MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , eq(0(), 0()) -> true() , eq(0(), s(y)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , minsort(nil()) -> nil() , minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) , min(nil()) -> 0() , min(cons(x, nil())) -> x , min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) , rm(x, nil()) -> nil() , rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) , if1(true(), x, y, xs) -> min(cons(x, xs)) , if1(false(), x, y, xs) -> min(cons(y, xs)) , if2(true(), x, y, xs) -> rm(x, xs) , if2(false(), x, y, xs) -> cons(y, rm(x, xs)) } Obligation: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , le^#(s(x), s(y)) -> c_3(le^#(x, y)) , eq^#(0(), 0()) -> c_4() , eq^#(0(), s(y)) -> c_5() , eq^#(s(x), 0()) -> c_6() , eq^#(s(x), s(y)) -> c_7(eq^#(x, y)) , minsort^#(nil()) -> c_8() , minsort^#(cons(x, xs)) -> c_9(min^#(cons(x, xs)), minsort^#(rm(min(cons(x, xs)), cons(x, xs))), rm^#(min(cons(x, xs)), cons(x, xs)), min^#(cons(x, xs))) , min^#(nil()) -> c_10() , min^#(cons(x, nil())) -> c_11() , min^#(cons(x, cons(y, xs))) -> c_12(if1^#(le(x, y), x, y, xs), le^#(x, y)) , rm^#(x, nil()) -> c_13() , rm^#(x, cons(y, xs)) -> c_14(if2^#(eq(x, y), x, y, xs), eq^#(x, y)) , if1^#(true(), x, y, xs) -> c_15(min^#(cons(x, xs))) , if1^#(false(), x, y, xs) -> c_16(min^#(cons(y, xs))) , if2^#(true(), x, y, xs) -> c_17(rm^#(x, xs)) , if2^#(false(), x, y, xs) -> c_18(rm^#(x, xs)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , le^#(s(x), s(y)) -> c_3(le^#(x, y)) , eq^#(0(), 0()) -> c_4() , eq^#(0(), s(y)) -> c_5() , eq^#(s(x), 0()) -> c_6() , eq^#(s(x), s(y)) -> c_7(eq^#(x, y)) , minsort^#(nil()) -> c_8() , minsort^#(cons(x, xs)) -> c_9(min^#(cons(x, xs)), minsort^#(rm(min(cons(x, xs)), cons(x, xs))), rm^#(min(cons(x, xs)), cons(x, xs)), min^#(cons(x, xs))) , min^#(nil()) -> c_10() , min^#(cons(x, nil())) -> c_11() , min^#(cons(x, cons(y, xs))) -> c_12(if1^#(le(x, y), x, y, xs), le^#(x, y)) , rm^#(x, nil()) -> c_13() , rm^#(x, cons(y, xs)) -> c_14(if2^#(eq(x, y), x, y, xs), eq^#(x, y)) , if1^#(true(), x, y, xs) -> c_15(min^#(cons(x, xs))) , if1^#(false(), x, y, xs) -> c_16(min^#(cons(y, xs))) , if2^#(true(), x, y, xs) -> c_17(rm^#(x, xs)) , if2^#(false(), x, y, xs) -> c_18(rm^#(x, xs)) } Weak Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , eq(0(), 0()) -> true() , eq(0(), s(y)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , minsort(nil()) -> nil() , minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) , min(nil()) -> 0() , min(cons(x, nil())) -> x , min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) , rm(x, nil()) -> nil() , rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) , if1(true(), x, y, xs) -> min(cons(x, xs)) , if1(false(), x, y, xs) -> min(cons(y, xs)) , if2(true(), x, y, xs) -> rm(x, xs) , if2(false(), x, y, xs) -> cons(y, rm(x, xs)) } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {1,2,4,5,6,8,10,11,13} by applications of Pre({1,2,4,5,6,8,10,11,13}) = {3,7,9,12,14,15,16,17,18}. Here rules are labeled as follows: DPs: { 1: le^#(0(), y) -> c_1() , 2: le^#(s(x), 0()) -> c_2() , 3: le^#(s(x), s(y)) -> c_3(le^#(x, y)) , 4: eq^#(0(), 0()) -> c_4() , 5: eq^#(0(), s(y)) -> c_5() , 6: eq^#(s(x), 0()) -> c_6() , 7: eq^#(s(x), s(y)) -> c_7(eq^#(x, y)) , 8: minsort^#(nil()) -> c_8() , 9: minsort^#(cons(x, xs)) -> c_9(min^#(cons(x, xs)), minsort^#(rm(min(cons(x, xs)), cons(x, xs))), rm^#(min(cons(x, xs)), cons(x, xs)), min^#(cons(x, xs))) , 10: min^#(nil()) -> c_10() , 11: min^#(cons(x, nil())) -> c_11() , 12: min^#(cons(x, cons(y, xs))) -> c_12(if1^#(le(x, y), x, y, xs), le^#(x, y)) , 13: rm^#(x, nil()) -> c_13() , 14: rm^#(x, cons(y, xs)) -> c_14(if2^#(eq(x, y), x, y, xs), eq^#(x, y)) , 15: if1^#(true(), x, y, xs) -> c_15(min^#(cons(x, xs))) , 16: if1^#(false(), x, y, xs) -> c_16(min^#(cons(y, xs))) , 17: if2^#(true(), x, y, xs) -> c_17(rm^#(x, xs)) , 18: if2^#(false(), x, y, xs) -> c_18(rm^#(x, xs)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { le^#(s(x), s(y)) -> c_3(le^#(x, y)) , eq^#(s(x), s(y)) -> c_7(eq^#(x, y)) , minsort^#(cons(x, xs)) -> c_9(min^#(cons(x, xs)), minsort^#(rm(min(cons(x, xs)), cons(x, xs))), rm^#(min(cons(x, xs)), cons(x, xs)), min^#(cons(x, xs))) , min^#(cons(x, cons(y, xs))) -> c_12(if1^#(le(x, y), x, y, xs), le^#(x, y)) , rm^#(x, cons(y, xs)) -> c_14(if2^#(eq(x, y), x, y, xs), eq^#(x, y)) , if1^#(true(), x, y, xs) -> c_15(min^#(cons(x, xs))) , if1^#(false(), x, y, xs) -> c_16(min^#(cons(y, xs))) , if2^#(true(), x, y, xs) -> c_17(rm^#(x, xs)) , if2^#(false(), x, y, xs) -> c_18(rm^#(x, xs)) } Weak