MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { eq(0(), 0()) -> true() , eq(0(), s(x)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , or(true(), y) -> true() , or(false(), y) -> y , union(empty(), h) -> h , union(edge(x, y, i), h) -> edge(x, y, union(i, h)) , reach(x, y, empty(), h) -> false() , reach(x, y, edge(u, v, i), h) -> if_reach_1(eq(x, u), x, y, edge(u, v, i), h) , if_reach_1(true(), x, y, edge(u, v, i), h) -> if_reach_2(eq(y, v), x, y, edge(u, v, i), h) , if_reach_1(false(), x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h)) , if_reach_2(true(), x, y, edge(u, v, i), h) -> true() , if_reach_2(false(), x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty())) } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 60.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 30 seconds) (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 30.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'bsearch-popstar (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 2) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 3) 'Fastest (timeout of 5 seconds) (timeout of 60 seconds)' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 3) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { eq^#(0(), 0()) -> c_1() , eq^#(0(), s(x)) -> c_2() , eq^#(s(x), 0()) -> c_3() , eq^#(s(x), s(y)) -> c_4(eq^#(x, y)) , or^#(true(), y) -> c_5() , or^#(false(), y) -> c_6(y) , union^#(empty(), h) -> c_7(h) , union^#(edge(x, y, i), h) -> c_8(x, y, union^#(i, h)) , reach^#(x, y, empty(), h) -> c_9() , reach^#(x, y, edge(u, v, i), h) -> c_10(if_reach_1^#(eq(x, u), x, y, edge(u, v, i), h)) , if_reach_1^#(true(), x, y, edge(u, v, i), h) -> c_11(if_reach_2^#(eq(y, v), x, y, edge(u, v, i), h)) , if_reach_1^#(false(), x, y, edge(u, v, i), h) -> c_12(reach^#(x, y, i, edge(u, v, h))) , if_reach_2^#(true(), x, y, edge(u, v, i), h) -> c_13() , if_reach_2^#(false(), x, y, edge(u, v, i), h) -> c_14(or^#(reach(x, y, i, h), reach(v, y, union(i, h), empty()))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { eq^#(0(), 0()) -> c_1() , eq^#(0(), s(x)) -> c_2() , eq^#(s(x), 0()) -> c_3() , eq^#(s(x), s(y)) -> c_4(eq^#(x, y)) , or^#(true(), y) -> c_5() , or^#(false(), y) -> c_6(y) , union^#(empty(), h) -> c_7(h) , union^#(edge(x, y, i), h) -> c_8(x, y, union^#(i, h)) , reach^#(x, y, empty(), h) -> c_9() , reach^#(x, y, edge(u, v, i), h) -> c_10(if_reach_1^#(eq(x, u), x, y, edge(u, v, i), h)) , if_reach_1^#(true(), x, y, edge(u, v, i), h) -> c_11(if_reach_2^#(eq(y, v), x, y, edge(u, v, i), h)) , if_reach_1^#(false(), x, y, edge(u, v, i), h) -> c_12(reach^#(x, y, i, edge(u, v, h))) , if_reach_2^#(true(), x, y, edge(u, v, i), h) -> c_13() , if_reach_2^#(false(), x, y, edge(u, v, i), h) -> c_14(or^#(reach(x, y, i, h), reach(v, y, union(i, h), empty()))) } Strict Trs: { eq(0(), 0()) -> true() , eq(0(), s(x)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , or(true(), y) -> true() , or(false(), y) -> y , union(empty(), h) -> h , union(edge(x, y, i), h) -> edge(x, y, union(i, h)) , reach(x, y, empty(), h) -> false() , reach(x, y, edge(u, v, i), h) -> if_reach_1(eq(x, u), x, y, edge(u, v, i), h) , if_reach_1(true(), x, y, edge(u, v, i), h) -> if_reach_2(eq(y, v), x, y, edge(u, v, i), h) , if_reach_1(false(), x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h)) , if_reach_2(true(), x, y, edge(u, v, i), h) -> true() , if_reach_2(false(), x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty())) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,2,3,5,9,13} by applications of Pre({1,2,3,5,9,13}) = {4,6,7,8,11,12,14}. Here rules are labeled as follows: DPs: { 1: eq^#(0(), 0()) -> c_1() , 2: eq^#(0(), s(x)) -> c_2() , 3: eq^#(s(x), 0()) -> c_3() , 4: eq^#(s(x), s(y)) -> c_4(eq^#(x, y)) , 5: or^#(true(), y) -> c_5() , 6: or^#(false(), y) -> c_6(y) , 7: union^#(empty(), h) -> c_7(h) , 8: union^#(edge(x, y, i), h) -> c_8(x, y, union^#(i, h)) , 9: reach^#(x, y, empty(), h) -> c_9() , 10: reach^#(x, y, edge(u, v, i), h) -> c_10(if_reach_1^#(eq(x, u), x, y, edge(u, v, i), h)) , 11: if_reach_1^#(true(), x, y, edge(u, v, i), h) -> c_11(if_reach_2^#(eq(y, v), x, y, edge(u, v, i), h)) , 12: if_reach_1^#(false(), x, y, edge(u, v, i), h) -> c_12(reach^#(x, y, i, edge(u, v, h))) , 13: if_reach_2^#(true(), x, y, edge(u, v, i), h) -> c_13() , 14: if_reach_2^#(false(), x, y, edge(u, v, i), h) -> c_14(or^#(reach(x, y, i, h), reach(v, y, union(i, h), empty()))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { eq^#(s(x), s(y)) -> c_4(eq^#(x, y)) , or^#(false(), y) -> c_6(y) , union^#(empty(), h) -> c_7(h) , union^#(edge(x, y, i), h) -> c_8(x, y, union^#(i, h)) , reach^#(x, y, edge(u, v, i), h) -> c_10(if_reach_1^#(eq(x, u), x, y, edge(u, v, i), h)) , if_reach_1^#(true(), x, y, edge(u, v, i), h) -> c_11(if_reach_2^#(eq(y, v), x, y, edge(u, v, i), h)) , if_reach_1^#(false(), x, y, edge(u, v, i), h) -> c_12(reach^#(x, y, i, edge(u, v, h))) , if_reach_2^#(false(), x, y, edge(u, v, i), h) -> c_14(or^#(reach(x, y, i, h), reach(v, y, union(i, h), empty()))) } Strict Trs: { eq(0(), 0()) -> true() , eq(0(), s(x)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , or(true(), y) -> true() , or(false(), y) -> y , union(empty(), h) -> h , union(edge(x, y, i), h) -> edge(x, y, union(i, h)) , reach(x, y, empty(), h) -> false() , reach(x, y, edge(u, v, i), h) -> if_reach_1(eq(x, u), x, y, edge(u, v, i), h) , if_reach_1(true(), x, y, edge(u, v, i), h) -> if_reach_2(eq(y, v), x, y, edge(u, v, i), h) , if_reach_1(false(), x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h)) , if_reach_2(true(), x, y, edge(u, v, i), h) -> true() , if_reach_2(false(), x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty())) } Weak DPs: { eq^#(0(), 0()) -> c_1() , eq^#(0(), s(x)) -> c_2() , eq^#(s(x), 0()) -> c_3() , or^#(true(), y) -> c_5() , reach^#(x, y, empty(), h) -> c_9() , if_reach_2^#(true(), x, y, edge(u, v, i), h) -> c_13() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..