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) , le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , app(nil(), y) -> y , app(add(n, x), y) -> add(n, app(x, y)) , min(add(n, nil())) -> n , min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x))) , if_min(true(), add(n, add(m, x))) -> min(add(n, x)) , if_min(false(), add(n, add(m, x))) -> min(add(m, x)) , rm(n, nil()) -> nil() , rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x)) , if_rm(true(), n, add(m, x)) -> rm(n, x) , if_rm(false(), n, add(m, x)) -> add(m, rm(n, x)) , minsort(nil(), nil()) -> nil() , minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y) , if_minsort(true(), add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil())) , if_minsort(false(), add(n, x), y) -> minsort(x, add(n, y)) } 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) '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) '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) '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)) , le^#(0(), y) -> c_5() , le^#(s(x), 0()) -> c_6() , le^#(s(x), s(y)) -> c_7(le^#(x, y)) , app^#(nil(), y) -> c_8(y) , app^#(add(n, x), y) -> c_9(n, app^#(x, y)) , min^#(add(n, nil())) -> c_10(n) , min^#(add(n, add(m, x))) -> c_11(if_min^#(le(n, m), add(n, add(m, x)))) , if_min^#(true(), add(n, add(m, x))) -> c_12(min^#(add(n, x))) , if_min^#(false(), add(n, add(m, x))) -> c_13(min^#(add(m, x))) , rm^#(n, nil()) -> c_14() , rm^#(n, add(m, x)) -> c_15(if_rm^#(eq(n, m), n, add(m, x))) , if_rm^#(true(), n, add(m, x)) -> c_16(rm^#(n, x)) , if_rm^#(false(), n, add(m, x)) -> c_17(m, rm^#(n, x)) , minsort^#(nil(), nil()) -> c_18() , minsort^#(add(n, x), y) -> c_19(if_minsort^#(eq(n, min(add(n, x))), add(n, x), y)) , if_minsort^#(true(), add(n, x), y) -> c_20(n, minsort^#(app(rm(n, x), y), nil())) , if_minsort^#(false(), add(n, x), y) -> c_21(minsort^#(x, add(n, y))) } 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)) , le^#(0(), y) -> c_5() , le^#(s(x), 0()) -> c_6() , le^#(s(x), s(y)) -> c_7(le^#(x, y)) , app^#(nil(), y) -> c_8(y) , app^#(add(n, x), y) -> c_9(n, app^#(x, y)) , min^#(add(n, nil())) -> c_10(n) , min^#(add(n, add(m, x))) -> c_11(if_min^#(le(n, m), add(n, add(m, x)))) , if_min^#(true(), add(n, add(m, x))) -> c_12(min^#(add(n, x))) , if_min^#(false(), add(n, add(m, x))) -> c_13(min^#(add(m, x))) , rm^#(n, nil()) -> c_14() , rm^#(n, add(m, x)) -> c_15(if_rm^#(eq(n, m), n, add(m, x))) , if_rm^#(true(), n, add(m, x)) -> c_16(rm^#(n, x)) , if_rm^#(false(), n, add(m, x)) -> c_17(m, rm^#(n, x)) , minsort^#(nil(), nil()) -> c_18() , minsort^#(add(n, x), y) -> c_19(if_minsort^#(eq(n, min(add(n, x))), add(n, x), y)) , if_minsort^#(true(), add(n, x), y) -> c_20(n, minsort^#(app(rm(n, x), y), nil())) , if_minsort^#(false(), add(n, x), y) -> c_21(minsort^#(x, add(n, y))) } Strict Trs: { eq(0(), 0()) -> true() , eq(0(), s(x)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , app(nil(), y) -> y , app(add(n, x), y) -> add(n, app(x, y)) , min(add(n, nil())) -> n , min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x))) , if_min(true(), add(n, add(m, x))) -> min(add(n, x)) , if_min(false(), add(n, add(m, x))) -> min(add(m, x)) , rm(n, nil()) -> nil() , rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x)) , if_rm(true(), n, add(m, x)) -> rm(n, x) , if_rm(false(), n, add(m, x)) -> add(m, rm(n, x)) , minsort(nil(), nil()) -> nil() , minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y) , if_minsort(true(), add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil())) , if_minsort(false(), add(n, x), y) -> minsort(x, add(n, y)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,2,3,5,6,14,18} by applications of Pre({1,2,3,5,6,14,18}) = {4,7,8,9,10,16,17,20}. 