YES(O(1),O(n^1)) 0.00/0.96 YES(O(1),O(n^1)) 0.00/0.96 0.00/0.96 We are left with following problem, upon which TcT provides the 0.00/0.96 certificate YES(O(1),O(n^1)). 0.00/0.96 0.00/0.96 Strict Trs: 0.00/0.96 { a(Z(), y, z) -> Z() 0.00/0.96 , a(C(x1, x2), y, z) -> C(a(x1, y, z), a(x2, y, y)) 0.00/0.96 , second(C(x1, x2)) -> x2 0.00/0.96 , eqZList(Z(), Z()) -> True() 0.00/0.96 , eqZList(Z(), C(y1, y2)) -> False() 0.00/0.96 , eqZList(C(x1, x2), Z()) -> False() 0.00/0.96 , eqZList(C(x1, x2), C(y1, y2)) -> 0.00/0.96 and(eqZList(x1, y1), eqZList(x2, y2)) 0.00/0.96 , first(C(x1, x2)) -> x1 } 0.00/0.96 Weak Trs: 0.00/0.96 { and(True(), True()) -> True() 0.00/0.96 , and(True(), False()) -> False() 0.00/0.96 , and(False(), True()) -> False() 0.00/0.96 , and(False(), False()) -> False() } 0.00/0.96 Obligation: 0.00/0.96 innermost runtime complexity 0.00/0.96 Answer: 0.00/0.96 YES(O(1),O(n^1)) 0.00/0.96 0.00/0.96 We add the following weak dependency pairs: 0.00/0.96 0.00/0.96 Strict DPs: 0.00/0.96 { a^#(Z(), y, z) -> c_1() 0.00/0.96 , a^#(C(x1, x2), y, z) -> c_2(a^#(x1, y, z), a^#(x2, y, y)) 0.00/0.96 , second^#(C(x1, x2)) -> c_3() 0.00/0.96 , eqZList^#(Z(), Z()) -> c_4() 0.00/0.96 , eqZList^#(Z(), C(y1, y2)) -> c_5() 0.00/0.96 , eqZList^#(C(x1, x2), Z()) -> c_6() 0.00/0.96 , eqZList^#(C(x1, x2), C(y1, y2)) -> 0.00/0.96 c_7(and^#(eqZList(x1, y1), eqZList(x2, y2))) 0.00/0.96 , first^#(C(x1, x2)) -> c_8() } 0.00/0.96 Weak DPs: 0.00/0.96 { and^#(True(), True()) -> c_9() 0.00/0.96 , and^#(True(), False()) -> c_10() 0.00/0.96 , and^#(False(), True()) -> c_11() 0.00/0.96 , and^#(False(), False()) -> c_12() } 0.00/0.96 0.00/0.96 and mark the set of starting terms. 0.00/0.96 0.00/0.96 We are left with following problem, upon which TcT provides the 0.00/0.96 certificate YES(O(1),O(n^1)). 0.00/0.96 0.00/0.96 Strict DPs: 0.00/0.96 { a^#(Z(), y, z) -> c_1() 0.00/0.96 , a^#(C(x1, x2), y, z) -> c_2(a^#(x1, y, z), a^#(x2, y, y)) 0.00/0.96 , second^#(C(x1, x2)) -> c_3() 0.00/0.96 , eqZList^#(Z(), Z()) -> c_4() 0.00/0.96 , eqZList^#(Z(), C(y1, y2)) -> c_5() 0.00/0.96 , eqZList^#(C(x1, x2), Z()) -> c_6() 0.00/0.96 , eqZList^#(C(x1, x2), C(y1, y2)) -> 0.00/0.96 c_7(and^#(eqZList(x1, y1), eqZList(x2, y2))) 0.00/0.96 , first^#(C(x1, x2)) -> c_8() } 0.00/0.96 Strict Trs: 0.00/0.96 { a(Z(), y, z) -> Z() 0.00/0.96 , a(C(x1, x2), y, z) -> C(a(x1, y, z), a(x2, y, y)) 0.00/0.96 , second(C(x1, x2)) -> x2 0.00/0.96 , eqZList(Z(), Z()) -> True() 