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: { -(x, 0()) -> x , -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0()) , -(0(), y) -> 0() , p(0()) -> 0() , p(s(x)) -> x } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) We add the following weak dependency pairs: Strict DPs: { -^#(x, 0()) -> c_1(x) , -^#(x, s(y)) -> c_2(x, y, -^#(x, p(s(y)))) , -^#(0(), y) -> c_3() , p^#(0()) -> c_4() , p^#(s(x)) -> c_5(x) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { -^#(x, 0()) -> c_1(x) , -^#(x, s(y)) -> c_2(x, y, -^#(x, p(s(y)))) , -^#(0(), y) -> c_3() , p^#(0()) -> c_4() , p^#(s(x)) -> c_5(x) } Strict Trs: { -(x, 0()) -> x , -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0()) , -(0(), y) -> 0() , p(0()) -> 0() , p(s(x)) -> x } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) We replace rewrite rules by usable rules: Strict Usable Rules: { p(0()) -> 0() , p(s(x)) -> x } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { -^#(x, 0()) -> c_1(x) , -^#(x, s(y)) -> c_2(x, y, -^#(x, p(s(y)))) , -^#(0(), y) -> c_3() , p^#(0()) -> c_4() , p^#(s(x)) -> c_5(x) } Strict Trs: { p(0()) -> 0() , p(s(x)) -> x } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following constant growth matrix-interpretation) The following argument positions are usable: Uargs(-^#) = {2}, Uargs(c_2) = {3} TcT has computed the following constructor-restricted matrix interpretation. [0] = [0] [2] [s](x1) = [1 2] x1 + [1] [0 0] [0] [p](x1) = [1 2] x1 + [0] [2 2] [0] [-^#](x1, x2) = [1 1] x1 + [2 0] x2 + [0] [2 1] [0 0] [0] [c_1](x1) = [1 1] x1 + [1] [1 1] [0] [c_2](x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [2] [1 1] [2 2] [0 1] [2] [c_3] = [1] [1] [p^#](x1) = [1 1] x1 + [1] [2 1] [2] [c_4] = [2] [0] [c_5](x1) = [1 1] x1 + [1] [1 0] [0] The following symbols are considered usable {p, -^#, p^#} The order satisfies the following ordering constraints: [p(0())] = [4] [4] > [0] [2] = [0()] [p(s(x))] = [1 2] x + [1] [2 4] [2] > [1 0] x + [0] [0 1] [0] = [x] [-^#(x, 0())] = [1 1] x + [0] [2 1] [0] ? [1 1] x + [1] [1 1] [0] = [c_1(x)] [-^#(x, s(y))] = [2 4] y + [1 1] x + [2] [0 0] [2 1] [0] ? [2 4] y + [1 1] x + [4] [2 2] [3 2] [2] = [c_2(x, y, -^#(x, p(s(y))))] [-^#(0(), y)] = [2 0] y + [2] [0 0] [2] > [1] [1] = [c_3()] [p^#(0())] = [3] [4] > [2] [0] = [c_4()] [p^#(s(x))] = [1 2] x + [2] [2 4] [4] > [1 1] x + [1] [1 0] [0] = [c_5(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 DPs: { -^#(x, 0()) -> c_1(x) , -^#(x, s(y)) -> c_2(x, y, -^#(x, p(s(y)))) } Weak DPs: { -^#(0(), y) -> c_3() , p^#(0()) -> c_4() , p^#(s(x)) -> c_5(x) } Weak Trs: { p(0()) -> 0() , p(s(x)) -> x } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { -^#(0(), y) -> c_3() , p^#(0()) -> c_4() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { -^#(x, 0()) -> c_1(x) , -^#(x, s(y)) -> c_2(x, y, -^#(x, p(s(y)))) } Weak DPs: { p^#(s(x)) -> c_5(x) } Weak Trs: { p(0()) -> 0() , p(s(x)) -> x } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: -^#(x, 0()) -> c_1(x) , 3: p^#(s(x)) -> c_5(x) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_2) = {3} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [-](x1, x2) = [7] x1 + [7] x2 + [0] [0] = [0] [s](x1) = [1] x1 + [0] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] [greater](x1, x2) = [1] x1 + [1] x2 + [0] [p](x1) = [1] x1 + [0] [-^#](x1, x2) = [7] x1 + [2] [c_1](x1) = [7] x1 + [0] [c_2](x1, x2, x3) = [1] x3 + [0] [c_3] = [0] [p^#](x1) = [7] x1 + [7] [c_4] = [0] [c_5](x1) = [7] x1 + [6] The following symbols are considered usable {p, -^#, p^#} The order satisfies the following ordering constraints: [p(0())] = [0] >= [0] = [0()] [p(s(x))] = [1] x + [0] >= [1] x + [0] = [x] [-^#(x, 0())] = [7] x + [2] > [7] x + [0] = [c_1(x)] [-^#(x, s(y))] = [7] x + [2] >= [7] x + [2] = [c_2(x, y, -^#(x, p(s(y))))] [p^#(s(x))] = [7] x + [7] > [7] x + [6] = [c_5(x)] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { -^#(x, s(y)) -> c_2(x, y, -^#(x, p(s(y)))) } Weak DPs: { -^#(x, 0()) -> c_1(x) , p^#(s(x)) -> c_5(x) } Weak Trs: { p(0()) -> 0() , p(s(x)) -> x } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 3' to orient following rules strictly. DPs: { 1: -^#(x, s(y)) -> c_2(x, y, -^#(x, p(s(y)))) , 3: p^#(s(x)) -> c_5(x) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_2) = {3} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA) and not(IDA(1)). [7 7 0] [7 7 0] [0] [-](x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] [0] [0] = [0] [0] [1 0 0] [1] [s](x1) = [1 0 0] x1 + [0] [0 1 1] [0] [0] [if](x1, x2, x3) = [0] [0] [0] [greater](x1, x2) = [0] [0] [0 1 0] [0] [p](x1) = [0 0 1] x1 + [0] [0 0 4] [0] [4 0 0] [0] [-^#](x1, x2) = [0 0 0] x2 + [1] [0 0 0] [4] [0] [c_1](x1) = [1] [4] [1 1 0] [1] [c_2](x1, x2, x3) = [0 0 0] x3 + [1] [0 0 0] [3] [0] [c_3] = [0] [0] [7] [p^#](x1) = [7] [7] [0] [c_4] = [0] [0] [6] [c_5](x1) = [7] [7] The following symbols are considered usable {p, -^#, p^#} The order satisfies the following ordering constraints: [p(0())] = [0] [0] [0] >= [0] [0] [0] = [0()] [p(s(x))] = [1 0 0] [0] [0 1 1] x + [0] [0 4 4] [0] >= [1 0 0] [0] [0 1 0] x + [0] [0 0 1] [0] = [x] [-^#(x, 0())] = [0] [1] [4] >= [0] [1] [4] = [c_1(x)] [-^#(x, s(y))] = [4 0 0] [4] [0 0 0] y + [1] [0 0 0] [4] > [4 0 0] [2] [0 0 0] y + [1] [0 0 0] [3] = [c_2(x, y, -^#(x, p(s(y))))] [p^#(s(x))] = [7] [7] [7] > [6] [7] [7] = [c_5(x)] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { -^#(x, 0()) -> c_1(x) , -^#(x, s(y)) -> c_2(x, y, -^#(x, p(s(y)))) , p^#(s(x)) -> c_5(x) } Weak Trs: { p(0()) -> 0() , p(s(x)) -> x } Obligation: runtime complexity Answer: YES(O(1),O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { -^#(x, 0()) -> c_1(x) , -^#(x, s(y)) -> c_2(x, y, -^#(x, p(s(y)))) , p^#(s(x)) -> c_5(x) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak Trs: { p(0()) -> 0() , p(s(x)) -> x } Obligation: runtime complexity Answer: YES(O(1),O(1)) No rule is usable, rules are removed from the input problem. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Hurray, we answered YES(O(1),O(n^1))