YES(O(1),O(n^1)) 12.97/3.92 YES(O(1),O(n^1)) 12.97/3.92 12.97/3.92 We are left with following problem, upon which TcT provides the 12.97/3.92 certificate YES(O(1),O(n^1)). 12.97/3.92 12.97/3.92 Strict Trs: 12.97/3.92 { circ(s, id()) -> s 12.97/3.92 , circ(circ(s, t), u) -> circ(s, circ(t, u)) 12.97/3.92 , circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) 12.97/3.92 , circ(cons(lift(), s), circ(cons(lift(), t), u)) -> 12.97/3.92 circ(cons(lift(), circ(s, t)), u) 12.97/3.92 , circ(cons(lift(), s), cons(a, t)) -> cons(a, circ(s, t)) 12.97/3.92 , circ(cons(lift(), s), cons(lift(), t)) -> 12.97/3.92 cons(lift(), circ(s, t)) 12.97/3.92 , circ(id(), s) -> s 12.97/3.92 , msubst(a, id()) -> a 12.97/3.92 , msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) 12.97/3.92 , subst(a, id()) -> a } 12.97/3.92 Obligation: 12.97/3.92 innermost runtime complexity 12.97/3.92 Answer: 12.97/3.92 YES(O(1),O(n^1)) 12.97/3.92 12.97/3.92 Arguments of following rules are not normal-forms: 12.97/3.92 12.97/3.92 { circ(cons(lift(), s), circ(cons(lift(), t), u)) -> 12.97/3.92 circ(cons(lift(), circ(s, t)), u) } 12.97/3.92 12.97/3.92 All above mentioned rules can be savely removed. 12.97/3.92 12.97/3.92 We are left with following problem, upon which TcT provides the 12.97/3.92 certificate YES(O(1),O(n^1)). 12.97/3.92 12.97/3.92 Strict Trs: 12.97/3.92 { circ(s, id()) -> s 12.97/3.92 , circ(circ(s, t), u) -> circ(s, circ(t, u)) 12.97/3.92 , circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) 12.97/3.92 , circ(cons(lift(), s), cons(a, t)) -> cons(a, circ(s, t)) 12.97/3.92 , circ(cons(lift(), s), cons(lift(), t)) -> 12.97/3.92 cons(lift(), circ(s, t)) 12.97/3.92 , circ(id(), s) -> s 12.97/3.92 , msubst(a, id()) -> a 12.97/3.92 , msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) 12.97/3.92 , subst(a, id()) -> a } 12.97/3.92 Obligation: 12.97/3.92 innermost runtime complexity 12.97/3.92 Answer: 12.97/3.92 YES(O(1),O(n^1)) 12.97/3.92 12.97/3.92 We add the following dependency tuples: 12.97/3.92 12.97/3.92 Strict DPs: 12.97/3.92 { circ^#(s, id()) -> c_1() 12.97/3.92 , circ^#(circ(s, t), u) -> c_2(circ^#(s, circ(t, u)), circ^#(t, u)) 12.97/3.92 , circ^#(cons(a, s), t) -> c_3(msubst^#(a, t), circ^#(s, t)) 12.97/3.92 , circ^#(cons(lift(), s), cons(a, t)) -> c_4(circ^#(s, t)) 12.97/3.92 , circ^#(cons(lift(), s), cons(lift(), t)) -> c_5(circ^#(s, t)) 12.97/3.92 , circ^#(id(), s) -> c_6() 12.97/3.92 , msubst^#(a, id()) -> c_7() 12.97/3.92 , msubst^#(msubst(a, s), t) -> 12.97/3.92 c_8(msubst^#(a, circ(s, t)), circ^#(s, t)) 12.97/3.92 , subst^#(a, id()) -> c_9() } 12.97/3.92 12.97/3.92 and mark the set of starting terms. 