DPs: { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , eq^#(0(), 0()) -> c_4() , eq^#(0(), s(y)) -> c_5() , eq^#(s(x), 0()) -> c_6() , minsort^#(nil()) -> c_8() , min^#(nil()) -> c_10() , min^#(cons(x, nil())) -> c_11() , rm^#(x, nil()) -> c_13() } Weak Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , eq(0(), 0()) -> true() , eq(0(), s(y)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , minsort(nil()) -> nil() , minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) , min(nil()) -> 0() , min(cons(x, nil())) -> x , min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) , rm(x, nil()) -> nil() , rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) , if1(true(), x, y, xs) -> min(cons(x, xs)) , if1(false(), x, y, xs) -> min(cons(y, xs)) , if2(true(), x, y, xs) -> rm(x, xs) , if2(false(), x, y, xs) -> cons(y, rm(x, xs)) } 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. { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , eq^#(0(), 0()) -> c_4() , eq^#(0(), s(y)) -> c_5() , eq^#(s(x), 0()) -> c_6() , minsort^#(nil()) -> c_8() , min^#(nil()) -> c_10() , min^#(cons(x, nil())) -> c_11() , rm^#(x, nil()) -> c_13() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { le^#(s(x), s(y)) -> c_3(le^#(x, y)) , eq^#(s(x), s(y)) -> c_7(eq^#(x, y)) , minsort^#(cons(x, xs)) -> c_9(min^#(cons(x, xs)), minsort^#(rm(min(cons(x, xs)), cons(x, xs))), rm^#(min(cons(x, xs)), cons(x, xs)), min^#(cons(x, xs))) , min^#(cons(x, cons(y, xs))) -> c_12(if1^#(le(x, y), x, y, xs), le^#(x, y)) , rm^#(x, cons(y, xs)) -> c_14(if2^#(eq(x, y), x, y, xs), eq^#(x, y)) , if1^#(true(), x, y, xs) -> c_15(min^#(cons(x, xs))) , if1^#(false(), x, y, xs) -> c_16(min^#(cons(y, xs))) , if2^#(true(), x, y, xs) -> c_17(rm^#(x, xs)) , if2^#(false(), x, y, xs) -> c_18(rm^#(x, xs)) } Weak Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , eq(0(), 0()) -> true() , eq(0(), s(y)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , minsort(nil()) -> nil() , minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) , min(nil()) -> 0() , min(cons(x, nil())) -> x , min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) , rm(x, nil()) -> nil() , rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) , if1(true(), x, y, xs) -> min(cons(x, xs)) , if1(false(), x, y, xs) -> min(cons(y, xs)) , if2(true(), x, y, xs) -> rm(x, xs) , if2(false(), x, y, xs) -> cons(y, rm(x, xs)) } Obligation: innermost runtime complexity Answer: MAYBE We replace rewrite rules by usable rules: Weak Usable Rules: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , eq(0(), 0()) -> true() , eq(0(), s(y)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , min(nil()) -> 0() , min(cons(x, nil())) -> x , min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) , rm(x, nil()) -> nil() , rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) , if1(true(), x, y, xs) -> min(cons(x, xs)) , if1(false(), x, y, xs) -> min(cons(y, xs)) , if2(true(), x, y, xs) -> rm(x, xs) , if2(false(), x, y, xs) -> cons(y, rm(x, xs)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { le^#(s(x), s(y)) -> c_3(le^#(x, y)) , eq^#(s(x), s(y)) -> c_7(eq^#(x, y)) , minsort^#(cons(x, xs)) -> c_9(min^#(cons(x, xs)), minsort^#(rm(min(cons(x, xs)), cons(x, xs))), rm^#(min(cons(x, xs)), cons(x, xs)), min^#(cons(x, xs))) , min^#(cons(x, cons(y, xs))) -> c_12(if1^#(le(x, y), x, y, xs), le^#(x, y)) , rm^#(x, cons(y, xs)) -> c_14(if2^#(eq(x, y), x, y, xs), eq^#(x, y)) , if1^#(true(), x, y, xs) -> c_15(min^#(cons(x, xs))) , if1^#(false(), x, y, xs) -> c_16(min^#(cons(y, xs))) , if2^#(true(), x, y, xs) -> c_17(rm^#(x, xs)) , if2^#(false(), x, y, xs) -> c_18(rm^#(x, xs)) } Weak Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , eq(0(), 0()) -> true() , eq(0(), s(y)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , min(nil()) -> 0() , min(cons(x, nil())) -> x , min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) , rm(x, nil()) -> nil() , rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) , if1(true(), x, y, xs) -> min(cons(x, xs)) , if1(false(), x, y, xs) -> min(cons(y, xs)) , if2(true(), x, y, xs) -> rm(x, xs) , if2(false(), x, y, xs) -> cons(y, rm(x, xs)) } Obligation: innermost runtime complexity Answer: MAYBE The input cannot be shown compatible Arrrr..