MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { cond1(true(), x, y) -> cond2(gr(y, 0()), x, y) , cond2(true(), x, y) -> cond2(gr(y, 0()), p(x), p(y)) , cond2(false(), x, y) -> cond1(and(eq(x, y), gr(x, 0())), x, y) , gr(0(), x) -> false() , gr(s(x), 0()) -> true() , gr(s(x), s(y)) -> gr(x, y) , p(0()) -> 0() , p(s(x)) -> x , and(x, false()) -> false() , and(true(), true()) -> true() , and(false(), x) -> false() , eq(0(), 0()) -> true() , eq(0(), s(x)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) } Obligation: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { cond1^#(true(), x, y) -> c_1(cond2^#(gr(y, 0()), x, y), gr^#(y, 0())) , cond2^#(true(), x, y) -> c_2(cond2^#(gr(y, 0()), p(x), p(y)), gr^#(y, 0()), p^#(x), p^#(y)) , cond2^#(false(), x, y) -> c_3(cond1^#(and(eq(x, y), gr(x, 0())), x, y), and^#(eq(x, y), gr(x, 0())), eq^#(x, y), gr^#(x, 0())) , gr^#(0(), x) -> c_4() , gr^#(s(x), 0()) -> c_5() , gr^#(s(x), s(y)) -> c_6(gr^#(x, y)) , p^#(0()) -> c_7() , p^#(s(x)) -> c_8() , and^#(x, false()) -> c_9() , and^#(true(), true()) -> c_10() , and^#(false(), x) -> c_11() , eq^#(0(), 0()) -> c_12() , eq^#(0(), s(x)) -> c_13() , eq^#(s(x), 0()) -> c_14() , eq^#(s(x), s(y)) -> c_15(eq^#(x, y)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { cond1^#(true(), x, y) -> c_1(cond2^#(gr(y, 0()), x, y), gr^#(y, 0())) , cond2^#(true(), x, y) -> c_2(cond2^#(gr(y, 0()), p(x), p(y)), gr^#(y, 0()), p^#(x), p^#(y)) , cond2^#(false(), x, y) -> c_3(cond1^#(and(eq(x, y), gr(x, 0())), x, y), and^#(eq(x, y), gr(x, 0())), eq^#(x, y), gr^#(x, 0())) , gr^#(0(), x) -> c_4() , gr^#(s(x), 0()) -> c_5() , gr^#(s(x), s(y)) -> c_6(gr^#(x, y)) , p^#(0()) -> c_7() , p^#(s(x)) -> c_8() , and^#(x, false()) -> c_9() , and^#(true(), true()) -> c_10() , and^#(false(), x) -> c_11() , eq^#(0(), 0()) -> c_12() , eq^#(0(), s(x)) -> c_13() , eq^#(s(x), 0()) -> c_14() , eq^#(s(x), s(y)) -> c_15(eq^#(x, y)) } Weak Trs: { cond1(true(), x, y) -> cond2(gr(y, 0()), x, y) , cond2(true(), x, y) -> cond2(gr(y, 0()), p(x), p(y)) , cond2(false(), x, y) -> cond1(and(eq(x, y), gr(x, 0())), x, y) , gr(0(), x) -> false() , gr(s(x), 0()) -> true() , gr(s(x), s(y)) -> gr(x, y) , p(0()) -> 0() , p(s(x)) -> x , and(x, false()) -> false() , and(true(), true()) -> true() , and(false(), x) -> false() , eq(0(), 0()) -> true() , eq(0(), s(x)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {4,5,7,8,9,10,11,12,13,14} by applications of Pre({4,5,7,8,9,10,11,12,13,14}) = {1,2,3,6,15}. Here rules are labeled as follows: DPs: { 1: cond1^#(true(), x, y) -> c_1(cond2^#(gr(y, 0()), x, y), gr^#(y, 0())) , 2: cond2^#(true(), x, y) -> c_2(cond2^#(gr(y, 0()), p(x), p(y)), gr^#(y, 0()), p^#(x), p^#(y)) , 3: cond2^#(false(), x, y) -> c_3(cond1^#(and(eq(x, y), gr(x, 0())), x, y), and^#(eq(x, y), gr(x, 0())), eq^#(x, y), gr^#(x, 0())) , 4: gr^#(0(), x) -> c_4() , 5: gr^#(s(x), 0()) -> c_5() , 6: gr^#(s(x), s(y)) -> c_6(gr^#(x, y)) , 7: p^#(0()) -> c_7() , 8: p^#(s(x)) -> c_8() , 9: and^#(x, false()) -> c_9() , 10: and^#(true(), true()) -> c_10() , 11: and^#(false(), x) -> c_11() , 12: eq^#(0(), 0()) -> c_12() , 13: eq^#(0(), s(x)) -> c_13() , 14: eq^#(s(x), 0()) -> c_14() , 15: eq^#(s(x), s(y)) -> c_15(eq^#(x, y)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { cond1^#(true(), x, y) -> c_1(cond2^#(gr(y, 0()), x, y), gr^#(y, 0())) , cond2^#(true(), x, y) -> c_2(cond2^#(gr(y, 0()), p(x), p(y)), gr^#(y, 0()), p^#(x), p^#(y)) , cond2^#(false(), x, y) -> c_3(cond1^#(and(eq(x, y), gr(x, 0())), x, y), and^#(eq(x, y), gr(x, 0())), eq^#(x, y), gr^#(x, 0())) , gr^#(s(x), s(y)) -> c_6(gr^#(x, y)) , eq^#(s(x), s(y)) -> c_15(eq^#(x, y)) } Weak DPs: { gr^#(0(), x) -> c_4() , gr^#(s(x), 0()) -> c_5() , p^#(0()) -> c_7() , p^#(s(x)) -> c_8() , and^#(x, false()) -> c_9() , and^#(true(), true()) -> c_10() , and^#(false(), x) -> c_11() , eq^#(0(), 0()) -> c_12() , eq^#(0(), s(x)) -> c_13() , eq^#(s(x), 0()) -> c_14() } Weak Trs: { cond1(true(), x, y) -> cond2(gr(y, 0()), x, y) , cond2(true(), x, y) -> cond2(gr(y, 0()), p(x), p(y)) , cond2(false(), x, y) -> cond1(and(eq(x, y), gr(x, 0())), x, y) , gr(0(), x) -> false() , gr(s(x), 0()) -> true() , gr(s(x), s(y)) -> gr(x, y) , p(0()) -> 0() , p(s(x)) -> x , and(x, false()) -> false() , and(true(), true()) -> true() , and(false(), x) -> false() , eq(0(), 0()) -> true() , eq(0(), s(x)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) } 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. { gr^#(0(), x) -> c_4() , gr^#(s(x), 0()) -> c_5() , p^#(0()) -> c_7() , p^#(s(x)) -> c_8() , and^#(x, false()) -> c_9() , and^#(true(), true()) -> c_10() , and^#(false(), x) -> c_11() , eq^#(0(), 0()) -> c_12() , eq^#(0(), s(x)) -> c_13() , eq^#(s(x), 0()) -> c_14() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { cond1^#(true(), x, y) -> c_1(cond2^#(gr(y, 0()), x, y), gr^#(y, 0())) , cond2^#(true(), x, y) -> c_2(cond2^#(gr(y, 0()), p(x), p(y)), gr^#(y, 0()), p^#(x), p^#(y)) , cond2^#(false(), x, y) -> c_3(cond1^#(and(eq(x, y), gr(x, 0())), x, y), and^#(eq(x, y), gr(x, 0())), eq^#(x, y), gr^#(x, 0())) , gr^#(s(x), s(y)) -> c_6(gr^#(x, y)) , eq^#(s(x), s(y)) -> c_15(eq^#(x, y)) } Weak Trs: { cond1(true(), x, y) -> cond2(gr(y, 0()), x, y) , cond2(true(), x, y) -> cond2(gr(y, 0()), p(x), p(y)) , cond2(false(), x, y) -> cond1(and(eq(x, y), gr(x, 0())), x, y) , gr(0(), x) -> false() , gr(s(x), 0()) -> true() , gr(s(x), s(y)) -> gr(x, y) , p(0()) -> 0() , p(s(x)) -> x , and(x, false()) -> false() , and(true(), true()) -> true() , and(false(), x) -> false() , eq(0(), 0()) -> true() , eq(0(), s(x)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) } Obligation: innermost runtime complexity Answer: MAYBE Due to missing edges in the dependency-graph, the right-hand sides of following rules could be simplified: { cond1^#(true(), x, y) -> c_1(cond2^#(gr(y, 0()), x, y), gr^#(y, 