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: le^#(0(), y) -> c_5() , 6: le^#(s(x), 0()) -> c_6() , 7: le^#(s(x), s(y)) -> c_7(le^#(x, y)) , 8: app^#(nil(), y) -> c_8(y) , 9: app^#(add(n, x), y) -> c_9(n, app^#(x, y)) , 10: min^#(add(n, nil())) -> c_10(n) , 11: min^#(add(n, add(m, x))) -> c_11(if_min^#(le(n, m), add(n, add(m, x)))) , 12: if_min^#(true(), add(n, add(m, x))) -> c_12(min^#(add(n, x))) , 13: if_min^#(false(), add(n, add(m, x))) -> c_13(min^#(add(m, x))) , 14: rm^#(n, nil()) -> c_14() , 15: rm^#(n, add(m, x)) -> c_15(if_rm^#(eq(n, m), n, add(m, x))) , 16: if_rm^#(true(), n, add(m, x)) -> c_16(rm^#(n, x)) , 17: if_rm^#(false(), n, add(m, x)) -> c_17(m, rm^#(n, x)) , 18: minsort^#(nil(), nil()) -> c_18() , 19: minsort^#(add(n, x), y) -> c_19(if_minsort^#(eq(n, min(add(n, x))), add(n, x), y)) , 20: if_minsort^#(true(), add(n, x), y) -> c_20(n, minsort^#(app(rm(n, x), y), nil())) , 21: if_minsort^#(false(), add(n, x), y) -> c_21(minsort^#(x, add(n, y))) } 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)) , le^#(s(x), s(y)) -> c_7(le^#(x, y)) , app^#(nil(), y) -> c_8(y) , app^#(add(n, x), y) -> c_9(n, app^#(x, y)) , min^#(add(n, nil())) -> c_10(n) , min^#(add(n, add(m, x))) -> c_11(if_min^#(le(n, m), add(n, add(m, x)))) , if_min^#(true(), add(n, add(m, x))) -> c_12(min^#(add(n, x))) , if_min^#(false(), add(n, add(m, x))) -> c_13(min^#(add(m, x))) , rm^#(n, add(m, x)) -> c_15(if_rm^#(eq(n, m), n, add(m, x))) , if_rm^#(true(), n, add(m, x)) -> c_16(rm^#(n, x)) , if_rm^#(false(), n, add(m, x)) -> c_17(m, rm^#(n, x)) , minsort^#(add(n, x), y) -> c_19(if_minsort^#(eq(n, min(add(n, x))), add(n, x), y)) , if_minsort^#(true(), add(n, x), y) -> c_20(n, minsort^#(app(rm(n, x), y), nil())) , if_minsort^#(false(), add(n, x), y) -> c_21(minsort^#(x, add(n, y))) } Strict Trs: { eq(0(), 0()) -> true() , eq(0(), s(x)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , app(nil(), y) -> y , app(add(n, x), y) -> add(n, app(x, y)) , min(add(n, nil())) -> n , min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x))) , if_min(true(), add(n, add(m, x))) -> min(add(n, x)) , if_min(false(), add(n, add(m, x))) -> min(add(m, x)) , rm(n, nil()) -> nil() , rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x)) , if_rm(true(), n, add(m, x)) -> rm(n, x) , if_rm(false(), n, add(m, x)) -> add(m, rm(n, x)) , minsort(nil(), nil()) -> nil() , minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y) , if_minsort(true(), add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil())) , if_minsort(false(), add(n, x), y) -> minsort(x, add(n, y)) } Weak DPs: { eq^#(0(), 0()) -> c_1() , eq^#(0(), s(x)) -> c_2() , eq^#(s(x), 0()) -> c_3() , le^#(0(), y) -> c_5() , le^#(s(x), 0()) -> c_6() , rm^#(n, nil()) -> c_14() , minsort^#(nil(), nil()) -> c_18() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..