) , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We use the processor 'matrix interpretation of dimension 2' to orient following rules strictly. Trs: { activate(n__s(X)) -> s(activate(X)) , activate(n__div(X1, X2)) -> div(activate(X1), X2) } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^2)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are usable: Uargs(minus) = {1, 2}, Uargs(geq) = {1, 2}, Uargs(div) = {1}, Uargs(s) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [minus](x1, x2) = [1 0] x1 + [1 4] x2 + [3] [0 0] [0 0] [4] [n__0] = [5] [4] [0] = [6] [4] [n__s](x1) = [1 4] x1 + [3] [0 1] [1] [activate](x1) = [1 1] x1 + [0] [0 1] [0] [geq](x1, x2) = [2 0] x1 + [1 1] x2 + [0] [0 0] [0 0] [4] [true] = [4] [0] [false] = [0] [0] [div](x1, x2) = [1 0] x1 + [0 3] x2 + [0] [0 1] [0 1] [4] [s](x1) = [1 4] x1 + [3] [0 1] [1] [if](x1, x2, x3) = [2 0] x1 + [7 7] x2 + [7 7] x3 + [0] [0 0] [7 7] [7 7] [4] [n__div](x1, x2) = [1 0] x1 + [0 2] x2 + [0] [0 1] [0 1] [4] [n__minus](x1, x2) = [1 0] x1 + [1 4] x2 + [0] [0 0] [0 0] [4] The following symbols are considered usable {minus, 0, activate, geq, div, s, if} The order satisfies the following ordering constraints: [minus(X1, X2)] = [1 0] X1 + [1 4] X2 + [3] [0 0] [0 0] [4] > [1 0] X1 + [1 4] X2 + [0] [0 0] [0 0] [4] = [n__minus(X1, X2)] [minus(n__0(), Y)] = [1 4] Y + [8] [0 0] [4] > [6] [4] = [0()] [minus(n__s(X), n__s(Y))] = [1 8] Y + [1 4] X + [13] [0 0] [0 0] [4] > [1 5] Y + [1 1] X + [3] [0 0] [0 0] [4] = [minus(activate(X), activate(Y))] [0()] = [6] [4] > [5] [4] = [n__0()] [activate(X)] = [1 1] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = [X] [activate(n__0())] = [9] [4] > [6] [4] = [0()] [activate(n__s(X))] = [1 5] X + [4] [0 1] [1] > [1 5] X + [3] [0 1] [1] = [s(activate(X))] [activate(n__div(X1, X2))] = [1 1] X1 + [0 3] X2 + [4] [0 1] [0 1] [4] > [1 1] X1 + [0 3] X2 + [0] [0 1] [0 1] [4] = [div(activate(X1), X2)] [activate(n__minus(X1, X2))] = [1 0] X1 + [1 4] X2 + [4] [0 0] [0 0] [4] > [1 0] X1 + [1 4] X2 + [3] [0 0] [0 0] [4] = [minus(X1, X2)] [geq(X, n__0())] = [2 0] X + [9] [0 0] [4] > [4] [0] = [true()] [geq(n__0(), n__s(Y))] = [1 5] Y + [14] [0 0] [4] > [0] [0] = [false()] [geq(n__s(X), n__s(Y))] = [1 5] Y + [2 8] X + [10] [0 0] [0 0] [4] > [1 2] Y + [2 2] X + [0] [0 0] [0 0] [4] = [geq(activate(X), activate(Y))] [div(X1, X2)] = [1 0] X1 + [0 3] X2 + [0] [0 1] [0 1] [4] >= [1 0] X1 + [0 2] X2 + [0] [0 1] [0 1] [4] = [n__div(X1, X2)] [s(X)] = [1 4] X + [3] [0 1] [1] >= [1 4] X + [3] [0 1] [1] = [n__s(X)] [if(true(), X, Y)] = [7 7] Y + [7 7] X + [8] [7 7] [7 7] [4] > [1 1] X + [0] [0 1] [0] = [activate(X)] [if(false(), X, Y)] = [7 7] Y + [7 7] X + [0] [7 7] [7 7] [4] >= [1 1] Y + [0] [0 1] [0] = [activate(Y)] 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: { minus(X1, X2) -> n__minus(X1, X2) , minus(n__0(), Y) -> 0() , minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) , 0() -> n__0() , activate(X) -> X , activate(n__0()) -> 0() , activate(n__s(X)) -> s(activate(X)) , activate(n__div(X1, X2)) -> div(activate(X1), X2) , activate(n__minus(X1, X2)) -> minus(X1, X2) , geq(X, n__0()) -> true() , geq(n__0(), n__s(Y)) -> false() , geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) , div(X1, X2) -> n__div(X1, X2) , s(X) -> n__s(X) , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Hurray, we answered YES(O(1),O(n^2))