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: { minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) , quot(0(), s(y)) -> 0() , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) We add the following weak dependency pairs: Strict DPs: { minus^#(x, 0()) -> c_1(x) , minus^#(s(x), s(y)) -> c_2(minus^#(x, y)) , quot^#(0(), s(y)) -> c_3() , quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y))) } 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: { minus^#(x, 0()) -> c_1(x) , minus^#(s(x), s(y)) -> c_2(minus^#(x, y)) , quot^#(0(), s(y)) -> c_3() , quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y))) } Strict Trs: { minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) , quot(0(), s(y)) -> 0() , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) We replace rewrite rules by usable rules: Strict Usable Rules: { minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { minus^#(x, 0()) -> c_1(x) , minus^#(s(x), s(y)) -> c_2(minus^#(x, y)) , quot^#(0(), s(y)) -> c_3() , quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y))) } Strict Trs: { minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) } 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(c_2) = {1}, Uargs(quot^#) = {1}, Uargs(c_4) = {1} TcT has computed the following constructor-restricted matrix interpretation. [minus](x1, x2) = [1 1] x1 + [2] [1 1] [0] [0] = [0] [0] [s](x1) = [1 1] x1 + [0] [0 0] [2] [minus^#](x1, x2) = [0 0] x1 + [0 0] x2 + [2] [1 1] [1 1] [2] [c_1](x1) = [0 0] x1 + [1] [1 1] [1] [c_2](x1) = [1 0] x1 + [1] [0 1] [1] [quot^#](x1, x2) = [1 0] x1 + [1] [0 0] [0] [c_3] = [0] [0] [c_4](x1) = [1 0] x1 + [1] [0 1] [2] The following symbols are considered usable {minus, minus^#, quot^#} The order satisfies the following ordering constraints: [minus(x, 0())] = [1 1] x + [2] [1 1] [0] > [1 0] x + [0] [0 1] [0] = [x] [minus(s(x), s(y))] = [1 1] x + [4] [1 1] [2] > [1 1] x + [2] [1 1] [0] = [minus(x, y)] [minus^#(x, 0())] = [0 0] x + [2] [1 1] [2] > [0 0] x + [1] [1 1] [1] = [c_1(x)] [minus^#(s(x), s(y))] = [0 0] x + [0 0] y + [2] [1 1] [1 1] [6] ? [0 0] x + [0 0] y + [3] [1 1] [1 1] [3] = [c_2(minus^#(x, y))] [quot^#(0(), s(y))] = [1] [0] > [0] [0] = [c_3()] [quot^#(s(x), s(y))] = [1 1] x + [1] [0 0] [0] ? [1 1] x + [4] [0 0] [2] = [c_4(quot^#(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 DPs: { minus^#(s(x), s(y)) -> c_2(minus^#(x, y)) , quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y))) } Weak DPs: { minus^#(x, 0()) -> c_1(x) , quot^#(0(), s(y)) -> c_3() } Weak Trs: { minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) } 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. { quot^#(0(), s(y)) -> c_3() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { minus^#(s(x), s(y)) -> c_2(minus^#(x, y)) , quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y))) } Weak DPs: { minus^#(x, 0()) -> c_1(x) } Weak Trs: { minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) } 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: { 2: quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y))) } Trs: { minus(s(x), s(y)) -> minus(x, y) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_2) = {1}, Uargs(c_4) = {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 + [2] [quot](x1, x2) = [7] x1 + [7] x2 + [0] [minus^#](x1, x2) = [0] [c_1](x1) = [0] [c_2](x1) = [2] x1 + [0] [quot^#](x1, x2) = [1] x1 + [1] x2 + [0] [c_3] = [0] [c_4](x1) = [1] x1 + [0] The following symbols are considered usable {minus, minus^#, quot^#} The order satisfies the following ordering constraints: [minus(x, 0())] = [1] x + [0] >= [1] x + [0] = [x] [minus(s(x), s(y))] = [1] x + [2] > [1] x + [0] = [minus(x, y)] [minus^#(x, 0())] = [0] >= [0] = [c_1(x)] [minus^#(s(x), s(y))] = [0] >= [0] = [c_2(minus^#(x, y))] [quot^#(s(x), s(y))] = [1] x + [1] y + [4] > [1] x + [1] y + [2] = [c_4(quot^#(minus(x, y), s(y)))] 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: { minus^#(s(x), s(y)) -> c_2(minus^#(x, y)) } Weak DPs: { minus^#(x, 0()) -> c_1(x) , quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y))) } Weak Trs: { minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) } 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. { quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y))) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { minus^#(s(x), s(y)) -> c_2(minus^#(x, y)) } Weak DPs: { minus^#(x, 0()) -> c_1(x) } Weak Trs: { minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) } Obligation: runtime complexity Answer: YES(O(1),O(n^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(n^1)). Strict DPs: { minus^#(s(x), s(y)) -> c_2(minus^#(x, y)) } Weak DPs: { minus^#(x, 0()) -> c_1(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: minus^#(s(x), s(y)) -> c_2(minus^#(x, y)) , 2: minus^#(x, 0()) -> c_1(x) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_2) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [minus](x1, x2) = [7] x1 + [7] x2 + [0] [0] = [4] [s](x1) = [1] x1 + [4] [quot](x1, x2) = [7] x1 + [7] x2 + [0] [minus^#](x1, x2) = [2] x2 + [0] [c_1](x1) = [6] [c_2](x1) = [1] x1 + [1] [quot^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_3] = [0] [c_4](x1) = [7] x1 + [0] The following symbols are considered usable {minus^#} The order satisfies the following ordering constraints: [minus^#(x, 0())] = [8] > [6] = [c_1(x)] [minus^#(s(x), s(y))] = [2] y + [8] > [2] y + [1] = [c_2(minus^#(x, y))] 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: { minus^#(x, 0()) -> c_1(x) , minus^#(s(x), s(y)) -> c_2(minus^#(x, y)) } 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. { minus^#(x, 0()) -> c_1(x) , minus^#(s(x), s(y)) -> c_2(minus^#(x, y)) } 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))