MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,Z)) 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,Z)) 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z)) from(X) -> cons(X,from(s(X))) pi(X) -> 2ndspos(X,from(0())) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) square(X) -> times(X,X) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,from/1,pi/1,plus/2,square/1,times/2} / {0/0,cons/2,cons2/2,negrecip/1,posrecip/1 ,rcons/2,rnil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,from,pi,plus,square ,times} and constructors {0,cons,cons2,negrecip,posrecip,rcons,rnil,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs 2ndsneg#(0(),Z) -> c_1() 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,Z))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,Z)) 2ndspos#(0(),Z) -> c_4() 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,Z))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,Z)) from#(X) -> c_7(from#(s(X))) pi#(X) -> c_8(2ndspos#(X,from(0())),from#(0())) plus#(0(),Y) -> c_9() plus#(s(X),Y) -> c_10(plus#(X,Y)) square#(X) -> c_11(times#(X,X)) times#(0(),Y) -> c_12() times#(s(X),Y) -> c_13(plus#(Y,times(X,Y)),times#(X,Y)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: 2ndsneg#(0(),Z) -> c_1() 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,Z))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,Z)) 2ndspos#(0(),Z) -> c_4() 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,Z))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,Z)) from#(X) -> c_7(from#(s(X))) pi#(X) -> c_8(2ndspos#(X,from(0())),from#(0())) plus#(0(),Y) -> c_9() plus#(s(X),Y) -> c_10(plus#(X,Y)) square#(X) -> c_11(times#(X,X)) times#(0(),Y) -> c_12() times#(s(X),Y) -> c_13(plus#(Y,times(X,Y)),times#(X,Y)) - Weak TRS: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,Z)) 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z)) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,Z)) 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z)) from(X) -> cons(X,from(s(X))) pi(X) -> 2ndspos(X,from(0())) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) square(X) -> times(X,X) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,from#/1,pi#/1,plus#/2 ,square#/1,times#/2} / {0/0,cons/2,cons2/2,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/1,c_6/1,c_7/1,c_8/2,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,negrecip,posrecip,rcons,rnil,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: from(X) -> cons(X,from(s(X))) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) 2ndsneg#(0(),Z) -> c_1() 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,Z))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,Z)) 2ndspos#(0(),Z) -> c_4() 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,Z))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,Z)) from#(X) -> c_7(from#(s(X))) pi#(X) -> c_8(2ndspos#(X,from(0())),from#(0())) plus#(0(),Y) -> c_9() plus#(s(X),Y) -> c_10(plus#(X,Y)) square#(X) -> c_11(times#(X,X)) times#(0(),Y) -> c_12() times#(s(X),Y) -> c_13(plus#(Y,times(X,Y)),times#(X,Y)) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: 2ndsneg#(0(),Z) -> c_1() 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,Z))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,Z)) 2ndspos#(0(),Z) -> c_4() 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,Z))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,Z)) from#(X) -> c_7(from#(s(X))) pi#(X) -> c_8(2ndspos#(X,from(0())),from#(0())) plus#(0(),Y) -> c_9() plus#(s(X),Y) -> c_10(plus#(X,Y)) square#(X) -> c_11(times#(X,X)) times#(0(),Y) -> c_12() times#(s(X),Y) -> c_13(plus#(Y,times(X,Y)),times#(X,Y)) - Weak TRS: from(X) -> cons(X,from(s(X))) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,from#/1,pi#/1,plus#/2 ,square#/1,times#/2} / {0/0,cons/2,cons2/2,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/1,c_6/1,c_7/1,c_8/2,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,negrecip,posrecip,rcons,rnil,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,4,9,12} by application of Pre({1,4,9,12}) = {3,6,8,10,11,13}. Here rules are labelled as follows: 1: 2ndsneg#(0(),Z) -> c_1() 