0.00/0.96 , eqZList(Z(), C(y1, y2)) -> False() 0.00/0.96 , eqZList(C(x1, x2), Z()) -> False() 0.00/0.96 , eqZList(C(x1, x2), C(y1, y2)) -> 0.00/0.96 and(eqZList(x1, y1), eqZList(x2, y2)) 0.00/0.96 , first(C(x1, x2)) -> x1 } 0.00/0.96 Weak DPs: 0.00/0.96 { and^#(True(), True()) -> c_9() 0.00/0.96 , and^#(True(), False()) -> c_10() 0.00/0.96 , and^#(False(), True()) -> c_11() 0.00/0.96 , and^#(False(), False()) -> c_12() } 0.00/0.96 Weak Trs: 0.00/0.96 { and(True(), True()) -> True() 0.00/0.96 , and(True(), False()) -> False() 0.00/0.96 , and(False(), True()) -> False() 0.00/0.96 , and(False(), False()) -> False() } 0.00/0.96 Obligation: 0.00/0.96 innermost runtime complexity 0.00/0.96 Answer: 0.00/0.96 YES(O(1),O(n^1)) 0.00/0.96 0.00/0.96 We replace rewrite rules by usable rules: 0.00/0.96 0.00/0.96 Strict Usable Rules: 0.00/0.96 { eqZList(Z(), Z()) -> True() 0.00/0.96 , eqZList(Z(), C(y1, y2)) -> False() 0.00/0.96 , eqZList(C(x1, x2), Z()) -> False() 0.00/0.96 , eqZList(C(x1, x2), C(y1, y2)) -> 0.00/0.96 and(eqZList(x1, y1), eqZList(x2, y2)) } 0.00/0.96 Weak Usable Rules: 0.00/0.96 { and(True(), True()) -> True() 0.00/0.96 , and(True(), False()) -> False() 0.00/0.96 , and(False(), True()) -> False() 0.00/0.96 , and(False(), False()) -> False() } 0.00/0.96 0.00/0.96 We are left with following problem, upon which TcT provides the 0.00/0.96 certificate YES(O(1),O(n^1)). 0.00/0.96 0.00/0.96 Strict DPs: 0.00/0.96 { a^#(Z(), y, z) -> c_1() 0.00/0.96 , a^#(C(x1, x2), y, z) -> c_2(a^#(x1, y, z), a^#(x2, y, y)) 0.00/0.96 , second^#(C(x1, x2)) -> c_3() 0.00/0.96 , eqZList^#(Z(), Z()) -> c_4() 0.00/0.96 , eqZList^#(Z(), C(y1, y2)) -> c_5() 0.00/0.96 , eqZList^#(C(x1, x2), Z()) -> c_6() 0.00/0.96 , eqZList^#(C(x1, x2), C(y1, y2)) -> 0.00/0.96 c_7(and^#(eqZList(x1, y1), eqZList(x2, y2))) 0.00/0.96 , first^#(C(x1, x2)) -> c_8() } 0.00/0.96 Strict Trs: 0.00/0.96 { eqZList(Z(), Z()) -> True() 0.00/0.96 , eqZList(Z(), C(y1, y2)) -> False() 0.00/0.96 , eqZList(C(x1, x2), Z()) -> False() 0.00/0.96 , eqZList(C(x1, x2), C(y1, y2)) -> 0.00/0.96 and(eqZList(x1, y1), eqZList(x2, y2)) } 0.00/0.96 Weak DPs: 0.00/0.96 { and^#(True(), True()) -> c_9() 0.00/0.96 , and^#(True(), False()) -> c_10() 0.00/0.96 , and^#(False(), True()) -> c_11() 0.00/0.96 , and^#(False(), False()) -> c_12() } 0.00/0.96 Weak Trs: 0.00/0.96 { and(True(), True()) -> True() 0.00/0.96 , and(True(), False()) -> False() 0.00/0.96 , and(False(), True()) -> False() 