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: { first(X1, X2) -> n__first(X1, X2) , first(0(), X) -> nil() , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) , s(X) -> n__s(X) , activate(X) -> X , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) , activate(n__from(X)) -> from(activate(X)) , activate(n__s(X)) -> s(activate(X)) , from(X) -> cons(X, n__from(n__s(X))) , from(X) -> n__from(X) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) Arguments of following rules are not normal-forms: { first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) } 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^1)). Strict Trs: { first(X1, X2) -> n__first(X1, X2) , first(0(), X) -> nil() , s(X) -> n__s(X) , activate(X) -> X , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) , activate(n__from(X)) -> from(activate(X)) , activate(n__s(X)) -> s(activate(X)) , from(X) -> cons(X, n__from(n__s(X))) , from(X) -> n__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(first) = {1, 2}, Uargs(s) = {1}, Uargs(from) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [first](x1, x2) = [1] x1 + [1] x2 + [1] [0] = [7] [nil] = [7] [s](x1) = [1] x1 + [7] [cons](x1, x2) = [1] x2 + [0] [n__first](x1, x2) = [1] x1 + [1] x2 + [7] [activate](x1) = [1] x1 + [7] [from](x1) = [1] x1 + [7] [n__from](x1) = [1] x1 + [7] [n__s](x1) = [1] x1 + [7] The following symbols are considered usable {first, s, activate, from} The order satisfies the following ordering constraints: [first(X1, X2)] = [1] X1 + [1] X2 + [1] ? [1] X1 + [1] X2 + [7] = [n__first(X1, X2)] [first(0(), X)] = [1] X + [8] > [7] = [nil()] [s(X)] = [1] X + [7] >= [1] X + [7] = [n__s(X)] [activate(X)] = [1] X + [7] > [1] X + [0] = [X] [activate(n__first(X1, X2))] = [1] X1 + [1] X2 + [14] ? [1] X1 + [1] X2 + [15] = [first(activate(X1), activate(X2))] [activate(n__from(X))] = [1] X + [14] >= [1] X + [14] = [from(activate(X))] [activate(n__s(X))] = [1] X + [14] >= [1] X + [14] = [s(activate(X))] [from(X)] = [1] X + [7] ? [1] X + [14] = [cons(X, n__from(n__s(X)))] [from(X)] = [1] X + [7] >= [1] X + [7] = [n__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: { first(X1, X2) -> n__first(X1, X2) , s(X) -> n__s(X) , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) , activate(n__from(X)) -> from(activate(X)) , activate(n__s(X)) -> s(activate(X)) , from(X) -> cons(X, n__from(n__s(X))) , from(X) -> n__from(X) } Weak Trs: { first(0(), X) -> nil() , activate(X) -> 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(first) = {1, 2}, Uargs(s) = {1}, Uargs(from) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [first](x1, x2) = [1] x1 + [1] x2 + [1] [0] = [7] [nil] = [0] [s](x1) = [1] x1 + [0] [cons](x1, x2) = [1] x1 + [7] [n__first](x1, x2) = [1] x1 + [1] x2 + [0] [activate](x1) = [1] x1 + [0] [from](x1) = [1] x1 + [0] [n__from](x1) = [1] x1 + [0] [n__s](x1) = [1] x1 + [0] The following symbols are considered usable {first, s, activate, from} The order satisfies the following ordering constraints: [first(X1, X2)] = [1] X1 + [1] X2 + [1] > [1] X1 + [1] X2 + [0] = [n__first(X1, X2)] [first(0(), X)] = [1] X + [8] > [0] = [nil()] [s(X)] = [1] X + [0] >= [1] X + [0] = [n__s(X)] [activate(X)] = [1] X + [0] >= [1] X + [0] = [X] [activate(n__first(X1, X2))] = [1] X1 + [1] X2 + [0] ? [1] X1 + [1] X2 + [1] = [first(activate(X1), activate(X2))] [activate(n__from(X))] = [1] X + [0] >= [1] X + [0] = [from(activate(X))] [activate(n__s(X))] = [1] X + [0] >= [1] X + [0] = [s(activate(X))] [from(X)] = [1] X + [0] ? [1] X + [7] = [cons(X, n__from(n__s(X)))] [from(X)] = [1] X + [0] >= [1] X + [0] = [n__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: { s(X) -> n__s(X) , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) , activate(n__from(X)) -> from(activate(X)) , activate(n__s(X)) -> s(activate(X)) , from(X) -> cons(X, n__from(n__s(X))) , from(X) -> n__from(X) } Weak Trs: { first(X1, X2) -> n__first(X1, X2) , first(0(), X) -> nil() , activate(X) -> 