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(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))) , plus(minus(x, s(0())), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0()))) , plus(0(), y) -> y , plus(s(x), y) -> s(plus(x, y)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We add the following dependency tuples: Strict DPs: { minus^#(x, 0()) -> c_1() , 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)), minus^#(x, y)) , plus^#(minus(x, s(0())), minus(y, s(s(z)))) -> c_5(plus^#(minus(y, s(s(z))), minus(x, s(0()))), minus^#(y, s(s(z))), minus^#(x, s(0()))) , plus^#(0(), y) -> c_6() , plus^#(s(x), y) -> c_7(plus^#(x, 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^2)). Strict DPs: { minus^#(x, 0()) -> c_1() , 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)), minus^#(x, y)) , plus^#(minus(x, s(0())), minus(y, s(s(z)))) -> c_5(plus^#(minus(y, s(s(z))), minus(x, s(0()))), minus^#(y, s(s(z))), minus^#(x, s(0()))) , plus^#(0(), y) -> c_6() , plus^#(s(x), y) -> c_7(plus^#(x, y)) } Weak 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))) , plus(minus(x, s(0())), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0()))) , plus(0(), y) -> y , plus(s(x), y) -> s(plus(x, y)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We estimate the number of application of {1,3,6} by applications of Pre({1,3,6}) = {2,4,5,7}. Here rules are labeled as follows: DPs: { 1: minus^#(x, 0()) -> c_1() , 2: minus^#(s(x), s(y)) -> c_2(minus^#(x, y)) , 3: quot^#(0(), s(y)) -> c_3() , 4: quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y)), minus^#(x, y)) , 5: plus^#(minus(x, s(0())), minus(y, s(s(z)))) -> c_5(plus^#(minus(y, s(s(z))), minus(x, s(0()))), minus^#(y, s(s(z))), minus^#(x, s(0()))) , 6: plus^#(0(), y) -> c_6() , 7: plus^#(s(x), y) -> c_7(plus^#(x, y)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). 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)), minus^#(x, y)) , plus^#(minus(x, s(0())), minus(y, s(s(z)))) -> c_5(plus^#(minus(y, s(s(z))), minus(x, s(0()))), minus^#(y, s(s(z))), minus^#(x, s(0()))) , plus^#(s(x), y) -> c_7(plus^#(x, y)) } Weak DPs: { minus^#(x, 0()) -> c_1() , quot^#(0(), s(y)) -> c_3() , plus^#(0(), y) -> c_6() } Weak 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))) , plus(minus(x, s(0())), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0()))) , plus(0(), y) -> y , plus(s(x), y) -> s(plus(x, y)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) 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() , quot^#(0(), s(y)) -> c_3() , plus^#(0(), y) -> c_6() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). 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)), minus^#(x, y)) , plus^#(minus(x, s(0())), minus(y, s(s(z)))) -> c_5(plus^#(minus(y, s(s(z))), minus(x, s(0()))), minus^#(y, s(s(z))), minus^#(x, s(0()))) , plus^#(s(x), y) -> c_7(plus^#(x, y)) } Weak 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))) , plus(minus(x, s(0())), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0()))) , plus(0(), y) -> y , plus(s(x), y) -> s(plus(x, y)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) Due to missing edges in the dependency-graph, the right-hand sides of following rules could be simplified: { plus^#(minus(x, s(0())), minus(y, s(s(z)))) -> c_5(plus^#(minus(y, s(s(z))), minus(x, s(0()))), minus^#(y, s(s(z))), minus^#(x, s(0()))) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { minus^#(s(x), s(y)) -> c_1(minus^#(x, y)) , quot^#(s(x), s(y)) -> c_2(quot^#(minus(x, y), s(y)), minus^#(x, y)) , plus^#(minus(x, s(0())), minus(y, s(s(z)))) -> c_3(plus^#(minus(y, s(s(z))), minus(x, s(0())))) , plus^#(s(x), y) -> c_4(plus^#(x, y)) } Weak 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))) , plus(minus(x, s(0())), minus(y, s(s(z)))) -> plus(minus(y, s(s(z))), minus(x, s(0()))) , plus(0(), y) -> y , plus(s(x), y) -> s(plus(x, y)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We replace rewrite rules by usable rules: Weak 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^2)). Strict DPs: { minus^#(s(x), s(y)) -> c_1(minus^#(x, y)) , quot^#(s(x), s(y)) -> c_2(quot^#(minus(x, y), s(y)), minus^#(x, y)) , plus^#(minus(x, s(0())), minus(y, s(s(z)))) -> c_3(plus^#(minus(y, s(s(z))), minus(x, s(0())))) , plus^#(s(x), y) -> c_4(plus^#(x, y)) } Weak Trs: { minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 3: plus^#(minus(x, s(0())), minus(y, s(s(z)))) -> c_3(plus^#(minus(y, s(s(z))), minus(x, s(0())))) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_2) = {1, 2}, Uargs(c_4) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [minus](x1, x2) = [4] x1 + [0] [0] = [0] [s](x1) = [1] x1 + [0] [quot](x1, x2) = [7] x1 + [7] x2 + [0] [plus](x1, x2) = [7] x1 + [7] x2 + [0] [minus^#](x1, x2) = [0] [c_1] = [0] [c_2](x1) = [7] x1 + [0] [quot^#](x1, x2) = [0] [c_3] = [0] [c_4](x1, x2) = [7] x1 + [7] x2 + [0] [plus^#](x1, x2) = [4] [c_5](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_6] = [0] [c_7](x1) = [7] x1 + [0] [c] = [0] [c_1](x1) = [4] x1 + [0] [c_2](x1, x2) = [4] x1 + [4] x2 + [0] [c_3](x1) = [1] [c_4](x1) = [1] x1 + [0] The following symbols are considered usable {minus, minus^#, quot^#, plus^#} The order satisfies the following ordering constraints: [minus(x, 0())] = [4] x + [0] >= [1] x + [0] = [x] [minus(s(x), s(y))] = [4] x + [0] >= [4] x + [0] = [minus(x, y)] [minus^#(s(x), s(y))] = [0] >= [0] = [c_1(minus^#(x, y))] [quot^#(s(x), s(y))] = [0] >= [0] = [c_2(quot^#(minus(x, y), s(y)), minus^#(x, y))] [plus^#(minus(x, s(0())), minus(y, s(s(z))))] = [4] > [1] = [c_3(plus^#(minus(y, s(s(z))), minus(x, s(0()))))] [plus^#(s(x), y)] = [4] >= [4] = [c_4(plus^#(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(n^2)). Strict DPs: { minus^#(s(x), s(y)) -> c_1(minus^#(x, y)) , quot^#(s(x), s(y)) -> c_2(quot^#(minus(x, y), s(y)), minus^#(x, y)) , plus^#(s(x), y) -> c_4(plus^#(x, y)) } Weak DPs: { plus^#(minus(x, s(0())), minus(y, s(s(z)))) -> c_3(plus^#(minus(y, s(s(z))), minus(x, s(0())))) } Weak Trs: { minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 3: plus^#(s(x), y) -> c_4(plus^#(x, y)) } Trs: { minus(s(x), s(y)) -> minus(x, y) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_2) = {1, 2}, Uargs(c_4) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [minus](x1, x2) = [2] x1 + [0] [0] = [0] [s](x1) = [1] x1 + [4] [quot](x1, x2) = [7] x1 + [7] x2 + [0] [plus](x1, x2) = [7] x1 + [7] x2 + [0] [minus^#](x1, x2) = [0] [c_1] = [0] [c_2](x1) = [7] x1 + [0] [quot^#](x1, x2) = [0] [c_3] = [0] [c_4](x1, x2) = [7] x1 + [7] x2 + [0] [plus^#](x1, x2) = [2] x1 + [4] x2 + [0] [c_5](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_6] = [0] [c_7](x1) = [7] x1 + [0] [c] = [0] [c_1](x1) = [1] x1 + [0] [c_2](x1, x2) = [4] x1 + [4] x2 + [0] [c_3](x1) = [0] [c_4](x1) = [1] x1 + [1] The following symbols are considered usable {minus, minus^#, quot^#, plus^#} The order satisfies the following ordering constraints: [minus(x, 0())] = [2] x + [0] >= [1] x + [0] = [x] [minus(s(x), s(y))] = [2] x + [8] > [2] x + [0] = [minus(x, y)] [minus^#(s(x), s(y))] = [0] >= [0] = [c_1(minus^#(x, y))] [quot^#(s(x), s(y))] = [0] >= [0] = [c_2(quot^#(minus(x, y), s(y)), minus^#(x, y))] [plus^#(minus(x, s(0())), minus(y, s(s(z))))] = [4] x + [8] y + [0] >= [0] = [c_3(plus^#(minus(y, s(s(z))), minus(x, s(0()))))] [plus^#(s(x), y)] = [2] x + [4] y + [8] > [2] x + [4] y + [1] = [c_4(plus^#(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(n^2)). Strict DPs: { minus^#(s(x), s(y)) -> c_1(minus^#(x, y)) , quot^#(s(x), s(y)) -> c_2(quot^#(minus(x, y), s(y)), minus^#(x, y)) } Weak DPs: { plus^#(minus(x, s(0())), minus(y, s(s(z)))) -> c_3(plus^#(minus(y, s(s(z))), minus(x, s(0())))) , plus^#(s(x), y) -> c_4(plus^#(x, y)) } Weak Trs: { minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { plus^#(minus(x, s(0())), minus(y, s(s(z)))) -> c_3(plus^#(minus(y, s(s(z))), minus(x, s(0())))) , plus^#(s(x), y) -> c_4(plus^#(x, y)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { minus^#(s(x), s(y)) -> c_1(minus^#(x, y)) , quot^#(s(x), s(y)) -> c_2(quot^#(minus(x, y), s(y)), minus^#(x, y)) } Weak Trs: { minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We use the processor 'Small Polynomial Path Order (PS,2-bounded)' to orient following rules strictly. DPs: { 1: minus^#(s(x), s(y)) -> c_1(minus^#(x, y)) , 2: quot^#(s(x), s(y)) -> c_2(quot^#(minus(x, y), s(y)), minus^#(x, y)) } Trs: { minus(s(x), s(y)) -> minus(x, y) } Sub-proof: ---------- The input was oriented with the instance of 'Small Polynomial Path Order (PS,2-bounded)' as induced by the safe mapping safe(minus) = {}, safe(0) = {}, safe(s) = {1}, safe(quot) = {}, safe(plus) = {}, safe(minus^#) = {}, safe(c_1) = {}, safe(c_2) = {}, safe(quot^#) = {}, safe(c_3) = {}, safe(c_4) = {}, safe(plus^#) = {}, safe(c_5) = {}, safe(c_6) = {}, safe(c_7) = {}, safe(c) = {}, safe(c_1) = {}, safe(c_2) = {}, safe(c_3) = {}, safe(c_4) = {} and precedence minus > minus^#, quot^# > minus^#, minus ~ quot^# . Following symbols are considered recursive: {minus^#, quot^#} The recursion depth is 2. Further, following argument filtering is employed: pi(minus) = 1, pi(0) = [], pi(s) = [1], pi(quot) = [], pi(plus) = [], pi(minus^#) = [1, 2], pi(c_1) = [], pi(c_2) = [], pi(quot^#) = [1, 2], pi(c_3) = [], pi(c_4) = [], pi(plus^#) = [], pi(c_5) = [], pi(c_6) = [], pi(c_7) = [], pi(c) = [], pi(c_1) = [1], pi(c_2) = [1, 2], pi(c_3) = [], pi(c_4) = [] Usable defined function symbols are a subset of: {minus, minus^#, quot^#, plus^#} For your convenience, here are the satisfied ordering constraints: pi(minus^#(s(x), s(y))) = minus^#(s(; x), s(; y);) > c_1(minus^#(x, y;);) = pi(c_1(minus^#(x, y))) pi(quot^#(s(x), s(y))) = quot^#(s(; x), s(; y);) > c_2(quot^#(x, s(; y);), minus^#(x, y;);) = pi(c_2(quot^#(minus(x, y), s(y)), minus^#(x, y))) pi(minus(x, 0())) = x >= x = pi(x) pi(minus(s(x), s(y))) = s(; x) > x = pi(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^#(s(x), s(y)) -> c_1(minus^#(x, y)) , quot^#(s(x), s(y)) -> c_2(quot^#(minus(x, y), s(y)), minus^#(x, y)) } Weak Trs: { minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) } Obligation: innermost 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^#(s(x), s(y)) -> c_1(minus^#(x, y)) , quot^#(s(x), s(y)) -> c_2(quot^#(minus(x, y), s(y)), minus^#(x, y)) } 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)) -> minus(x, y) } Obligation: innermost 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: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Hurray, we answered YES(O(1),O(n^2))