12.97/3.92 12.97/3.92 We are left with following problem, upon which TcT provides the 12.97/3.92 certificate YES(O(1),O(n^1)). 12.97/3.92 12.97/3.92 Strict DPs: 12.97/3.92 { circ^#(s, id()) -> c_1() 12.97/3.92 , circ^#(circ(s, t), u) -> c_2(circ^#(s, circ(t, u)), circ^#(t, u)) 12.97/3.92 , circ^#(cons(a, s), t) -> c_3(msubst^#(a, t), circ^#(s, t)) 12.97/3.92 , circ^#(cons(lift(), s), cons(a, t)) -> c_4(circ^#(s, t)) 12.97/3.92 , circ^#(cons(lift(), s), cons(lift(), t)) -> c_5(circ^#(s, t)) 12.97/3.92 , circ^#(id(), s) -> c_6() 12.97/3.92 , msubst^#(a, id()) -> c_7() 12.97/3.92 , msubst^#(msubst(a, s), t) -> 12.97/3.92 c_8(msubst^#(a, circ(s, t)), circ^#(s, t)) 12.97/3.92 , subst^#(a, id()) -> c_9() } 12.97/3.92 Weak Trs: 12.97/3.92 { circ(s, id()) -> s 12.97/3.92 , circ(circ(s, t), u) -> circ(s, circ(t, u)) 12.97/3.92 , circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) 12.97/3.92 , circ(cons(lift(), s), cons(a, t)) -> cons(a, circ(s, t)) 12.97/3.92 , circ(cons(lift(), s), cons(lift(), t)) -> 12.97/3.92 cons(lift(), circ(s, t)) 12.97/3.92 , circ(id(), s) -> s 12.97/3.92 , msubst(a, id()) -> a 12.97/3.92 , msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) 12.97/3.92 , subst(a, id()) -> a } 12.97/3.92 Obligation: 12.97/3.92 innermost runtime complexity 12.97/3.92 Answer: 12.97/3.92 YES(O(1),O(n^1)) 12.97/3.92 12.97/3.92 We estimate the number of application of {1,6,7,9} by applications 12.97/3.92 of Pre({1,6,7,9}) = {2,3,4,5,8}. Here rules are labeled as follows: 12.97/3.92 12.97/3.92 DPs: 12.97/3.92 { 1: circ^#(s, id()) -> c_1() 12.97/3.92 , 2: circ^#(circ(s, t), u) -> 12.97/3.92 c_2(circ^#(s, circ(t, u)), circ^#(t, u)) 12.97/3.92 , 3: circ^#(cons(a, s), t) -> c_3(msubst^#(a, t), circ^#(s, t)) 12.97/3.92 , 4: circ^#(cons(lift(), s), cons(a, t)) -> c_4(circ^#(s, t)) 12.97/3.92 , 5: circ^#(cons(lift(), s), cons(lift(), t)) -> c_5(circ^#(s, t)) 12.97/3.92 , 6: circ^#(id(), s) -> c_6() 12.97/3.92 , 7: msubst^#(a, id()) -> c_7() 12.97/3.92 , 8: msubst^#(msubst(a, s), t) -> 12.97/3.92 c_8(msubst^#(a, circ(s, t)), circ^#(s, t)) 12.97/3.92 , 9: subst^#(a, id()) -> c_9() } 12.97/3.92 12.97/3.92 We are left with following problem, upon which TcT provides the 12.97/3.92 certificate YES(O(1),O(n^1)). 12.97/3.92 12.97/3.92 Strict DPs: 12.97/3.92 { circ^#(circ(s, t), u) -> c_2(circ^#(s, circ(t, u)), circ^#(t, u)) 12.97/3.92 , circ^#(cons(a, s), t) -> c_3(msubst^#(a, t), circ^#(s, t)) 12.97/3.92 , circ^#(cons(lift(), s), cons(a, t)) -> c_4(circ^#(s, t)) 12.97/3.92 , circ^#(cons(lift(), s), cons(lift(), t)) -> c_5(circ^#(s, t)) 12.97/3.92 , msubst^#(msubst(a, s), t) -> 12.97/3.93 c_8(msubst^#(a, circ(s, t)), circ^#(s, t)) } 12.97/3.93 Weak DPs: 12.97/3.93 { circ^#(s, id()) -> c_1() 12.97/3.93 , circ^#(id(), s) -> c_6() 12.97/3.93 , msubst^#(a, id()) -> c_7() 12.97/3.93 , subst^#(a, id()) -> c_9() } 12.97/3.93 Weak Trs: 12.97/3.93 { circ(s, id()) -> s 12.97/3.93 , circ(circ(s, t), u) -> circ(s, circ(t, u)) 12.97/3.93 , circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) 12.97/3.93 , circ(cons(lift(), s), cons(a, t)) -> cons(a, circ(s, t)) 12.97/3.93 , circ(cons(lift(), s), cons(lift(), t)) -> 12.97/3.93 cons(lift(), circ(s, t)) 12.97/3.93 , circ(id(), s) -> s 12.97/3.93 , msubst(a, id()) -> a 12.97/3.93 , msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) 12.97/3.93 , subst(a, id()) -> a } 12.97/3.93 Obligation: 12.97/3.93 innermost runtime complexity 12.97/3.93 Answer: 12.97/3.93 YES(O(1),O(n^1)) 12.97/3.93 12.97/3.93 The following weak DPs constitute a sub-graph of the DG that is 12.97/3.93 closed under successors. The DPs are removed. 12.97/3.93 12.97/3.93 { circ^#(s, id()) -> c_1() 12.97/3.93 , circ^#(id(), s) -> c_6() 12.97/3.93 , msubst^#(a, id()) -> c_7() 12.97/3.93 , subst^#(a, id()) -> c_9() } 12.97/3.93 12.97/3.93 We are left with following problem, upon which TcT provides the 12.97/3.93 certificate YES(O(1),O(n^1)). 12.97/3.93 12.97/3.93 Strict DPs: 12.97/3.93 { circ^#(circ(s, t), u) -> c_2(circ^#(s, circ(t, u)), circ^#(t, u)) 12.97/3.93 , circ^#(cons(a, s), t) -> c_3(msubst^#(a, t), circ^#(s, t)) 12.97/3.93 , circ^#(cons(lift(), s), cons(a, t)) -> c_4(circ^#(s, t)) 12.97/3.93 , circ^#(cons(lift(), s), cons(lift(), t)) -> c_5(circ^#(s, t)) 12.97/3.93 , msubst^#(msubst(a, s), t) -> 12.97/3.93 c_8(msubst^#(a, circ(s, t)), circ^#(s, t)) } 12.97/3.93 Weak Trs: 12.97/3.93 { circ(s, id()) -> s 12.97/3.93 , circ(circ(s, t), u) -> circ(s, circ(t, u)) 12.97/3.93 , circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) 12.97/3.93 , circ(cons(lift(), s), cons(a, t)) -> cons(a, circ(s, t)) 12.97/3.93 , circ(cons(lift(), s), cons(lift(), t)) -> 12.97/3.93 cons(lift(), circ(s, t)) 12.97/3.93 , circ(id(), s) -> s 12.97/3.93 , msubst(a, id()) -> a 12.97/3.93 , msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) 12.97/3.93 , subst(a, id()) -> a } 12.97/3.93 Obligation: 12.97/3.93 innermost runtime complexity 12.97/3.93 Answer: 12.97/3.93 YES(O(1),O(n^1)) 12.97/3.93 12.97/3.93 Due to missing edges in the dependency-graph, the right-hand sides 12.97/3.93 of following rules could be simplified: 12.97/3.93 12.97/3.93 { circ^#(circ(s, t), u) -> c_2(circ^#(s, circ(t, u)), circ^#(t, u)) 12.97/3.93 , msubst^#(msubst(a, s), t) -> 12.97/3.93 c_8(msubst^#(a, circ(s, t)), circ^#(s, t)) } 12.97/3.93 12.97/3.93 We are left with following problem, upon which TcT provides the 12.97/3.93 certificate YES(O(1),O(n^1)). 