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: runtime complexity Answer: YES(O(1),O(n^2)) We add the following weak dependency pairs: Strict DPs: { min^#(X, 0()) -> c_1(X) , 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(X) , 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: 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(X) , 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: 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(min) = {1}, 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] [0] [s](x1) = [1 2] x1 + [1] [0 1] [1] [quot](x1, x2) = [1 2] x1 + [1] [0 1] [0] [min^#](x1, x2) = [0 0] x1 + [0 0] x2 + [2] [1 2] [2 2] [0] [c_1](x1) = [0 0] x1 + [1] [1 1] [1] [c_2](x1) = [1 0] x1 + [1] [0 1] [2] [quot^#](x1, x2) = [1 0] x1 + [0] [0 0] [0] [c_3] = [1] [1] [c_4](x1) = [1 0] x1 + [1] [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 + [2] [0 1] [1] > [1 0] X + [1] [0 1] [0] = [min(X, Y)] [quot(0(), s(Y))] = [1] [0] > [0] [0] = [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 2] [0] ? [0 0] X + [1] [1 1] [1] = [c_1(X)] [min^#(s(X), s(Y))] = [0 0] X + [0 0] Y + [2] [1 4] [2 6] [7] ? [0 0] X + [0 0] Y + [3] [1 2] [2 2] [2] = [c_2(min^#(X, Y))] [quot^#(0(), s(Y))] = [0] [0] ? [1] [1] = [c_3()] [quot^#(s(X), s(Y))] = [1 2] X + [1] [0 0] [0] ? [1 0] X + [2] [0 0] [2] = [c_4(quot^#(min(X, Y), s(Y)))] [log^#(s(0()))] = [1] [0] ? [1] [1] = [c_5()] [log^#(s(s(X)))] = [1 4] X + [4] [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(n^1)). Strict DPs: { min^#(X, 0()) -> c_1(X) , 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 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: runtime complexity Answer: YES(?,O(n^1)) We estimate the number of application of {3,5} by applications of Pre({3,5}) = {1,4,6}. Here rules are labeled as follows: DPs: { 1: min^#(X, 0()) -> c_1(X) , 2: min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) , 3: quot^#(0(), s(Y)) -> c_3() , 4: quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y))) , 5: log^#(s(0())) -> c_5() , 6: 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(n^1)). Strict DPs: { min^#(X, 0()) -> c_1(X) , 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: { 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: runtime complexity Answer: YES(?,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() , log^#(s(0())) -> c_5() } We are left with following problem, upon which TcT provides the certificate YES(?,O(n^1)). Strict DPs: { min^#(X, 0()) -> c_1(X) , 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: runtime complexity Answer: YES(?,O(n^1)) We employ 'linear path analysis' using the following approximated dependency graph: ->{1,2} [ ? ] | |->{3} [ YES(O(1),O(n^1)) ] | `->{4} [ YES(O(1),O(n^1)) ] Here dependency-pairs are as follows: Strict DPs: { 1: min^#(X, 0()) -> c_1(X) , 2: min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) , 3: quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y))) , 4: log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) } * Path {1,2}->{3}: 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 DPs: { min^#(X, 0()) -> c_1(X) , min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) , quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y))) } 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: runtime complexity Answer: YES(O(1),O(n^1)) We replace rewrite rules by usable rules: Weak Usable Rules: { min(X, 0()) -> X , min(s(X), s(Y)) -> min(X, Y) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { min^#(X, 0()) -> c_1(X) , min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) , quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y))) } Weak Trs: { min(X, 0()) -> X , min(s(X), s(Y)) -> min(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: { 3: quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y))) } Trs: { min(s(X), s(Y)) -> min(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). [min](x1, x2) = [1] x1 + [0] [0] = [0] [s](x1) = [1] x1 + [1] [quot](x1, x2) = [7] x1 + [7] x2 + [0] [log](x1) = [7] x1 + [0] [min^#](x1, x2) = [0] [c_1](x1) = [0] [c_2](x1) = [4] x1 + [0] [quot^#](x1, x2) = [3] x1 + [1] x2 + [4] [c_3] = [0] [c_4](x1) = [1] x1 + [1] [log^#](x1) = [7] x1 + [0] [c_5] = [0] [c_6](x1) = [7] x1 + [0] The following symbols are considered usable {min, min^#, quot^#} The order satisfies the following ordering constraints: [min(X, 0())] = [1] X + [0] >= [1] X + [0] = [X] [min(s(X), s(Y))] = [1] X + [1] > [1] X + [0] = [min(X, Y)] [min^#(X, 0())] = [0] >= [0] = [c_1(X)] [min^#(s(X), s(Y))] = [0] >= [0] = [c_2(min^#(X, Y))] [quot^#(s(X), s(Y))] = [3] X + [1] Y + [8] > [3] X + [1] Y + [6] = [c_4(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(n^1)). Strict DPs: { min^#(X, 0()) -> c_1(X) , min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) } Weak DPs: { quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y))) } Weak Trs: { min(X, 0()) -> X , min(s(X), s(Y)) -> min(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^#(min(X, Y), s(Y))) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { min^#(X, 0()) -> c_1(X) , min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) } Weak Trs: { min(X, 0()) -> X , min(s(X), s(Y)) -> min(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: { min^#(X, 0()) -> c_1(X) , min^#(s(X), s(Y)) -> c_2(min^#(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: { 1: min^#(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). [min](x1, x2) = [7] x1 + [7] x2 + [0] [0] = [1] [s](x1) = [1] x1 + [0] [quot](x1, x2) = [7] x1 + [7] x2 + [0] [log](x1) = [7] x1 + [0] [min^#](x1, x2) = [2] x2 + [0] [c_1](x1) = [0] [c_2](x1) = [1] x1 + [0] [quot^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_3] = [0] [c_4](x1) = [7] x1 + [0] [log^#](x1) = [7] x1 + [0] [c_5] = [0] [c_6](x1) = [7] x1 + [0] The following symbols are considered usable {min^#} The order satisfies the following ordering constraints: [min^#(X, 0())] = [2] > [0] = [c_1(X)] [min^#(s(X), s(Y))] = [2] Y + [0] >= [2] Y + [0] = [c_2(min^#(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^1)). Strict DPs: { min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) } Weak DPs: { min^#(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: min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_2) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [min](x1, x2) = [7] x1 + [7] x2 + [0] [0] = [7] [s](x1) = [1] x1 + [4] [quot](x1, x2) = [7] x1 + [7] x2 + [0] [log](x1) = [7] x1 + [0] [min^#](x1, x2) = [2] x1 + [0] [c_1](x1) = [1] x1 + [0] [c_2](x1) = [1] x1 + [5] [quot^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_3] = [0] [c_4](x1) = [7] x1 + [0] [log^#](x1) = [7] x1 + [0] [c_5] = [0] [c_6](x1) = [7] x1 + [0] The following symbols are considered usable {min^#} The order satisfies the following ordering constraints: [min^#(X, 0())] = [2] X + [0] >= [1] X + [0] = [c_1(X)] [min^#(s(X), s(Y))] = [2] X + [8] > [2] X + [5] = [c_2(min^#(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: { min^#(X, 0()) -> c_1(X) , min^#(s(X), s(Y)) -> c_2(min^#(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. { min^#(X, 0()) -> c_1(X) , min^#(s(X), s(Y)) -> c_2(min^#(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 * Path {1,2}->{4}: 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 DPs: { min^#(X, 0()) -> c_1(X) , min^#(s(X), s(Y)) -> c_2(min^#(X, 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: runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 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 following argument positions are usable: Uargs(c_2) = {1}, Uargs(c_6) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [min](x1, x2) = [1] x1 + [0] [0] = [0] [s](x1) = [1] x1 + [1] [quot](x1, x2) = [1] x1 + [0] [log](x1) = [7] x1 + [0] [min^#](x1, x2) = [0] [c_1](x1) = [0] [c_2](x1) = [1] x1 + [0] [quot^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_3] = [0] [c_4](x1) = [7] x1 + [0] [log^#](x1) = [4] x1 + [0] [c_5] = [0] [c_6](x1) = [1] x1 + [1] The following symbols are considered usable {min, quot, min^#, log^#} The order satisfies the following ordering constraints: [min(X, 0())] = [1] X + [0] >= [1] X + [0] = [X] [min(s(X), s(Y))] = [1] X + [1] > [1] X + [0] = [min(X, Y)] [quot(0(), s(Y))] = [0] >= [0] = [0()] [quot(s(X), s(Y))] = [1] X + [1] >= [1] X + [1] = [s(quot(min(X, Y), s(Y)))] [min^#(X, 0())] = [0] >= [0] = [c_1(X)] [min^#(s(X), s(Y))] = [0] >= [0] = [c_2(min^#(X, Y))] [log^#(s(s(X)))] = [4] X + [8] > [4] X + [5] = [c_6(log^#(s(quot(X, s(s(0()))))))] 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: { min^#(X, 0()) -> c_1(X) , min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) } Weak DPs: { 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: 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. { 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(n^1)). Strict DPs: { min^#(X, 0()) -> c_1(X) , min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) } 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: 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: { min^#(X, 0()) -> c_1(X) , min^#(s(X), s(Y)) -> c_2(min^#(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: { 1: min^#(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). [min](x1, x2) = [7] x1 + [7] x2 + [0] [0] = [1] [s](x1) = [1] x1 + [0] [quot](x1, x2) = [7] x1 + [7] x2 + [0] [log](x1) = [7] x1 + [0] [min^#](x1, x2) = [2] x2 + [0] [c_1](x1) = [0] [c_2](x1) = [1] x1 + [0] [quot^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_3] = [0] [c_4](x1) = [7] x1 + [0] [log^#](x1) = [7] x1 + [0] [c_5] = [0] [c_6](x1) = [7] x1 + [0] The following symbols are considered usable {min^#} The order satisfies the following ordering constraints: [min^#(X, 0())] = [2] > [0] = [c_1(X)] [min^#(s(X), s(Y))] = [2] Y + [0] >= [2] Y + [0] = [c_2(min^#(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^1)). Strict DPs: { min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) } Weak DPs: { min^#(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: min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_2) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [min](x1, x2) = [7] x1 + [7] x2 + [0] [0] = [7] [s](x1) = [1] x1 + [4] [quot](x1, x2) = [7] x1 + [7] x2 + [0] [log](x1) = [7] x1 + [0] [min^#](x1, x2) = [2] x1 + [0] [c_1](x1) = [1] x1 + [0] [c_2](x1) = [1] x1 + [5] [quot^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_3] = [0] [c_4](x1) = [7] x1 + [0] [log^#](x1) = [7] x1 + [0] [c_5] = [0] [c_6](x1) = [7] x1 + [0] The following symbols are considered usable {min^#} The order satisfies the following ordering constraints: [min^#(X, 0())] = [2] X + [0] >= [1] X + [0] = [c_1(X)] [min^#(s(X), s(Y))] = [2] X + [8] > [2] X + [5] = [c_2(min^#(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: { min^#(X, 0()) -> c_1(X) , min^#(s(X), s(Y)) -> c_2(min^#(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. { min^#(X, 0()) -> c_1(X) , min^#(s(X), s(Y)) -> c_2(min^#(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^2))