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: { min(X, 0()) -> X , min(s(X), s(Y)) -> min(X, Y) , quot(0(), s(Y)) -> 0() , quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) , log(s(0())) -> 0() , log(s(s(X))) -> s(log(s(quot(X, s(s(0())))))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We add the following innermost weak dependency pairs: Strict DPs: { min^#(X, 0()) -> c_1() , min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) , quot^#(0(), s(Y)) -> c_3() , quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y))) , log^#(s(0())) -> c_5() , log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) } 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: { min^#(X, 0()) -> c_1() , min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) , quot^#(0(), s(Y)) -> c_3() , quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y))) , log^#(s(0())) -> c_5() , log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) } Strict Trs: { min(X, 0()) -> X , min(s(X), s(Y)) -> min(X, Y) , quot(0(), s(Y)) -> 0() , quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) , log(s(0())) -> 0() , log(s(s(X))) -> s(log(s(quot(X, s(s(0())))))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We replace rewrite rules by usable rules: Strict Usable Rules: { min(X, 0()) -> X , min(s(X), s(Y)) -> min(X, Y) , quot(0(), s(Y)) -> 0() , quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { min^#(X, 0()) -> c_1() , min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) , quot^#(0(), s(Y)) -> c_3() , quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y))) , log^#(s(0())) -> c_5() , log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) } Strict Trs: { min(X, 0()) -> X , min(s(X), s(Y)) -> min(X, Y) , quot(0(), s(Y)) -> 0() , quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following constant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(quot) = {1}, Uargs(c_2) = {1}, Uargs(quot^#) = {1}, Uargs(c_4) = {1}, Uargs(log^#) = {1}, Uargs(c_6) = {1} TcT has computed the following constructor-restricted matrix interpretation. [min](x1, x2) = [1 0] x1 + [1] [0 1] [0] [0] = [0] [2] [s](x1) = [1 2] x1 + [2] [0 1] [1] [quot](x1, x2) = [1 2] x1 + [0] [0 1] [0] [min^#](x1, x2) = [0 0] x1 + [0 0] x2 + [2] [1 1] [1 1] [1] [c_1] = [1] [1] [c_2](x1) = [1 0] x1 + [1] [0 1] [1] [quot^#](x1, x2) = [2 0] x1 + [0] [0 0] [0] [c_3] = [1] [1] [c_4](x1) = [1 0] x1 + [2] [0 1] [2] [log^#](x1) = [1 0] x1 + [0] [0 0] [0] [c_5] = [1] [1] [c_6](x1) = [1 0] x1 + [2] [0 1] [2] The following symbols are considered usable {min, quot, min^#, quot^#, log^#} The order satisfies the following ordering constraints: [min(X, 0())] = [1 0] X + [1] [0 1] [0] > [1 0] X + [0] [0 1] [0] = [X] [min(s(X), s(Y))] = [1 2] X + [3] [0 1] [1] > [1 0] X + [1] [0 1] [0] = [min(X, Y)] [quot(0(), s(Y))] = [4] [2] > [0] [2] = [0()] [quot(s(X), s(Y))] = [1 4] X + [4] [0 1] [1] > [1 4] X + [3] [0 1] [1] = [s(quot(min(X, Y), s(Y)))] [min^#(X, 0())] = [0 0] X + [2] [1 1] [3] > [1] [1] = [c_1()] [min^#(s(X), s(Y))] = [0 0] X + [0 0] Y + [2] [1 3] [1 3] [7] ? [0 0] X + [0 0] Y + [3] [1 1] [1 1] [2] = [c_2(min^#(X, Y))] [quot^#(0(), s(Y))] = [0] [0] ? [1] [1] = [c_3()] [quot^#(s(X), s(Y))] = [2 4] X + [4] [0 0] [0] ? [2 0] X + [4] [0 0] [2] = [c_4(quot^#(min(X, Y), s(Y)))] [log^#(s(0()))] = [6] [0] ? [1] [1] = [c_5()] [log^#(s(s(X)))] = [1 4] X + [6] [0 0] [0] ? [1 4] X + [4] [0 0] [2] = [c_6(log^#(s(quot(X, s(s(0()))))))] 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: { min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) , quot^#(0(), s(Y)) -> c_3() , quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y))) , log^#(s(0())) -> c_5() , log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) } Weak DPs: { min^#(X, 0()) -> c_1() } Weak Trs: { min(X, 0()) -> X , min(s(X), s(Y)) -> min(X, Y) , quot(0(), s(Y)) -> 0() , quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We estimate the number of application of {2,4} by applications of Pre({2,4}) = {3,5}. Here rules are labeled as follows: DPs: { 1: min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) , 2: quot^#(0(), s(Y)) -> c_3() , 3: quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y))) , 4: log^#(s(0())) -> c_5() , 5: log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) , 6: min^#(X, 0()) -> c_1() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) , quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y))) , log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) } Weak DPs: { min^#(X, 0()) -> c_1() , quot^#(0(), s(Y)) -> c_3() , log^#(s(0())) -> c_5() } Weak Trs: { min(X, 0()) -> X , min(s(X), s(Y)) -> min(X, Y) , quot(0(), s(Y)) -> 0() , quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) } Obligation: innermost 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. { min^#(X, 0()) -> c_1() , quot^#(0(), s(Y)) -> c_3() , log^#(s(0())) -> c_5() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) , quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y))) , log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) } Weak Trs: { min(X, 0()) -> X , min(s(X), s(Y)) -> min(X, Y) , quot(0(), s(Y)) -> 0() , quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'Small Polynomial Path Order (PS,1-bounded)' to orient following rules strictly. DPs: { 1: min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) , 2: quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y))) , 3: log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) } Trs: { min(s(X), s(Y)) -> min(X, Y) } Sub-proof: ---------- The input was oriented with the instance of 'Small Polynomial Path Order (PS,1-bounded)' as induced by the safe mapping safe(min) = {}, safe(0) = {}, safe(s) = {1}, safe(quot) = {}, safe(log) = {}, safe(min^#) = {2}, safe(c_1) = {}, safe(c_2) = {}, safe(quot^#) = {2}, safe(c_3) = {}, safe(c_4) = {}, safe(log^#) = {}, safe(c_5) = {}, safe(c_6) = {} and precedence quot > min^#, quot ~ quot^#, quot ~ log^#, quot^# ~ log^# . Following symbols are considered recursive: {min^#, quot^#, log^#} The recursion depth is 1. Further, following argument filtering is employed: pi(min) = 1, pi(0) = [], pi(s) = [1], pi(quot) = 1, pi(log) = [], pi(min^#) = [1], pi(c_1) = [], pi(c_2) = [1], pi(quot^#) = [1], pi(c_3) = [], pi(c_4) = [1], pi(log^#) = [1], pi(c_5) = [], pi(c_6) = [1] Usable defined function symbols are a subset of: {min, quot, min^#, quot^#, log^#} For your convenience, here are the satisfied ordering constraints: pi(min^#(s(X), s(Y))) = min^#(s(; X);) > c_2(min^#(X;);) = pi(c_2(min^#(X, Y))) pi(quot^#(s(X), s(Y))) = quot^#(s(; X);) > c_4(quot^#(X;);) = pi(c_4(quot^#(min(X, Y), s(Y)))) pi(log^#(s(s(X)))) = log^#(s(; s(; X));) > c_6(log^#(s(; X););) = pi(c_6(log^#(s(quot(X, s(s(0()))))))) pi(min(X, 0())) = X >= X = pi(X) pi(min(s(X), s(Y))) = s(; X) > X = pi(min(X, Y)) pi(quot(0(), s(Y))) = 0() >= 0() = pi(0()) pi(quot(s(X), s(Y))) = s(; X) >= s(; X) = pi(s(quot(min(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(1)). Weak DPs: { min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) , quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y))) , log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) } Weak Trs: { min(X, 0()) -> X , min(s(X), s(Y)) -> min(X, Y) , quot(0(), s(Y)) -> 0() , quot(s(X), s(Y)) -> s(quot(min(X, Y), s(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. { min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) , quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y))) , log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak Trs: { min(X, 0()) -> X , min(s(X), s(Y)) -> min(X, Y) , quot(0(), s(Y)) -> 0() , quot(s(X), s(Y)) -> s(quot(min(X, Y), s(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))