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(first) = {1, 2}, Uargs(s) = {1}, Uargs(from) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [first](x1, x2) = [1] x1 + [1] x2 + [4] [0] = [3] [nil] = [7] [s](x1) = [1] x1 + [1] [cons](x1, x2) = [1] x1 + [7] [n__first](x1, x2) = [1] x1 + [1] x2 + [1] [activate](x1) = [1] x1 + [0] [from](x1) = [1] x1 + [0] [n__from](x1) = [1] x1 + [0] [n__s](x1) = [1] x1 + [0] The following symbols are considered usable {first, s, activate, from} The order satisfies the following ordering constraints: [first(X1, X2)] = [1] X1 + [1] X2 + [4] > [1] X1 + [1] X2 + [1] = [n__first(X1, X2)] [first(0(), X)] = [1] X + [7] >= [7] = [nil()] [s(X)] = [1] X + [1] > [1] X + [0] = [n__s(X)] [activate(X)] = [1] X + [0] >= [1] X + [0] = [X] [activate(n__first(X1, X2))] = [1] X1 + [1] X2 + [1] ? [1] X1 + [1] X2 + [4] = [first(activate(X1), activate(X2))] [activate(n__from(X))] = [1] X + [0] >= [1] X + [0] = [from(activate(X))] [activate(n__s(X))] = [1] X + [0] ? [1] X + [1] = [s(activate(X))] [from(X)] = [1] X + [0] ? [1] X + [7] = [cons(X, n__from(n__s(X)))] [from(X)] = [1] X + [0] >= [1] X + [0] = [n__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: { activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) , activate(n__from(X)) -> from(activate(X)) , activate(n__s(X)) -> s(activate(X)) , from(X) -> cons(X, n__from(n__s(X))) , from(X) -> n__from(X) } Weak Trs: { first(X1, X2) -> n__first(X1, X2) , first(0(), X) -> nil() , s(X) -> n__s(X) , activate(X) -> 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(first) = {1, 2}, Uargs(s) = {1}, Uargs(from) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [first](x1, x2) = [1] x1 + [1] x2 + [0] [0] = [7] [nil] = [7] [s](x1) = [1] x1 + [0] [cons](x1, x2) = [1] x1 + [7] [n__first](x1, x2) = [1] x1 + [1] x2 + [0] [activate](x1) = [1] x1 + [0] [from](x1) = [1] x1 + [0] [n__from](x1) = [1] x1 + [1] [n__s](x1) = [1] x1 + [0] The following symbols are considered usable {first, s, activate, from} The order satisfies the following ordering constraints: [first(X1, X2)] = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = [n__first(X1, X2)] [first(0(), X)] = [1] X + [7] >= [7] = [nil()] [s(X)] = [1] X + [0] >= [1] X + [0] = [n__s(X)] [activate(X)] = [1] X + [0] >= [1] X + [0] = [X] [activate(n__first(X1, X2))] = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = [first(activate(X1), activate(X2))] [activate(n__from(X))] = [1] X + [1] > [1] X + [0] = [from(activate(X))] [activate(n__s(X))] = [1] X + [0] >= [1] X + [0] = [s(activate(X))] [from(X)] = [1] X + [0] ? [1] X + [7] = [cons(X, n__from(n__s(X)))] [from(X)] = [1] X + [0] ? [1] X + [1] = [n__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: { activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) , activate(n__s(X)) -> s(activate(X)) , from(X) -> cons(X, n__from(n__s(X))) , from(X) -> n__from(X) } Weak Trs: { first(X1, X2) -> n__first(X1, X2) , first(0(), X) -> nil() , s(X) -> n__s(X) , activate(X) -> X , activate(n__from(X)) -> from(activate(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(first) = {1, 2}, Uargs(s) = {1}, Uargs(from) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [first](x1, x2) = [1] x1 + [1] x2 + [0] [0] = [7] [nil] = [7] [s](x1) = [1] x1 + [0] [cons](x1, x2) = [1] x1 + [0] [n__first](x1, x2) = [1] x1 + [1] x2 + [0] [activate](x1) = [1] x1 + [0] [from](x1) = [1] x1 + [1] [n__from](x1) = [1] x1 + [1] [n__s](x1) = [1] x1 + [0] The following symbols are considered usable {first, s, activate, from} The order satisfies the following ordering constraints: [first(X1, X2)] = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = [n__first(X1, X2)] [first(0(), X)] = [1] X + [7] >= [7] = [nil()] [s(X)] = [1] X + [0] >= [1] X + [0] = [n__s(X)] [activate(X)] = [1] X + [0] >= [1] X + [0] = [X] [activate(n__first(X1, X2))] = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = [first(activate(X1), activate(X2))] [activate(n__from(X))] = [1] X + [1] >= [1] X + [1] = [from(activate(X))] [activate(n__s(X))] = [1] X + [0] >= [1] X + [0] = [s(activate(X))] [from(X)] = [1] X + [1] > [1] X + [0] = [cons(X, n__from(n__s(X)))] [from(X)] = [1] X + [1] >= [1] X + [1] = [n__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: { activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) , activate(n__s(X)) -> s(activate(X)) , from(X) -> n__from(X) } Weak Trs: { first(X1, X2) -> n__first(X1, X2) , first(0(), X) -> nil() , s(X) -> n__s(X) , activate(X) -> X , activate(n__from(X)) -> from(activate(X)) , from(X) -> cons(X, n__from(n__s(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: { activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) , activate(n__s(X)) -> s(activate(X)) } 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(first) = {1, 2}, Uargs(s) = {1}, Uargs(from) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [first](x1, x2) = [1] x1 + [1] x2 + [4] [0] = [1] [nil] = [3] [s](x1) = [1] x1 + [4] [cons](x1, x2) = [1] x1 + [2] [n__first](x1, x2) = [1] x1 + [1] x2 + [4] [activate](x1) = [2] x1 + [0] [from](x1) = [1] x1 + [2] [n__from](x1) = [1] x1 + [2] [n__s](x1) = [1] x1 + [4] The following symbols are considered usable {first, s, activate, from} The order satisfies the following ordering constraints: [first(X1, X2)] = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [4] = [n__first(X1, X2)] [first(0(), X)] = [1] X + [5] > [3] = [nil()] [s(X)] = [1] X + [4] >= [1] X + [4] = [n__s(X)] [activate(X)] = [2] X + [0] >= [1] X + [0] = [X] [activate(n__first(X1, X2))] = [2] X1 + [2] X2 + [8] > [2] X1 + [2] X2 + [4] = [first(activate(X1), activate(X2))] [activate(n__from(X))] = [2] X + [4] > [2] X + [2] = [from(activate(X))] [activate(n__s(X))] = [2] X + [8] > [2] X + [4] = [s(activate(X))] [from(X)] = [1] X + [2] >= [1] X + [2] = [cons(X, n__from(n__s(X)))] [from(X)] = [1] X + [2] >= [1] X + [2] = [n__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: { from(X) -> n__from(X) } Weak Trs: { first(X1, X2) -> n__first(X1, X2) , first(0(), X) -> nil() , s(X) -> n__s(X) , activate(X) -> X , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) , activate(n__from(X)) -> from(activate(X)) , activate(n__s(X)) -> s(activate(X)) , from(X) -> cons(X, n__from(n__s(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: { from(X) -> n__from(X) } 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(first) = {1, 2}, Uargs(s) = {1}, Uargs(from) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [first](x1, x2) = [1] x1 + [1] x2 + [0] [0] = [1] [nil] = [1] [s](x1) = [1] x1 + [0] [cons](x1, x2) = [1] x1 + [4] [n__first](x1, x2) = [1] x1 + [1] x2 + [0] [activate](x1) = [4] x1 + [0] [from](x1) = [1] x1 + [4] [n__from](x1) = [1] x1 + [1] [n__s](x1) = [1] x1 + [0] The following symbols are considered usable {first, s, activate, from} The order satisfies the following ordering constraints: [first(X1, X2)] = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = [n__first(X1, X2)] [first(0(), X)] = [1] X + [1] >= [1] = [nil()] [s(X)] = [1] X + [0] >= [1] X + [0] = [n__s(X)] [activate(X)] = [4] X + [0] >= [1] X + [0] = [X] [activate(n__first(X1, X2))] = [4] X1 + [4] X2 + [0] >= [4] X1 + [4] X2 + [0] = [first(activate(X1), activate(X2))] [activate(n__from(X))] = [4] X + [4] >= [4] X + [4] = [from(activate(X))] [activate(n__s(X))] = [4] X + [0] >= [4] X + [0] = [s(activate(X))] [from(X)] = [1] X + [4] >= [1] X + [4] = [cons(X, n__from(n__s(X)))] [from(X)] = [1] X + [4] > [1] X + [1] = [n__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: { first(X1, X2) -> n__first(X1, X2) , first(0(), X) -> nil() , s(X) -> n__s(X) , activate(X) -> X , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) , activate(n__from(X)) -> from(activate(X)) , activate(n__s(X)) -> s(activate(X)) , from(X) -> cons(X, n__from(n__s(X))) , from(X) -> n__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))