2: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,Z))) 3: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,Z)) 4: 2ndspos#(0(),Z) -> c_4() 5: 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,Z))) 6: 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,Z)) 7: from#(X) -> c_7(from#(s(X))) 8: pi#(X) -> c_8(2ndspos#(X,from(0())),from#(0())) 9: plus#(0(),Y) -> c_9() 10: plus#(s(X),Y) -> c_10(plus#(X,Y)) 11: square#(X) -> c_11(times#(X,X)) 12: times#(0(),Y) -> c_12() 13: times#(s(X),Y) -> c_13(plus#(Y,times(X,Y)),times#(X,Y)) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,Z))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,Z)) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,Z))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,Z)) from#(X) -> c_7(from#(s(X))) pi#(X) -> c_8(2ndspos#(X,from(0())),from#(0())) plus#(s(X),Y) -> c_10(plus#(X,Y)) square#(X) -> c_11(times#(X,X)) times#(s(X),Y) -> c_13(plus#(Y,times(X,Y)),times#(X,Y)) - Weak DPs: 2ndsneg#(0(),Z) -> c_1() 2ndspos#(0(),Z) -> c_4() plus#(0(),Y) -> c_9() times#(0(),Y) -> c_12() - Weak TRS: from(X) -> cons(X,from(s(X))) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,from#/1,pi#/1,plus#/2 ,square#/1,times#/2} / {0/0,cons/2,cons2/2,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/1,c_6/1,c_7/1,c_8/2,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,negrecip,posrecip,rcons,rnil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,Z))) -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,Z)):2 2:S:2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,Z)) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,Z)):4 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,Z))):3 -->_1 2ndspos#(0(),Z) -> c_4():11 3:S:2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,Z))) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,Z)):4 4:S:2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,Z)) -->_1 2ndsneg#(0(),Z) -> c_1():10 -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,Z)):2 -->_1 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,Z))):1 5:S:from#(X) -> c_7(from#(s(X))) -->_1 from#(X) -> c_7(from#(s(X))):5 6:S:pi#(X) -> c_8(2ndspos#(X,from(0())),from#(0())) -->_1 2ndspos#(0(),Z) -> c_4():11 -->_2 from#(X) -> c_7(from#(s(X))):5 -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,Z)):4 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,Z))):3 7:S:plus#(s(X),Y) -> c_10(plus#(X,Y)) -->_1 plus#(0(),Y) -> c_9():12 -->_1 plus#(s(X),Y) -> c_10(plus#(X,Y)):7 8:S:square#(X) -> c_11(times#(X,X)) -->_1 times#(s(X),Y) -> c_13(plus#(Y,times(X,Y)),times#(X,Y)):9 -->_1 times#(0(),Y) -> c_12():13 9:S:times#(s(X),Y) -> c_13(plus#(Y,times(X,Y)),times#(X,Y)) -->_2 times#(0(),Y) -> c_12():13 -->_1 plus#(0(),Y) -> c_9():12 -->_2 times#(s(X),Y) -> c_13(plus#(Y,times(X,Y)),times#(X,Y)):9 -->_1 plus#(s(X),Y) -> c_10(plus#(X,Y)):7 10:W:2ndsneg#(0(),Z) -> c_1() 11:W:2ndspos#(0(),Z) -> c_4() 12:W:plus#(0(),Y) -> c_9() 13:W:times#(0(),Y) -> c_12() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 13: times#(0(),Y) -> c_12() 12: plus#(0(),Y) -> c_9() 11: 2ndspos#(0(),Z) -> c_4() 10: 2ndsneg#(0(),Z) -> c_1() * Step 5: RemoveHeads MAYBE + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,Z))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,Z)) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,Z))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,Z)) from#(X) -> c_7(from#(s(X))) pi#(X) -> c_8(2ndspos#(X,from(0())),from#(0())) plus#(s(X),Y) -> c_10(plus#(X,Y)) square#(X) -> c_11(times#(X,X)) times#(s(X),Y) -> c_13(plus#(Y,times(X,Y)),times#(X,Y)) - Weak TRS: from(X) -> cons(X,from(s(X))) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,from#/1,pi#/1,plus#/2 ,square#/1,times#/2} / {0/0,cons/2,cons2/2,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/1,c_6/1,c_7/1,c_8/2,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,negrecip,posrecip,rcons,rnil,s} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,Z))) -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,Z)):2 2:S:2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,Z)) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,Z)):4 