MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { 0(#()) -> #() , +(x, #()) -> x , +(0(x), 0(y)) -> 0(+(x, y)) , +(0(x), 1(y)) -> 1(+(x, y)) , +(#(), x) -> x , +(1(x), 0(y)) -> 1(+(x, y)) , +(1(x), 1(y)) -> 0(+(+(x, y), 1(#()))) , *(0(x), y) -> 0(*(x, y)) , *(#(), x) -> #() , *(1(x), y) -> +(0(*(x, y)), y) , sum(nil()) -> 0(#()) , sum(cons(x, l)) -> +(x, sum(l)) , prod(nil()) -> 1(#()) , prod(cons(x, l)) -> *(x, prod(l)) } 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 minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with perSymbol-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) '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 2) '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 3) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { 0^#(#()) -> c_1() , +^#(x, #()) -> c_2(x) , +^#(0(x), 0(y)) -> c_3(0^#(+(x, y))) , +^#(0(x), 1(y)) -> c_4(+^#(x, y)) , +^#(#(), x) -> c_5(x) , +^#(1(x), 0(y)) -> c_6(+^#(x, y)) , +^#(1(x), 1(y)) -> c_7(0^#(+(+(x, y), 1(#())))) , *^#(0(x), y) -> c_8(0^#(*(x, y))) , *^#(#(), x) -> c_9() , *^#(1(x), y) -> c_10(+^#(0(*(x, y)), y)) , sum^#(nil()) -> c_11(0^#(#())) , sum^#(cons(x, l)) -> c_12(+^#(x, sum(l))) , prod^#(nil()) -> c_13() , prod^#(cons(x, l)) -> c_14(*^#(x, prod(l))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { 0^#(#()) -> c_1() , +^#(x, #()) -> c_2(x) , +^#(0(x), 0(y)) -> c_3(0^#(+(x, y))) , +^#(0(x), 1(y)) -> c_4(+^#(x, y)) , +^#(#(), x) -> c_5(x) , +^#(1(x), 0(y)) -> c_6(+^#(x, y)) , +^#(1(x), 1(y)) -> c_7(0^#(+(+(x, y), 1(#())))) , *^#(0(x), y) -> c_8(0^#(*(x, y))) , *^#(#(), x) -> c_9() , *^#(1(x), y) -> c_10(+^#(0(*(x, y)), y)) , sum^#(nil()) -> c_11(0^#(#())) , sum^#(cons(x, l)) -> c_12(+^#(x, sum(l))) , prod^#(nil()) -> c_13() , prod^#(cons(x, l)) -> c_14(*^#(x, prod(l))) } Strict Trs: { 0(#()) -> #() , +(x, #()) -> x , +(0(x), 0(y)) -> 0(+(x, y)) , +(0(x), 1(y)) -> 1(+(x, y)) , +(#(), x) -> x , +(1(x), 0(y)) -> 1(+(x, y)) , +(1(x), 1(y)) -> 0(+(+(x, y), 1(#()))) , *(0(x), y) -> 0(*(x, y)) , *(#(), x) -> #() , *(1(x), y) -> +(0(*(x, y)), y) , sum(nil()) -> 0(#()) , sum(cons(x, l)) -> +(x, sum(l)) , prod(nil()) -> 1(#()) , prod(cons(x, l)) -> *(x, prod(l)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,9,13} by applications of Pre({1,9,13}) = {2,3,5,7,8,11,14}. Here rules are labeled as follows: DPs: { 1: 0^#(#()) -> c_1() , 2: +^#(x, #()) -> c_2(x) , 3: +^#(0(x), 0(y)) -> c_3(0^#(+(x, y))) , 4: +^#(0(x), 1(y)) -> c_4(+^#(x, y)) , 5: +^#(#(), x) -> c_5(x) , 6: +^#(1(x), 0(y)) -> c_6(+^#(x, y)) , 7: +^#(1(x), 1(y)) -> c_7(0^#(+(+(x, y), 1(#())))) , 8: *^#(0(x), y) -> c_8(0^#(*(x, y))) , 9: *^#(#(), x) -> c_9() , 10: *^#(1(x), y) -> c_10(+^#(0(*(x, y)), y)) , 11: sum^#(nil()) -> c_11(0^#(#())) , 