0.00/0.96 , and(False(), False()) -> False() } 0.00/0.96 Obligation: 0.00/0.96 innermost runtime complexity 0.00/0.96 Answer: 0.00/0.96 YES(O(1),O(n^1)) 0.00/0.96 0.00/0.96 The weightgap principle applies (using the following constant 0.00/0.96 growth matrix-interpretation) 0.00/0.96 0.00/0.96 The following argument positions are usable: 0.00/0.96 Uargs(and) = {1, 2}, Uargs(c_2) = {1, 2}, Uargs(c_7) = {1}, 0.00/0.96 Uargs(and^#) = {1, 2} 0.00/0.96 0.00/0.96 TcT has computed the following constructor-restricted matrix 0.00/0.96 interpretation. 0.00/0.96 0.00/0.96 [eqZList](x1, x2) = [1 1] x1 + [1] 0.00/0.96 [0 0] [1] 0.00/0.96 0.00/0.96 [Z] = [0] 0.00/0.96 [0] 0.00/0.96 0.00/0.96 [True] = [0] 0.00/0.96 [0] 0.00/0.96 0.00/0.96 [C](x1, x2) = [1 0] x1 + [1 0] x2 + [2] 0.00/0.96 [0 1] [0 1] [2] 0.00/0.96 0.00/0.96 [and](x1, x2) = [1 0] x1 + [1 2] x2 + [0] 0.00/0.96 [0 0] [0 0] [0] 0.00/0.96 0.00/0.96 [False] = [0] 0.00/0.96 [0] 0.00/0.96 0.00/0.96 [a^#](x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [2] 0.00/0.96 [1 1] [1 1] [2 2] [2] 0.00/0.96 0.00/0.96 [c_1] = [1] 0.00/0.96 [1] 0.00/0.96 0.00/0.96 [c_2](x1, x2) = [1 0] x1 + [1 0] x2 + [2] 0.00/0.96 [0 1] [0 1] [2] 0.00/0.96 0.00/0.96 [second^#](x1) = [1 1] x1 + [2] 0.00/0.96 [1 1] [2] 0.00/0.96 0.00/0.96 [c_3] = [1] 0.00/0.96 [1] 0.00/0.96 0.00/0.96 [eqZList^#](x1, x2) = [1 1] x1 + [0] 0.00/0.96 [0 0] [0] 0.00/0.96 0.00/0.96 [c_4] = [1] 0.00/0.96 [1] 0.00/0.96 0.00/0.96 [c_5] = [1] 0.00/0.96 [0] 0.00/0.96 0.00/0.96 [c_6] = [1] 0.00/0.96 [0] 0.00/0.96 0.00/0.96 [c_7](x1) = [1 0] x1 + [1] 0.00/0.96 [0 1] [2] 0.00/0.96 0.00/0.96 [and^#](x1, x2) = [1 2] x1 + [1 1] x2 + [0] 0.00/0.96 [0 0] [0 0] [0] 0.00/0.96 0.00/0.96 [first^#](x1) = [1 1] x1 + [2] 0.00/0.96 [1 1] [2] 0.00/0.96 0.00/0.96 [c_8] = [1] 0.00/0.96 [1] 0.00/0.96 0.00/0.96 [c_9] = [0] 0.00/0.96 [0] 0.00/0.96 0.00/0.96 [c_10] = [0] 0.00/0.96 [0] 0.00/0.96 0.00/0.96 [c_11] = [0] 0.00/0.96 [0] 0.00/0.96 0.00/0.96 [c_12] = [0] 0.00/0.96 [0] 0.00/0.96 0.00/0.96 The order satisfies the following ordering constraints: 0.00/0.96 0.00/0.96 [eqZList(Z(), Z())] = [1] 0.00/0.96 [1] 0.00/0.96 > [0] 0.00/0.96 [0] 0.00/0.96 = [True()] 0.00/0.96 0.00/0.96 [eqZList(Z(), C(y1, y2))] = [1] 0.00/0.96 [1] 0.00/0.97 > [0] 0.00/0.97 [0] 0.00/0.97 = [False()] 0.00/0.97 0.00/0.97 [eqZList(C(x1, x2), Z())] = [1 1] x1 + [1 1] x2 + [5] 0.00/0.97 [0 0] [0 0] [1] 0.00/0.97 > [0] 0.00/0.97 [0] 0.00/0.97 = [False()] 0.00/0.97 0.00/0.97 [eqZList(C(x1, x2), C(y1, y2))] = [1 1] x1 + [1 1] x2 + [5] 0.00/0.97 [0 0] [0 0] [1] 0.00/0.97 > [1 1] x1 + [1 1] x2 + [4] 0.00/0.97 [0 0] [0 0] [0] 0.00/0.97 = [and(eqZList(x1, y1), eqZList(x2, y2))] 0.00/0.97 0.00/0.97 [and(True(), True())] = [0] 0.00/0.97 [0] 0.00/0.97 >= [0] 0.00/0.97 [0] 0.00/0.97 = [True()] 0.00/0.97 0.00/0.97 [and(True(), False())] = [0] 0.00/0.97 [0] 0.00/0.97 >= [0] 0.00/0.97 [0] 0.00/0.97 = [False()] 0.00/0.97 0.00/0.97 [and(False(), True())] = [0] 0.00/0.97 [0] 0.00/0.97 >= [0] 0.00/0.97 [0] 0.00/0.97 = [False()] 0.00/0.97 0.00/0.97 [and(False(), False())] = [0] 0.00/0.97 [0] 0.00/0.97 >= [0] 0.00/0.97 [0] 0.00/0.97 = [False()] 0.00/0.97 0.00/0.97 [a^#(Z(), y, z)] = [0 0] y + [0 0] z + [2] 0.00/0.97 [1 1] [2 2] [2] 0.00/0.97 > [1] 0.00/0.97 [1] 0.00/0.97 = [c_1()] 0.00/0.97 0.00/0.97 [a^#(C(x1, x2), y, z)] = [0 0] x1 + [0 0] x2 + [0 0] y + [0 0] z + [2] 0.00/0.97 [1 1] [1 1] [1 1] [2 2] [6] 0.00/0.97 ? [0 0] x1 + [0 0] x2 + [0 0] y + [0 0] z + [6] 0.00/0.97 [1 1] [1 1] [4 4] [2 2] [6] 0.00/0.97 = [c_2(a^#(x1, y, z), a^#(x2, y, y))] 0.00/0.97 0.00/0.97 [second^#(C(x1, x2))] = [1 1] x1 + [1 1] x2 + [6] 0.00/0.97 [1 1] [1 1] [6] 0.00/0.97 > [1] 0.00/0.97 [1] 0.00/0.97 = [c_3()] 0.00/0.97 0.00/0.97 [eqZList^#(Z(), Z())] = [0] 0.00/0.97 [0] 0.00/0.97 ? [1] 0.00/0.97 [1] 0.00/0.97 = [c_4()] 0.00/0.97 0.00/0.97 [eqZList^#(Z(), C(y1, y2))] = [0] 0.00/0.97 [0] 0.00/0.97 ? [1] 0.00/0.97 [0] 0.00/0.97 = [c_5()] 0.00/0.97 0.00/0.97 [eqZList^#(C(x1, x2), Z())] = [1 1] x1 + [1 1] x2 + [4] 0.00/0.97 [0 0] [0 0] [0] 0.00/0.97 > [1] 0.00/0.97 [0] 0.00/0.97 = [c_6()] 0.00/0.97 0.00/0.97 [eqZList^#(C(x1, x2), C(y1, y2))] = [1 1] x1 + [1 1] x2 + [4] 0.00/0.97 [0 0] [0 0] [0] 0.00/0.97 ? [1 1] x1 + [1 1] x2 + [6] 0.00/0.97 [0 0] [0 0] [2] 0.00/0.97 = [c_7(and^#(eqZList(x1, y1), eqZList(x2, y2)))] 0.00/0.97 0.00/0.97 [and^#(True(), True())] = [0] 0.00/0.97 [0] 0.00/0.97 >= [0] 0.00/0.97 [0] 0.00/0.97 = [c_9()] 0.00/0.97 0.00/0.97 [and^#(True(), False())] = [0] 0.00/0.97 [0] 0.00/0.97 >= [0] 0.00/0.97 [0] 0.00/0.97 = [c_10()] 0.00/0.97 0.00/0.97 [and^#(False(), True())] = [0] 0.00/0.97 [0] 0.00/0.97 >= [0] 0.00/0.97 [0] 0.00/0.97 = [c_11()] 0.00/0.97 0.00/0.97 [and^#(False(), False())] = [0] 0.00/0.97 [0] 0.00/0.97 >= [0] 0.00/0.97 [0] 0.00/0.97 = [c_12()] 0.00/0.97 0.00/0.97 [first^#(C(x1, x2))] = [1 1] x1 + [1 1] x2 + [6] 0.00/0.97 [1 1] [1 1] [6] 0.00/0.97 > [1] 0.00/0.97 [1] 0.00/0.97 = [c_8()] 0.00/0.97 0.00/0.97 0.00/0.97 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.97 0.00/0.97 We are left with following problem, upon which TcT provides the 0.00/0.97 certificate YES(O(1),O(n^1)). 0.00/0.97 0.00/0.97 Strict DPs: 0.00/0.97 { a^#(C(x1, x2), y, z) -> c_2(a^#(x1, y, z), a^#(x2, y, y)) 0.00/0.97 , eqZList^#(Z(), Z()) -> c_4() 0.00/0.97 , eqZList^#(Z(), C(y1, y2)) -> c_5() 0.00/0.97 , eqZList^#(C(x1, x2), C(y1, y2)) -> 0.00/0.97 c_7(and^#(eqZList(x1, y1), eqZList(x2, y2))) } 0.00/0.97 Weak DPs: 0.00/0.97 { a^#(Z(), y, z) -> c_1() 0.00/0.97 , second^#(C(x1, x2)) -> c_3() 0.00/0.97 , eqZList^#(C(x1, x2), Z()) -> c_6() 0.00/0.97 , and^#(True(), True()) -> c_9() 0.00/0.97 , and^#(True(), False()) -> c_10() 0.00/0.97 , and^#(False(), True()) -> c_11() 0.00/0.97 , and^#(False(), False()) -> c_12() 0.00/0.97 , first^#(C(x1, x2)) -> c_8() } 0.00/0.97 Weak Trs: 0.00/0.97 { eqZList(Z(), Z()) -> True() 0.00/0.97 , eqZList(Z(), C(y1, y2)) -> False() 0.00/0.97 , eqZList(C(x1, x2), Z()) -> False() 0.00/0.97 , eqZList(C(x1, x2), C(y1, y2)) -> 0.00/0.97 and(eqZList(x1, y1), eqZList(x2, y2)) 0.00/0.97 , and(True(), True()) -> True() 0.00/0.97 , and(True(), False()) -> False() 0.00/0.97 , and(False(), True()) -> False() 0.00/0.97 , and(False(), False()) -> False() } 0.00/0.97 Obligation: 0.00/0.97 innermost runtime complexity 0.00/0.97 Answer: 0.00/0.97 YES(O(1),O(n^1)) 0.00/0.97 0.00/0.97 We estimate the number of application of {2,3,4} by applications of 0.00/0.97 Pre({2,3,4}) = {}. Here rules are labeled as follows: 0.00/0.97 0.00/0.97 DPs: 0.00/0.97 { 1: a^#(C(x1, x2), y, z) -> c_2(a^#(x1, y, z), a^#(x2, y, y)) 0.00/0.97 , 2: eqZList^#(Z(), Z()) -> c_4() 0.00/0.97 , 3: eqZList^#(Z(), C(y1, y2)) -> c_5() 0.00/0.97 , 4: eqZList^#(C(x1, x2), C(y1, y2)) -> 0.00/0.97 c_7(and^#(eqZList(x1, y1), eqZList(x2, y2))) 0.00/0.97 , 5: a^#(Z(), y, z) -> c_1() 0.00/0.97 , 6: second^#(C(x1, x2)) -> c_3() 0.00/0.97 , 7: eqZList^#(C(x1, x2), Z()) -> c_6() 0.00/0.97 , 8: and^#(True(), True()) -> c_9() 0.00/0.97 , 9: and^#(True(), False()) -> c_10() 0.00/0.97 , 10: and^#(False(), True()) -> c_11() 0.00/0.97 , 11: and^#(False(), False()) -> c_12() 0.00/0.97 , 12: first^#(C(x1, x2)) -> c_8() } 0.00/0.97 0.00/0.97 We are left with following problem, upon which TcT provides the 0.00/0.97 certificate YES(O(1),O(n^1)). 