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: { circ(s, id()) -> s , circ(circ(s, t), u) -> circ(s, circ(t, u)) , circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) , circ(cons(lift(), s), circ(cons(lift(), t), u)) -> circ(cons(lift(), circ(s, t)), u) , circ(cons(lift(), s), cons(a, t)) -> cons(a, circ(s, t)) , circ(cons(lift(), s), cons(lift(), t)) -> cons(lift(), circ(s, t)) , circ(id(), s) -> s , msubst(a, id()) -> a , msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) , subst(a, id()) -> a } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) Arguments of following rules are not normal-forms: { circ(cons(lift(), s), circ(cons(lift(), t), u)) -> circ(cons(lift(), circ(s, t)), u) } All above mentioned rules can be savely removed. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { circ(s, id()) -> s , circ(circ(s, t), u) -> circ(s, circ(t, u)) , circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) , circ(cons(lift(), s), cons(a, t)) -> cons(a, circ(s, t)) , circ(cons(lift(), s), cons(lift(), t)) -> cons(lift(), circ(s, t)) , circ(id(), s) -> s , msubst(a, id()) -> a , msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) , subst(a, id()) -> a } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We add the following dependency tuples: Strict DPs: { circ^#(s, id()) -> c_1() , circ^#(circ(s, t), u) -> c_2(circ^#(s, circ(t, u)), circ^#(t, u)) , circ^#(cons(a, s), t) -> c_3(msubst^#(a, t), circ^#(s, t)) , circ^#(cons(lift(), s), cons(a, t)) -> c_4(circ^#(s, t)) , circ^#(cons(lift(), s), cons(lift(), t)) -> c_5(circ^#(s, t)) , circ^#(id(), s) -> c_6() , msubst^#(a, id()) -> c_7() , msubst^#(msubst(a, s), t) -> c_8(msubst^#(a, circ(s, t)), circ^#(s, t)) , subst^#(a, id()) -> c_9() } 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: { circ^#(s, id()) -> c_1() , circ^#(circ(s, t), u) -> c_2(circ^#(s, circ(t, u)), circ^#(t, u)) , circ^#(cons(a, s), t) -> c_3(msubst^#(a, t), circ^#(s, t)) , circ^#(cons(lift(), s), cons(a, t)) -> c_4(circ^#(s, t)) , circ^#(cons(lift(), s), cons(lift(), t)) -> c_5(circ^#(s, t)) , circ^#(id(), s) -> c_6() , msubst^#(a, id()) -> c_7() , msubst^#(msubst(a, s), t) -> c_8(msubst^#(a, circ(s, t)), circ^#(s, t)) , subst^#(a, id()) -> c_9() } Weak Trs: { circ(s, id()) -> s , circ(circ(s, t), u) -> circ(s, circ(t, u)) , circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) , circ(cons(lift(), s), cons(a, t)) -> cons(a, circ(s, t)) , circ(cons(lift(), s), cons(lift(), t)) -> cons(lift(), circ(s, t)) , circ(id(), s) -> s , msubst(a, id()) -> a , msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) , subst(a, id()) -> a } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We estimate the number of application of {1,6,7,9} by applications of Pre({1,6,7,9}) = {2,3,4,5,8}. Here rules are labeled as follows: DPs: { 1: circ^#(s, id()) -> c_1() , 2: circ^#(circ(s, t), u) -> c_2(circ^#(s, circ(t, u)), circ^#(t, u)) , 3: circ^#(cons(a, s), t) -> c_3(msubst^#(a, t), circ^#(s, t)) , 4: circ^#(cons(lift(), s), cons(a, t)) -> c_4(circ^#(s, t)) , 5: circ^#(cons(lift(), s), cons(lift(), t)) -> c_5(circ^#(s, t)) , 6: circ^#(id(), s) -> c_6() , 7: msubst^#(a, id()) -> c_7() , 8: msubst^#(msubst(a, s), t) -> c_8(msubst^#(a, circ(s, t)), circ^#(s, t)) , 9: subst^#(a, id()) -> c_9() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { circ^#(circ(s, t), u) -> c_2(circ^#(s, circ(t, u)), circ^#(t, u)) , circ^#(cons(a, s), t) -> c_3(msubst^#(a, t), circ^#(s, t)) , circ^#(cons(lift(), s), cons(a, t)) -> c_4(circ^#(s, t)) , circ^#(cons(lift(), s), cons(lift(), t)) -> c_5(circ^#(s, t)) , msubst^#(msubst(a, s), t) -> c_8(msubst^#(a, circ(s, t)), circ^#(s, t)) } Weak DPs: { circ^#(s, id()) -> c_1() , circ^#(id(), s) -> c_6() , msubst^#(a, id()) -> c_7() , subst^#(a, id()) -> c_9() } Weak Trs: { circ(s, id()) -> s , circ(circ(s, t), u) -> circ(s, circ(t, u)) , circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) , circ(cons(lift(), s), cons(a, t)) -> cons(a, circ(s, t)) , circ(cons(lift(), s), cons(lift(), t)) -> cons(lift(), circ(s, t)) , circ(id(), s) -> s , msubst(a, id()) -> a , msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) , subst(a, id()) -> a } 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. { circ^#(s, id()) -> c_1() , circ^#(id(), s) -> c_6() , msubst^#(a, id()) -> c_7() , subst^#(a, id()) -> c_9() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { circ^#(circ(s, t), u) -> c_2(circ^#(s, circ(t, u)), circ^#(t, u)) , circ^#(cons(a, s), t) -> c_3(msubst^#(a, t), circ^#(s, t)) , circ^#(cons(lift(), s), cons(a, t)) -> c_4(circ^#(s, t)) , circ^#(cons(lift(), s), cons(lift(), t)) -> c_5(circ^#(s, t)) , msubst^#(msubst(a, s), t) -> c_8(msubst^#(a, circ(s, t)), circ^#(s, t)) } Weak Trs: { circ(s, id()) -> s , circ(circ(s, t), u) -> circ(s, circ(t, u)) , circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) , circ(cons(lift(), s), cons(a, t)) -> cons(a, circ(s, t)) , circ(cons(lift(), s), cons(lift(), t)) -> cons(lift(), circ(s, t)) , circ(id(), s) -> s , msubst(a, id()) -> a , msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) , subst(a, id()) -> a } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) Due to missing edges in the dependency-graph, the right-hand sides of following rules could be simplified: { circ^#(circ(s, t), u) -> c_2(circ^#(s, circ(t, u)), circ^#(t, u)) , msubst^#(msubst(a, s), t) -> c_8(msubst^#(a, circ(s, t)), circ^#(s, t)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { circ^#(circ(s, t), u) -> c_1(circ^#(t, u)) , circ^#(cons(a, s), t) -> c_2(msubst^#(a, t), circ^#(s, t)) , circ^#(cons(lift(), s), cons(a, t)) -> c_3(circ^#(s, t)) , circ^#(cons(lift(), s), cons(lift(), t)) -> c_4(circ^#(s, t)) , msubst^#(msubst(a, s), t) -> c_5(circ^#(s, t)) } Weak Trs: { circ(s, id()) -> s , circ(circ(s, t), u) -> circ(s, circ(t, u)) , circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) , circ(cons(lift(), s), cons(a, t)) -> cons(a, circ(s, t)) , circ(cons(lift(), s), cons(lift(), t)) -> cons(lift(), circ(s, t)) , circ(id(), s) -> s , msubst(a, id()) -> a , msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) , subst(a, id()) -> a } Obligation: innermost 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: { circ^#(circ(s, t), u) -> c_1(circ^#(t, u)) , circ^#(cons(a, s), t) -> c_2(msubst^#(a, t), circ^#(s, t)) , circ^#(cons(lift(), s), cons(a, t)) -> c_3(circ^#(s, t)) , circ^#(cons(lift(), s), cons(lift(), t)) -> c_4(circ^#(s, t)) , msubst^#(msubst(a, s), t) -> c_5(circ^#(s, t)) } Obligation: innermost 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: circ^#(circ(s, t), u) -> c_1(circ^#(t, u)) , 2: circ^#(cons(a, s), t) -> c_2(msubst^#(a, t), circ^#(s, t)) , 3: circ^#(cons(lift(), s), cons(a, t)) -> c_3(circ^#(s, t)) , 4: circ^#(cons(lift(), s), cons(lift(), t)) -> c_4(circ^#(s, t)) , 5: msubst^#(msubst(a, s), t) -> c_5(circ^#(s, t)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_2) = {1, 2}, Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_5) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [circ](x1, x2) = [7] x1 + [2] x2 + [2] [cons](x1, x2) = [1] x1 + [1] x2 + [2] [msubst](x1, x2) = [7] x1 + [4] x2 + [4] [lift] = [0] [id] = [0] [subst](x1, x2) = [7] x1 + [7] x2 + [0] [circ^#](x1, x2) = [4] x1 + [0] [c_1] = [0] [c_2](x1, x2) = [7] x1 + [7] x2 + [0] [c_3](x1, x2) = [7] x1 + [7] x2 + [0] [msubst^#](x1, x2) = [2] x1 + [0] [c_4](x1) = [7] x1 + [0] [c_5](x1) = [7] x1 + [0] [c_6] = [0] [c_7] = [0] [c_8](x1, x2) = [7] x1 + [7] x2 + [0] [subst^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_9] = [0] [c] = [0] [c_1](x1) = [1] x1 + [5] [c_2](x1, x2) = [2] x1 + [1] x2 + [7] [c_3](x1) = [1] x1 + [7] [c_4](x1) = [1] x1 + [7] [c_5](x1) = [1] x1 + [3] The following symbols are considered usable {circ^#, msubst^#} The order satisfies the following ordering constraints: [circ^#(circ(s, t), u)] = [28] s + [8] t + [8] > [4] t + [5] = [c_1(circ^#(t, u))] [circ^#(cons(a, s), t)] = [4] a + [4] s + [8] > [4] a + [4] s + [7] = [c_2(msubst^#(a, t), circ^#(s, t))] [circ^#(cons(lift(), s), cons(a, t))] = [4] s + [8] > [4] s + [7] = [c_3(circ^#(s, t))] [circ^#(cons(lift(), s), cons(lift(), t))] = [4] s + [8] > [4] s + [7] = [c_4(circ^#(s, t))] [msubst^#(msubst(a, s), t)] = [14] a + [8] s + [8] > [4] s + [3] = [c_5(circ^#(s, t))] 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: { circ^#(circ(s, t), u) -> c_1(circ^#(t, u)) , circ^#(cons(a, s), t) -> c_2(msubst^#(a, t), circ^#(s, t)) , circ^#(cons(lift(), s), cons(a, t)) -> c_3(circ^#(s, t)) , circ^#(cons(lift(), s), cons(lift(), t)) -> c_4(circ^#(s, t)) , msubst^#(msubst(a, s), t) -> c_5(circ^#(s, t)) } 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. { circ^#(circ(s, t), u) -> c_1(circ^#(t, u)) , circ^#(cons(a, s), t) -> c_2(msubst^#(a, t), circ^#(s, t)) , circ^#(cons(lift(), s), cons(a, t)) -> c_3(circ^#(s, t)) , circ^#(cons(lift(), s), cons(lift(), t)) -> c_4(circ^#(s, t)) , msubst^#(msubst(a, s), t) -> c_5(circ^#(s, t)) } 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^1))