12: sum^#(cons(x, l)) -> c_12(+^#(x, sum(l))) , 13: prod^#(nil()) -> c_13() , 14: prod^#(cons(x, l)) -> c_14(*^#(x, prod(l))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { +^#(x, #()) -> c_2(x) , +^#(0(x), 0(y)) -> c_3(0^#(+(x, y))) , +^#(0(x), 1(y)) -> c_4(+^#(x, y)) , +^#(#(), x) -> c_5(x) , +^#(1(x), 0(y)) -> c_6(+^#(x, y)) , +^#(1(x), 1(y)) -> c_7(0^#(+(+(x, y), 1(#())))) , *^#(0(x), y) -> c_8(0^#(*(x, y))) , *^#(1(x), y) -> c_10(+^#(0(*(x, y)), y)) , sum^#(nil()) -> c_11(0^#(#())) , sum^#(cons(x, l)) -> c_12(+^#(x, sum(l))) , prod^#(cons(x, l)) -> c_14(*^#(x, prod(l))) } Strict Trs: { 0(#()) -> #() , +(x, #()) -> x , +(0(x), 0(y)) -> 0(+(x, y)) , +(0(x), 1(y)) -> 1(+(x, y)) , +(#(), x) -> x , +(1(x), 0(y)) -> 1(+(x, y)) , +(1(x), 1(y)) -> 0(+(+(x, y), 1(#()))) , *(0(x), y) -> 0(*(x, y)) , *(#(), x) -> #() , *(1(x), y) -> +(0(*(x, y)), y) , sum(nil()) -> 0(#()) , sum(cons(x, l)) -> +(x, sum(l)) , prod(nil()) -> 1(#()) , prod(cons(x, l)) -> *(x, prod(l)) } Weak DPs: { 0^#(#()) -> c_1() , *^#(#(), x) -> c_9() , prod^#(nil()) -> c_13() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {2,6,7,9} by applications of Pre({2,6,7,9}) = {1,3,4,5,8,10,11}. Here rules are labeled as follows: DPs: { 1: +^#(x, #()) -> c_2(x) , 2: +^#(0(x), 0(y)) -> c_3(0^#(+(x, y))) , 3: +^#(0(x), 1(y)) -> c_4(+^#(x, y)) , 4: +^#(#(), x) -> c_5(x) , 5: +^#(1(x), 0(y)) -> c_6(+^#(x, y)) , 6: +^#(1(x), 1(y)) -> c_7(0^#(+(+(x, y), 1(#())))) , 7: *^#(0(x), y) -> c_8(0^#(*(x, y))) , 8: *^#(1(x), y) -> c_10(+^#(0(*(x, y)), y)) , 9: sum^#(nil()) -> c_11(0^#(#())) , 10: sum^#(cons(x, l)) -> c_12(+^#(x, sum(l))) , 11: prod^#(cons(x, l)) -> c_14(*^#(x, prod(l))) , 12: 0^#(#()) -> c_1() , 13: *^#(#(), x) -> c_9() , 14: prod^#(nil()) -> c_13() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { +^#(x, #()) -> c_2(x) , +^#(0(x), 1(y)) -> c_4(+^#(x, y)) , +^#(#(), x) -> c_5(x) , +^#(1(x), 0(y)) -> c_6(+^#(x, y)) , *^#(1(x), y) -> c_10(+^#(0(*(x, y)), y)) , sum^#(cons(x, l)) -> c_12(+^#(x, sum(l))) , prod^#(cons(x, l)) -> c_14(*^#(x, prod(l))) } Strict Trs: { 0(#()) -> #() , +(x, #()) -> x , +(0(x), 0(y)) -> 0(+(x, y)) , +(0(x), 1(y)) -> 1(+(x, y)) , +(#(), x) -> x , +(1(x), 0(y)) -> 1(+(x, y)) , +(1(x), 1(y)) -> 0(+(+(x, y), 1(#()))) , *(0(x), y) -> 0(*(x, y)) , *(#(), x) -> #() , *(1(x), y) -> +(0(*(x, y)), y) , sum(nil()) -> 0(#()) , sum(cons(x, l)) -> +(x, sum(l)) , prod(nil()) -> 1(#()) , prod(cons(x, l)) -> *(x, prod(l)) } Weak DPs: { 0^#(#()) -> c_1() , +^#(0(x), 0(y)) -> c_3(0^#(+(x, y))) , +^#(1(x), 1(y)) -> c_7(0^#(+(+(x, y), 1(#())))) , *^#(0(x), y) -> c_8(0^#(*(x, y))) , *^#(#(), x) -> c_9() , sum^#(nil()) -> c_11(0^#(#())) , prod^#(nil()) -> c_13() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..