0())) , cond2^#(true(), x, y) -> c_2(cond2^#(gr(y, 0()), p(x), p(y)), gr^#(y, 0()), p^#(x), p^#(y)) , cond2^#(false(), x, y) -> c_3(cond1^#(and(eq(x, y), gr(x, 0())), x, y), and^#(eq(x, y), gr(x, 0())), eq^#(x, y), gr^#(x, 0())) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { cond1^#(true(), x, y) -> c_1(cond2^#(gr(y, 0()), x, y)) , cond2^#(true(), x, y) -> c_2(cond2^#(gr(y, 0()), p(x), p(y))) , cond2^#(false(), x, y) -> c_3(cond1^#(and(eq(x, y), gr(x, 0())), x, y), eq^#(x, y)) , gr^#(s(x), s(y)) -> c_4(gr^#(x, y)) , eq^#(s(x), s(y)) -> c_5(eq^#(x, y)) } Weak Trs: { cond1(true(), x, y) -> cond2(gr(y, 0()), x, y) , cond2(true(), x, y) -> cond2(gr(y, 0()), p(x), p(y)) , cond2(false(), x, y) -> cond1(and(eq(x, y), gr(x, 0())), x, y) , gr(0(), x) -> false() , gr(s(x), 0()) -> true() , gr(s(x), s(y)) -> gr(x, y) , p(0()) -> 0() , p(s(x)) -> x , and(x, false()) -> false() , and(true(), true()) -> true() , and(false(), x) -> false() , eq(0(), 0()) -> true() , eq(0(), s(x)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) } Obligation: innermost runtime complexity Answer: MAYBE We replace rewrite rules by usable rules: Weak Usable Rules: { gr(0(), x) -> false() , gr(s(x), 0()) -> true() , gr(s(x), s(y)) -> gr(x, y) , p(0()) -> 0() , p(s(x)) -> x , and(x, false()) -> false() , and(true(), true()) -> true() , and(false(), x) -> false() , eq(0(), 0()) -> true() , eq(0(), s(x)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { cond1^#(true(), x, y) -> c_1(cond2^#(gr(y, 0()), x, y)) , cond2^#(true(), x, y) -> c_2(cond2^#(gr(y, 0()), p(x), p(y))) , cond2^#(false(), x, y) -> c_3(cond1^#(and(eq(x, y), gr(x, 0())), x, y), eq^#(x, y)) , gr^#(s(x), s(y)) -> c_4(gr^#(x, y)) , eq^#(s(x), s(y)) -> c_5(eq^#(x, y)) } Weak Trs: { gr(0(), x) -> false() , gr(s(x), 0()) -> true() , gr(s(x), s(y)) -> gr(x, y) , p(0()) -> 0() , p(s(x)) -> x , and(x, false()) -> false() , and(true(), true()) -> true() , and(false(), x) -> false() , eq(0(), 0()) -> true() , eq(0(), s(x)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) } Obligation: innermost runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'matrices' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'matrix interpretation of dimension 4' failed due to the following reason: Following exception was raised: stack overflow 2) 'matrix interpretation of dimension 3' failed due to the following reason: The input cannot be shown compatible 3) 'matrix interpretation of dimension 3' failed due to the following reason: The input cannot be shown compatible 4) 'matrix interpretation of dimension 2' failed due to the following reason: The input cannot be shown compatible 5) 'matrix interpretation of dimension 2' failed due to the following reason: The input cannot be shown compatible 6) 'matrix interpretation of dimension 1' failed due to the following reason: The input cannot be shown compatible 2) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. Arrrr..