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,Z))):3 3:S:2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,Z))) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,Z)):4 4:S:2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,Z)) -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,Z)):2 -->_1 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,Z))):1 5:S:from#(X) -> c_7(from#(s(X))) -->_1 from#(X) -> c_7(from#(s(X))):5 6:S:pi#(X) -> c_8(2ndspos#(X,from(0())),from#(0())) -->_2 from#(X) -> c_7(from#(s(X))):5 -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,Z)):4 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,Z))):3 7:S:plus#(s(X),Y) -> c_10(plus#(X,Y)) -->_1 plus#(s(X),Y) -> c_10(plus#(X,Y)):7 8:S:square#(X) -> c_11(times#(X,X)) -->_1 times#(s(X),Y) -> c_13(plus#(Y,times(X,Y)),times#(X,Y)):9 9:S:times#(s(X),Y) -> c_13(plus#(Y,times(X,Y)),times#(X,Y)) -->_2 times#(s(X),Y) -> c_13(plus#(Y,times(X,Y)),times#(X,Y)):9 -->_1 plus#(s(X),Y) -> c_10(plus#(X,Y)):7 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(8,square#(X) -> c_11(times#(X,X)))] * Step 6: NaturalMI MAYBE + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,Z))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,Z)) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,Z))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,Z)) from#(X) -> c_7(from#(s(X))) pi#(X) -> c_8(2ndspos#(X,from(0())),from#(0())) plus#(s(X),Y) -> c_10(plus#(X,Y)) times#(s(X),Y) -> c_13(plus#(Y,times(X,Y)),times#(X,Y)) - Weak TRS: from(X) -> cons(X,from(s(X))) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,from#/1,pi#/1,plus#/2 ,square#/1,times#/2} / {0/0,cons/2,cons2/2,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/1,c_6/1,c_7/1,c_8/2,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,negrecip,posrecip,rcons,rnil,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1,2}, uargs(c_10) = {1}, uargs(c_13) = {1,2} Following symbols are considered usable: {2ndsneg#,2ndspos#,from#,pi#,plus#,square#,times#} TcT has computed the following interpretation: p(0) = [1] p(2ndsneg) = [2] x1 + [1] p(2ndspos) = [1] x1 + [2] p(cons) = [1] p(cons2) = [0] p(from) = [1] p(negrecip) = [1] p(pi) = [1] x1 + [2] p(plus) = [4] p(posrecip) = [2] p(rcons) = [0] p(rnil) = [1] p(s) = [0] p(square) = [4] x1 + [0] p(times) = [1] x2 + [10] p(2ndsneg#) = [0] p(2ndspos#) = [0] p(from#) = [2] x1 + [0] p(pi#) = [1] x1 + [10] p(plus#) = [0] p(square#) = [1] x1 + [1] p(times#) = [0] p(c_1) = [1] p(c_2) = [1] x1 + [0] p(c_3) = [8] x1 + [0] p(c_4) = [2] p(c_5) = [8] x1 + [0] p(c_6) = [4] x1 + [0] p(c_7) = [8] x1 + [0] p(c_8) = [1] x1 + [1] x2 + [0] p(c_9) = [0] p(c_10) = [4] x1 + [0] p(c_11) = [8] x1 + [0] p(c_12) = [1] p(c_13) = [1] x1 + [8] x2 + [0] Following rules are strictly oriented: pi#(X) = [1] X + [10] > [2] = c_8(2ndspos#(X,from(0())),from#(0())) Following rules are (at-least) weakly oriented: 2ndsneg#(s(N),cons(X,Z)) = [0] >= [0] = c_2(2ndsneg#(s(N),cons2(X,Z))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) = [0] >= [0] = c_3(2ndspos#(N,Z)) 2ndspos#(s(N),cons(X,Z)) = [0] >= [0] = c_5(2ndspos#(s(N),cons2(X,Z))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) = [0] >= [0] = c_6(2ndsneg#(N,Z)) from#(X) = [2] X + [0] >= [0] = c_7(from#(s(X))) plus#(s(X),Y) = [0] >= [0] = c_10(plus#(X,Y)) times#(s(X),Y) = [0] >= [0] = c_13(plus#(Y,times(X,Y)),times#(X,Y)) * Step 7: Failure MAYBE + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,Z))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,Z)) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,Z))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,Z)) from#(X) -> c_7(from#(s(X))) plus#(s(X),Y) -> c_10(plus#(X,Y)) times#(s(X),Y) -> c_13(plus#(Y,times(X,Y)),times#(X,Y)) - Weak DPs: pi#(X) -> c_8(2ndspos#(X,from(0())),from#(0())) - Weak TRS: from(X) -> cons(X,from(s(X))) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,from#/1,pi#/1,plus#/2 ,square#/1,times#/2} / {0/0,cons/2,cons2/2,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0 ,c_5/1,c_6/1,c_7/1,c_8/2,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,negrecip,posrecip,rcons,rnil,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE