MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { min(x, 0()) -> 0() , min(0(), y) -> 0() , min(s(x), s(y)) -> s(min(x, y)) , max(x, 0()) -> x , max(0(), y) -> y , max(s(x), s(y)) -> s(max(x, y)) , +(0(), y) -> y , +(s(x), y) -> s(+(x, y)) , -(x, 0()) -> x , -(s(x), s(y)) -> -(x, y) , *(x, 0()) -> 0() , *(x, s(y)) -> +(x, *(x, y)) , f(s(x)) -> f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0()))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0())))))))) } Obligation: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { min^#(x, 0()) -> c_1() , min^#(0(), y) -> c_2() , min^#(s(x), s(y)) -> c_3(min^#(x, y)) , max^#(x, 0()) -> c_4() , max^#(0(), y) -> c_5() , max^#(s(x), s(y)) -> c_6(max^#(x, y)) , +^#(0(), y) -> c_7() , +^#(s(x), y) -> c_8(+^#(x, y)) , -^#(x, 0()) -> c_9() , -^#(s(x), s(y)) -> c_10(-^#(x, y)) , *^#(x, 0()) -> c_11() , *^#(x, s(y)) -> c_12(+^#(x, *(x, y)), *^#(x, y)) , f^#(s(x)) -> c_13(f^#(-(max(*(s(x), s(x)), +(s(x), s(s(s(0()))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0())))))))), -^#(max(*(s(x), s(x)), +(s(x), s(s(s(0()))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0()))))))), max^#(*(s(x), s(x)), +(s(x), s(s(s(0()))))), *^#(s(x), s(x)), +^#(s(x), s(s(s(0())))), max^#(s(*(s(x), s(x))), +(s(x), s(s(s(s(0())))))), *^#(s(x), s(x)), +^#(s(x), s(s(s(s(0())))))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { min^#(x, 0()) -> c_1() , min^#(0(), y) -> c_2() , min^#(s(x), s(y)) -> c_3(min^#(x, y)) , max^#(x, 0()) -> c_4() , max^#(0(), y) -> c_5() , max^#(s(x), s(y)) -> c_6(max^#(x, y)) , +^#(0(), y) -> c_7() , +^#(s(x), y) -> c_8(+^#(x, y)) , -^#(x, 0()) -> c_9() , -^#(s(x), s(y)) -> c_10(-^#(x, y)) , *^#(x, 0()) -> c_11() , *^#(x, s(y)) -> c_12(+^#(x, *(x, y)), *^#(x, y)) , f^#(s(x)) -> c_13(f^#(-(max(*(s(x), s(x)), +(s(x), s(s(s(0()))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0())))))))), -^#(max(*(s(x), s(x)), +(s(x), s(s(s(0()))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0()))))))), max^#(*(s(x), s(x)), +(s(x), s(s(s(0()))))), *^#(s(x), s(x)), +^#(s(x), s(s(s(0())))), max^#(s(*(s(x), s(x))), +(s(x), s(s(s(s(0())))))), *^#(s(x), s(x)), +^#(s(x), s(s(s(s(0())))))) } Weak Trs: { min(x, 0()) -> 0() , min(0(), y) -> 0() , min(s(x), s(y)) -> s(min(x, y)) , max(x, 0()) -> x , max(0(), y) -> y , max(s(x), s(y)) -> s(max(x, y)) , +(0(), y) -> y , +(s(x), y) -> s(+(x, y)) , -(x, 0()) -> x , -(s(x), s(y)) -> -(x, y) , *(x, 0()) -> 0() , *(x, s(y)) -> +(x, *(x, y)) , f(s(x)) -> f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0()))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0())))))))) } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {1,2,4,5,7,9,11} by applications of Pre({1,2,4,5,7,9,11}) = {3,6,8,10,12,13}. Here rules are labeled as follows: DPs: { 1: min^#(x, 0()) -> c_1() , 2: min^#(0(), y) -> c_2() , 3: min^#(s(x), s(y)) -> c_3(min^#(x, y)) , 4: max^#(x, 0()) -> c_4() , 5: max^#(0(), y) -> c_5() , 6: max^#(s(x), s(y)) -> c_6(max^#(x, y)) , 7: +^#(0(), y) -> c_7() , 8: +^#(s(x), y) -> c_8(+^#(x, y)) , 9: -^#(x, 0()) -> c_9() , 10: -^#(s(x), s(y)) -> c_10(-^#(x, y)) , 11: *^#(x, 0()) -> c_11() , 12: *^#(x, s(y)) -> c_12(+^#(x, *(x, y)), *^#(x, y)) , 13: f^#(s(x)) -> c_13(f^#(-(max(*(s(x), s(x)), +(s(x), s(s(s(0()))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0())))))))), -^#(max(*(s(x), s(x)), +(s(x), s(s(s(0()))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0()))))))), max^#(*(s(x), s(x)), +(s(x), s(s(s(0()))))), *^#(s(x), s(x)), +^#(s(x), s(s(s(0())))), max^#(s(*(s(x), s(x))), +(s(x), s(s(s(s(0())))))), *^#(s(x), s(x)), +^#(s(x), s(s(s(s(0())))))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { min^#(s(x), s(y)) -> c_3(min^#(x, y)) , max^#(s(x), s(y)) -> c_6(max^#(x, y)) , +^#(s(x), y) -> c_8(+^#(x, y)) , -^#(s(x), s(y)) -> c_10(-^#(x, y)) , *^#(x, s(y)) -> c_12(+^#(x, *(x, y)), *^#(x, y)) , f^#(s(x)) -> c_13(f^#(-(max(*(s(x), s(x)), +(s(x), s(s(s(0()))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0())))))))), -^#(max(*(s(x), s(x)), +(s(x), s(s(s(0()))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0()))))))), max^#(*(s(x), s(x)), +(s(x), s(s(s(0()))))), *^#(s(x), s(x)), +^#(s(x), s(s(s(0())))), max^#(s(*(s(x), s(x))), +(s(x), s(s(s(s(0())))))), *^#(s(x), s(x)), +^#(s(x), s(s(s(s(0())))))) } Weak DPs: { min^#(x, 0()) -> c_1() , min^#(0(), y) -> c_2() , max^#(x, 0()) -> c_4() , max^#(0(), y) -> c_5() , +^#(0(), y) -> c_7() , -^#(x, 0()) -> c_9() , *^#(x, 0()) -> c_11() } Weak Trs: { min(x, 0()) -> 0() , min(0(), y) -> 0() , min(s(x), s(y)) -> s(min(x, y)) , max(x, 0()) -> x , max(0(), y) -> y , max(s(x), s(y)) -> s(max(x, y)) , +(0(), y) -> y , +(s(x), y) -> s(+(x, y)) , -(x, 0()) -> x , -(s(x), s(y)) -> -(x, y) , *(x, 0()) -> 0() , *(x, s(y)) -> +(x, *(x, y)) , f(s(x)) -> f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0()))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0())))))))) } Obligation: innermost runtime complexity Answer: MAYBE 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() , min^#(0(), y) -> c_2() , max^#(x, 0()) -> c_4() , max^#(0(), y) -> c_5() , +^#(0(), y) -> c_7() , -^#(x, 0()) -> c_9() , *^#(x, 0()) -> c_11() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { min^#(s(x), s(y)) -> c_3(min^#(x, y)) , max^#(s(x), s(y)) -> c_6(max^#(x, y)) , +^#(s(x), y) -> c_8(+^#(x, y)) , -^#(s(x), s(y)) -> c_10(-^#(x, y)) , *^#(x, s(y)) -> c_12(+^#(x, *(x, y)), *^#(x, y)) , f^#(s(x)) -> c_13(f^#(-(max(*(s(x), s(x)), +(s(x), s(s(s(0()))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0())))))))), -^#(max(*(s(x), s(x)), +(s(x), s(s(s(0()))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0()))))))), max^#(*(s(x), s(x)), +(s(x), s(s(s(0()))))), *^#(s(x), s(x)), +^#(s(x), s(s(s(0())))), max^#(s(*(s(x), s(x))), +(s(x), s(s(s(s(0())))))), *^#(s(x), s(x)), +^#(s(x), s(s(s(s(0())))))) } Weak Trs: { min(x, 0()) -> 0() , min(0(), y) -> 0() , min(s(x), s(y)) -> s(min(x, y)) , max(x, 0()) -> x , max(0(), y) -> y , max(s(x), s(y)) -> s(max(x, y)) , +(0(), y) -> y , +(s(x), y) -> s(+(x, y)) , -(x, 0()) -> x , -(s(x), s(y)) -> -(x, y) , *(x, 0()) -> 0() , *(x, s(y)) -> +(x, *(x, y)) , f(s(x)) -> f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0()))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0())))))))) } Obligation: innermost runtime complexity Answer: MAYBE We replace rewrite rules by usable rules: Weak Usable Rules: { max(x, 0()) -> x , max(0(), y) -> y , max(s(x), s(y)) -> s(max(x, y)) , +(0(), y) -> y , +(s(x), y) -> s(+(x, y)) , -(x, 0()) -> x , -(s(x), s(y)) -> -(x, y) , *(x, 0()) -> 0() , *(x, s(y)) -> +(x, *(x, y)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { min^#(s(x), s(y)) -> c_3(min^#(x, y)) , max^#(s(x), s(y)) -> c_6(max^#(x, y)) , +^#(s(x), y) -> c_8(+^#(x, y)) , -^#(s(x), s(y)) -> c_10(-^#(x, y)) , *^#(x, s(y)) -> c_12(+^#(x, *(x, y)), *^#(x, y)) , f^#(s(x)) -> c_13(f^#(-(max(*(s(x), s(x)), +(s(x), s(s(s(0()))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0())))))))), -^#(max(*(s(x), s(x)), +(s(x), s(s(s(0()))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0()))))))), max^#(*(s(x), s(x)), +(s(x), s(s(s(0()))))), *^#(s(x), s(x)), +^#(s(x), s(s(s(0())))), max^#(s(*(s(x), s(x))), +(s(x), s(s(s(s(0())))))), *^#(s(x), s(x)), +^#(s(x), s(s(s(s(0())))))) } Weak Trs: { max(x, 0()) -> x , max(0(), y) -> y , max(s(x), s(y)) -> s(max(x, y)) , +(0(), y) -> y , +(s(x), y) -> s(+(x, y)) , -(x, 0()) -> x , -(s(x), s(y)) -> -(x, y) , *(x, 0()) -> 0() , *(x, s(y)) -> +(x, *(x, y)) } Obligation: innermost runtime complexity Answer: MAYBE The input cannot be shown compatible Arrrr..