0.00/0.97 0.00/0.97 Strict DPs: 0.00/0.97 { a^#(C(x1, x2), y, z) -> c_2(a^#(x1, y, z), a^#(x2, y, y)) } 0.00/0.97 Weak DPs: 0.00/0.97 { a^#(Z(), y, z) -> c_1() 0.00/0.97 , second^#(C(x1, x2)) -> c_3() 0.00/0.97 , eqZList^#(Z(), Z()) -> c_4() 0.00/0.97 , eqZList^#(Z(), C(y1, y2)) -> c_5() 0.00/0.97 , eqZList^#(C(x1, x2), Z()) -> c_6() 0.00/0.97 , eqZList^#(C(x1, x2), C(y1, y2)) -> 0.00/0.97 c_7(and^#(eqZList(x1, y1), eqZList(x2, y2))) 0.00/0.97 , and^#(True(), True()) -> c_9() 0.00/0.97 , and^#(True(), False()) -> c_10() 0.00/0.97 , and^#(False(), True()) -> c_11() 0.00/0.97 , and^#(False(), False()) -> c_12() 0.00/0.97 , first^#(C(x1, x2)) -> c_8() } 0.00/0.97 Weak Trs: 0.00/0.97 { eqZList(Z(), Z()) -> True() 0.00/0.97 , eqZList(Z(), C(y1, y2)) -> False() 0.00/0.97 , eqZList(C(x1, x2), Z()) -> False() 0.00/0.97 , eqZList(C(x1, x2), C(y1, y2)) -> 0.00/0.97 and(eqZList(x1, y1), eqZList(x2, y2)) 0.00/0.97 , and(True(), True()) -> True() 0.00/0.97 , and(True(), False()) -> False() 0.00/0.97 , and(False(), True()) -> False() 0.00/0.97 , and(False(), False()) -> False() } 0.00/0.97 Obligation: 0.00/0.97 innermost runtime complexity 0.00/0.97 Answer: 0.00/0.97 YES(O(1),O(n^1)) 0.00/0.97 0.00/0.97 The following weak DPs constitute a sub-graph of the DG that is 0.00/0.97 closed under successors. The DPs are removed. 0.00/0.97 0.00/0.97 { a^#(Z(), y, z) -> c_1() 0.00/0.97 , second^#(C(x1, x2)) -> c_3() 0.00/0.97 , eqZList^#(Z(), Z()) -> c_4() 0.00/0.97 , eqZList^#(Z(), C(y1, y2)) -> c_5() 0.00/0.97 , eqZList^#(C(x1, x2), Z()) -> c_6() 0.00/0.97 , eqZList^#(C(x1, x2), C(y1, y2)) -> 0.00/0.97 c_7(and^#(eqZList(x1, y1), eqZList(x2, y2))) 0.00/0.97 , and^#(True(), True()) -> c_9() 0.00/0.97 , and^#(True(), False()) -> c_10() 0.00/0.97 , and^#(False(), True()) -> c_11() 0.00/0.97 , and^#(False(), False()) -> c_12() 0.00/0.97 , first^#(C(x1, x2)) -> c_8() } 0.00/0.97 0.00/0.97 We are left with following problem, upon which TcT provides the 0.00/0.97 certificate YES(O(1),O(n^1)). 0.00/0.97 0.00/0.97 Strict DPs: 0.00/0.97 { a^#(C(x1, x2), y, z) -> c_2(a^#(x1, y, z), a^#(x2, y, y)) } 0.00/0.97 Weak Trs: 0.00/0.97 { eqZList(Z(), Z()) -> True() 0.00/0.97 , eqZList(Z(), C(y1, y2)) -> False() 0.00/0.97 , eqZList(C(x1, x2), Z()) -> False() 0.00/0.97 , eqZList(C(x1, x2), C(y1, y2)) -> 0.00/0.97 and(eqZList(x1, y1), eqZList(x2, y2)) 0.00/0.97 , and(True(), True()) -> True() 0.00/0.97 , and(True(), False()) -> False() 0.00/0.97 , and(False(), True()) -> False() 0.00/0.97 , and(False(), False()) -> False() } 0.00/0.97 Obligation: 0.00/0.97 innermost runtime complexity 0.00/0.97 Answer: 0.00/0.97 YES(O(1),O(n^1)) 0.00/0.97 0.00/0.97 No rule is usable, rules are removed from the input problem. 0.00/0.97 0.00/0.97 We are left with following problem, upon which TcT provides the 0.00/0.97 certificate YES(O(1),O(n^1)). 0.00/0.97 0.00/0.97 Strict DPs: 0.00/0.97 { a^#(C(x1, x2), y, z) -> c_2(a^#(x1, y, z), a^#(x2, y, y)) } 0.00/0.97 Obligation: 0.00/0.97 innermost runtime complexity 0.00/0.97 Answer: 0.00/0.97 YES(O(1),O(n^1)) 0.00/0.97 0.00/0.97 We use the processor 'matrix interpretation of dimension 1' to 0.00/0.97 orient following rules strictly. 0.00/0.97 0.00/0.97 DPs: 0.00/0.97 { 1: a^#(C(x1, x2), y, z) -> c_2(a^#(x1, y, z), a^#(x2, y, y)) } 0.00/0.97 0.00/0.97 Sub-proof: 0.00/0.97 ---------- 0.00/0.97 The following argument positions are usable: 0.00/0.97 Uargs(c_2) = {1, 2} 0.00/0.97 0.00/0.97 TcT has computed the following constructor-based matrix 0.00/0.97 interpretation satisfying not(EDA). 0.00/0.97 0.00/0.97 [C](x1, x2) = [1] x1 + [1] x2 + [1] 0.00/0.97 0.00/0.97 [a^#](x1, x2, x3) = [4] x1 + [0] 0.00/0.97 0.00/0.97 [c_2](x1, x2) = [1] x1 + [1] x2 + [1] 0.00/0.97 0.00/0.97 The order satisfies the following ordering constraints: 0.00/0.97 0.00/0.97 [a^#(C(x1, x2), y, z)] = [4] x1 + [4] x2 + [4] 0.00/0.97 > [4] x1 + [4] x2 + [1] 0.00/0.97 = [c_2(a^#(x1, y, z), a^#(x2, y, y))] 0.00/0.97 0.00/0.97 0.00/0.97 The strictly oriented rules are moved into the weak component. 0.00/0.97 0.00/0.97 We are left with following problem, upon which TcT provides the 0.00/0.97 certificate YES(O(1),O(1)). 0.00/0.97 0.00/0.97 Weak DPs: 0.00/0.97 { a^#(C(x1, x2), y, z) -> c_2(a^#(x1, y, z), a^#(x2, y, y)) } 0.00/0.97 Obligation: 0.00/0.97 innermost runtime complexity 0.00/0.97 Answer: 0.00/0.97 YES(O(1),O(1)) 0.00/0.97 0.00/0.97 The following weak DPs constitute a sub-graph of the DG that is 0.00/0.97 closed under successors. The DPs are removed. 0.00/0.97 0.00/0.97 { a^#(C(x1, x2), y, z) -> c_2(a^#(x1, y, z), a^#(x2, y, y)) } 0.00/0.97 0.00/0.97 We are left with following problem, upon which TcT provides the 0.00/0.97 certificate YES(O(1),O(1)). 0.00/0.97 0.00/0.97 Rules: Empty 0.00/0.97 Obligation: 0.00/0.97 innermost runtime complexity 0.00/0.97 Answer: 0.00/0.97 YES(O(1),O(1)) 0.00/0.97 0.00/0.97 Empty rules are trivially bounded 0.00/0.97 0.00/0.97 Hurray, we answered YES(O(1),O(n^1)) 0.00/0.97 EOF