12.97/3.93 12.97/3.93 Strict DPs: 12.97/3.93 { circ^#(circ(s, t), u) -> c_1(circ^#(t, u)) 12.97/3.93 , circ^#(cons(a, s), t) -> c_2(msubst^#(a, t), circ^#(s, t)) 12.97/3.93 , circ^#(cons(lift(), s), cons(a, t)) -> c_3(circ^#(s, t)) 12.97/3.93 , circ^#(cons(lift(), s), cons(lift(), t)) -> c_4(circ^#(s, t)) 12.97/3.93 , msubst^#(msubst(a, s), t) -> c_5(circ^#(s, t)) } 12.97/3.93 Weak Trs: 12.97/3.93 { circ(s, id()) -> s 12.97/3.93 , circ(circ(s, t), u) -> circ(s, circ(t, u)) 12.97/3.93 , circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) 12.97/3.93 , circ(cons(lift(), s), cons(a, t)) -> cons(a, circ(s, t)) 12.97/3.93 , circ(cons(lift(), s), cons(lift(), t)) -> 12.97/3.93 cons(lift(), circ(s, t)) 12.97/3.93 , circ(id(), s) -> s 12.97/3.93 , msubst(a, id()) -> a 12.97/3.93 , msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) 12.97/3.93 , subst(a, id()) -> a } 12.97/3.93 Obligation: 12.97/3.93 innermost runtime complexity 12.97/3.93 Answer: 12.97/3.93 YES(O(1),O(n^1)) 12.97/3.93 12.97/3.93 No rule is usable, rules are removed from the input problem. 12.97/3.93 12.97/3.93 We are left with following problem, upon which TcT provides the 12.97/3.93 certificate YES(O(1),O(n^1)). 12.97/3.93 12.97/3.93 Strict DPs: 12.97/3.93 { circ^#(circ(s, t), u) -> c_1(circ^#(t, u)) 12.97/3.93 , circ^#(cons(a, s), t) -> c_2(msubst^#(a, t), circ^#(s, t)) 12.97/3.93 , circ^#(cons(lift(), s), cons(a, t)) -> c_3(circ^#(s, t)) 12.97/3.93 , circ^#(cons(lift(), s), cons(lift(), t)) -> c_4(circ^#(s, t)) 12.97/3.93 , msubst^#(msubst(a, s), t) -> c_5(circ^#(s, t)) } 12.97/3.93 Obligation: 12.97/3.93 innermost runtime complexity 12.97/3.93 Answer: 12.97/3.93 YES(O(1),O(n^1)) 12.97/3.93 12.97/3.93 We use the processor 'matrix interpretation of dimension 1' to 12.97/3.93 orient following rules strictly. 12.97/3.93 12.97/3.93 DPs: 12.97/3.93 { 1: circ^#(circ(s, t), u) -> c_1(circ^#(t, u)) 12.97/3.93 , 2: circ^#(cons(a, s), t) -> c_2(msubst^#(a, t), circ^#(s, t)) 12.97/3.93 , 3: circ^#(cons(lift(), s), cons(a, t)) -> c_3(circ^#(s, t)) 12.97/3.93 , 4: circ^#(cons(lift(), s), cons(lift(), t)) -> c_4(circ^#(s, t)) 12.97/3.93 , 5: msubst^#(msubst(a, s), t) -> c_5(circ^#(s, t)) } 12.97/3.93 12.97/3.93 Sub-proof: 12.97/3.93 ---------- 12.97/3.93 The following argument positions are usable: 12.97/3.93 Uargs(c_1) = {1}, Uargs(c_2) = {1, 2}, Uargs(c_3) = {1}, 12.97/3.93 Uargs(c_4) = {1}, Uargs(c_5) = {1} 12.97/3.93 12.97/3.93 TcT has computed the following constructor-based matrix 12.97/3.93 interpretation satisfying not(EDA). 