We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { minus(X, 0()) -> X , minus(s(X), s(Y)) -> p(minus(X, Y)) , p(s(X)) -> X , div(0(), s(Y)) -> 0() , div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) } 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(s) = {1}, Uargs(p) = {1}, Uargs(div) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [minus](x1, x2) = [1] x1 + [0] [0] = [5] [s](x1) = [1] x1 + [1] [p](x1) = [1] x1 + [0] [div](x1, x2) = [1] x1 + [0] The order satisfies the following ordering constraints: [minus(X, 0())] = [1] X + [0] >= [1] X + [0] = [X] [minus(s(X), s(Y))] = [1] X + [1] > [1] X + [0] = [p(minus(X, Y))] [p(s(X))] = [1] X + [1] > [1] X + [0] = [X] [div(0(), s(Y))] = [5] >= [5] = [0()] [div(s(X), s(Y))] = [1] X + [1] >= [1] X + [1] = [s(div(minus(X, Y), s(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^1)). Strict Trs: { minus(X, 0()) -> X , div(0(), s(Y)) -> 0() , div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) } Weak Trs: { minus(s(X), s(Y)) -> p(minus(X, Y)) , p(s(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(s) = {1}, Uargs(p) = {1}, Uargs(div) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [minus](x1, x2) = [1] x1 + [1] [0] = [4] [s](x1) = [1] x1 + [0] [p](x1) = [1] x1 + [0] [div](x1, x2) = [1] x1 + [0] The order satisfies the following ordering constraints: [minus(X, 0())] = [1] X + [1] > [1] X + [0] = [X] [minus(s(X), s(Y))] = [1] X + [1] >= [1] X + [1] = [p(minus(X, Y))] [p(s(X))] = [1] X + [0] >= [1] X + [0] = [X] [div(0(), s(Y))] = [4] >= [4] = [0()] [div(s(X), s(Y))] = [1] X + [0] ? [1] X + [1] = [s(div(minus(X, Y), s(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^1)). Strict Trs: { div(0(), s(Y)) -> 0() , div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) } Weak Trs: { minus(X, 0()) -> X , minus(s(X), s(Y)) -> p(minus(X, Y)) , p(s(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(s) = {1}, Uargs(p) = {1}, Uargs(div) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [minus](x1, x2) = [1] x1 + [4] [0] = [0] [s](x1) = [1] x1 + [4] [p](x1) = [1] x1 + [4] [div](x1, x2) = [1] x1 + [1] x2 + [0] The order satisfies the following ordering constraints: [minus(X, 0())] = [1] X + [4] > [1] X + [0] = [X] [minus(s(X), s(Y))] = [1] X + [8] >= [1] X + [8] = [p(minus(X, Y))] [p(s(X))] = [1] X + [8] > [1] X + [0] = [X] [div(0(), s(Y))] = [1] Y + [4] > [0] = [0()] [div(s(X), s(Y))] = [1] X + [1] Y + [8] ? [1] X + [1] Y + [12] = [s(div(minus(X, Y), s(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^1)). Strict Trs: { div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) } Weak Trs: { minus(X, 0()) -> X , minus(s(X), s(Y)) -> p(minus(X, Y)) , p(s(X)) -> X , div(0(), s(Y)) -> 0() } 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: { div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) } 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(s) = {1}, Uargs(p) = {1}, Uargs(div) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [minus](x1, x2) = [1] x1 + [0] [0] = [0] [s](x1) = [1] x1 + [4] [p](x1) = [1] x1 + [0] [div](x1, x2) = [2] x1 + [0] The order satisfies the following ordering constraints: [minus(X, 0())] = [1] X + [0] >= [1] X + [0] = [X] [minus(s(X), s(Y))] = [1] X + [4] > [1] X + [0] = [p(minus(X, Y))] [p(s(X))] = [1] X + [4] > [1] X + [0] = [X] [div(0(), s(Y))] = [0] >= [0] = [0()] [div(s(X), s(Y))] = [2] X + [8] > [2] X + [4] = [s(div(minus(X, Y), s(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(X, 0()) -> X , minus(s(X), s(Y)) -> p(minus(X, Y)) , p(s(X)) -> X , div(0(), s(Y)) -> 0() , div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Hurray, we answered YES(O(1),O(n^1))