12.97/3.93 12.97/3.93 [circ](x1, x2) = [7] x1 + [4] x2 + [4] 12.97/3.93 12.97/3.93 [cons](x1, x2) = [1] x1 + [1] x2 + [4] 12.97/3.93 12.97/3.93 [msubst](x1, x2) = [7] x1 + [5] x2 + [4] 12.97/3.93 12.97/3.93 [lift] = [0] 12.97/3.93 12.97/3.93 [circ^#](x1, x2) = [2] x1 + [0] 12.97/3.93 12.97/3.93 [msubst^#](x1, x2) = [1] x1 + [0] 12.97/3.93 12.97/3.93 [c_1](x1) = [4] x1 + [3] 12.97/3.93 12.97/3.93 [c_2](x1, x2) = [2] x1 + [1] x2 + [3] 12.97/3.93 12.97/3.93 [c_3](x1) = [1] x1 + [7] 12.97/3.93 12.97/3.93 [c_4](x1) = [1] x1 + [1] 12.97/3.93 12.97/3.93 [c_5](x1) = [1] x1 + [3] 12.97/3.93 12.97/3.93 The order satisfies the following ordering constraints: 12.97/3.93 12.97/3.93 [circ^#(circ(s, t), u)] = [14] s + [8] t + [8] 12.97/3.93 > [8] t + [3] 12.97/3.93 = [c_1(circ^#(t, u))] 12.97/3.93 12.97/3.93 [circ^#(cons(a, s), t)] = [2] a + [2] s + [8] 12.97/3.93 > [2] a + [2] s + [3] 12.97/3.93 = [c_2(msubst^#(a, t), circ^#(s, t))] 12.97/3.93 12.97/3.93 [circ^#(cons(lift(), s), cons(a, t))] = [2] s + [8] 12.97/3.93 > [2] s + [7] 12.97/3.93 = [c_3(circ^#(s, t))] 12.97/3.93 12.97/3.93 [circ^#(cons(lift(), s), cons(lift(), t))] = [2] s + [8] 12.97/3.93 > [2] s + [1] 12.97/3.93 = [c_4(circ^#(s, t))] 12.97/3.93 12.97/3.93 [msubst^#(msubst(a, s), t)] = [7] a + [5] s + [4] 12.97/3.93 > [2] s + [3] 12.97/3.93 = [c_5(circ^#(s, t))] 12.97/3.93 12.97/3.93 12.97/3.93 The strictly oriented rules are moved into the weak component. 12.97/3.93 12.97/3.93 We are left with following problem, upon which TcT provides the 12.97/3.93 certificate YES(O(1),O(1)). 12.97/3.93 12.97/3.93 Weak DPs: 12.97/3.93 { circ^#(circ(s, t), u) -> c_1(circ^#(t, u)) 12.97/3.93 , circ^#(cons(a, s), t) -> c_2(msubst^#(a, t), circ^#(s, t)) 12.97/3.93 , circ^#(cons(lift(), s), cons(a, t)) -> c_3(circ^#(s, t)) 12.97/3.93 , circ^#(cons(lift(), s), cons(lift(), t)) -> c_4(circ^#(s, t)) 12.97/3.93 , msubst^#(msubst(a, s), t) -> c_5(circ^#(s, t)) } 12.97/3.93 Obligation: 12.97/3.93 innermost runtime complexity 12.97/3.93 Answer: 12.97/3.93 YES(O(1),O(1)) 12.97/3.93 12.97/3.93 The following weak DPs constitute a sub-graph of the DG that is 12.97/3.93 closed under successors. The DPs are removed. 12.97/3.93 12.97/3.93 { circ^#(circ(s, t), u) -> c_1(circ^#(t, u)) 12.97/3.93 , circ^#(cons(a, s), t) -> c_2(msubst^#(a, t), circ^#(s, t)) 12.97/3.93 , circ^#(cons(lift(), s), cons(a, t)) -> c_3(circ^#(s, t)) 12.97/3.93 , circ^#(cons(lift(), s), cons(lift(), t)) -> c_4(circ^#(s, t)) 12.97/3.93 , msubst^#(msubst(a, s), t) -> c_5(circ^#(s, t)) } 12.97/3.93 12.97/3.93 We are left with following problem, upon which TcT provides the 12.97/3.93 certificate YES(O(1),O(1)). 12.97/3.93 12.97/3.93 Rules: Empty 12.97/3.93 Obligation: 12.97/3.93 innermost runtime complexity 12.97/3.93 Answer: 12.97/3.93 YES(O(1),O(1)) 12.97/3.93 12.97/3.93 Empty rules are trivially bounded 12.97/3.93 12.97/3.93 Hurray, we answered YES(O(1),O(n^1)) 12.97/3.94 EOF