WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^2)) + Considered Problem: - Strict TRS: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z))) 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z))) 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z))) activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(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,activate/1,from/1,pi/1,plus/2,square/1,times/2} / {0/0,cons/2,cons2/2,n__from/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,from,pi,plus,square ,times} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z))) 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z))) 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z))) activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(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,activate/1,from/1,pi/1,plus/2,square/1,times/2} / {0/0,cons/2,cons2/2,n__from/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,from,pi,plus,square ,times} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: plus(x,y){x -> s(x)} = plus(s(x),y) ->^+ s(plus(x,y)) = C[plus(x,y) = plus(x,y){}] ** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z))) 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z))) 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z))) activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(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,activate/1,from/1,pi/1,plus/2,square/1,times/2} / {0/0,cons/2,cons2/2,n__from/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,from,pi,plus,square ,times} and constructors {0,cons,cons2,n__from,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,activate(Z))),activate#(Z)) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)) 2ndspos#(0(),Z) -> c_4() 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)) activate#(X) -> c_7() activate#(n__from(X)) -> c_8(from#(X)) from#(X) -> c_9() from#(X) -> c_10() pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) plus#(0(),Y) -> c_12() plus#(s(X),Y) -> c_13(plus#(X,Y)) square#(X) -> c_14(times#(X,X)) times#(0(),Y) -> c_15() times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 2ndsneg#(0(),Z) -> c_1() 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)) 2ndspos#(0(),Z) -> c_4() 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)) activate#(X) -> c_7() activate#(n__from(X)) -> c_8(from#(X)) from#(X) -> c_9() from#(X) -> c_10() pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) plus#(0(),Y) -> c_12() plus#(s(X),Y) -> c_13(plus#(X,Y)) square#(X) -> c_14(times#(X,X)) times#(0(),Y) -> c_15() times#(s(X),Y) -> c_16(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,activate(Z))) 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z))) 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z))) activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(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,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/2,c_6/2,c_7/0,c_8/1,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,4,7,9,10,12,15} by application of Pre({1,4,7,9,10,12,15}) = {2,3,5,6,8,11,13,14,16}. 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,activate(Z))),activate#(Z)) 3: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)) 4: 2ndspos#(0(),Z) -> c_4() 5: 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)) 6: 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)) 7: activate#(X) -> c_7() 8: activate#(n__from(X)) -> c_8(from#(X)) 9: from#(X) -> c_9() 10: from#(X) -> c_10() 11: pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) 12: plus#(0(),Y) -> c_12() 13: plus#(s(X),Y) -> c_13(plus#(X,Y)) 14: square#(X) -> c_14(times#(X,X)) 15: times#(0(),Y) -> c_15() 16: times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) ** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)) activate#(n__from(X)) -> c_8(from#(X)) pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) plus#(s(X),Y) -> c_13(plus#(X,Y)) square#(X) -> c_14(times#(X,X)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak DPs: 2ndsneg#(0(),Z) -> c_1() 2ndspos#(0(),Z) -> c_4() activate#(X) -> c_7() from#(X) -> c_9() from#(X) -> c_10() plus#(0(),Y) -> c_12() times#(0(),Y) -> c_15() - Weak TRS: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z))) 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z))) 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z))) activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(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,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/2,c_6/2,c_7/0,c_8/1,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {5} by application of Pre({5}) = {1,2,3,4}. Here rules are labelled as follows: 1: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)) 2: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)) 3: 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)) 4: 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)) 5: activate#(n__from(X)) -> c_8(from#(X)) 6: pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) 7: plus#(s(X),Y) -> c_13(plus#(X,Y)) 8: square#(X) -> c_14(times#(X,X)) 9: times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) 10: 2ndsneg#(0(),Z) -> c_1() 11: 2ndspos#(0(),Z) -> c_4() 12: activate#(X) -> c_7() 13: from#(X) -> c_9() 14: from#(X) -> c_10() 15: plus#(0(),Y) -> c_12() 16: times#(0(),Y) -> c_15() ** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)) pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) plus#(s(X),Y) -> c_13(plus#(X,Y)) square#(X) -> c_14(times#(X,X)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak DPs: 2ndsneg#(0(),Z) -> c_1() 2ndspos#(0(),Z) -> c_4() activate#(X) -> c_7() activate#(n__from(X)) -> c_8(from#(X)) from#(X) -> c_9() from#(X) -> c_10() plus#(0(),Y) -> c_12() times#(0(),Y) -> c_15() - Weak TRS: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z))) 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z))) 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z))) activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(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,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/2,c_6/2,c_7/0,c_8/1,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,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,activate(Z))),activate#(Z)) -->_2 activate#(n__from(X)) -> c_8(from#(X)):12 -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)):2 -->_2 activate#(X) -> c_7():11 2:S:2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)) -->_2 activate#(n__from(X)) -> c_8(from#(X)):12 -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)):4 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)):3 -->_2 activate#(X) -> c_7():11 -->_1 2ndspos#(0(),Z) -> c_4():10 3:S:2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)) -->_2 activate#(n__from(X)) -> c_8(from#(X)):12 -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)):4 -->_2 activate#(X) -> c_7():11 4:S:2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)) -->_2 activate#(n__from(X)) -> c_8(from#(X)):12 -->_2 activate#(X) -> c_7():11 -->_1 2ndsneg#(0(),Z) -> c_1():9 -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)):2 -->_1 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)):1 5:S:pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) -->_2 from#(X) -> c_10():14 -->_2 from#(X) -> c_9():13 -->_1 2ndspos#(0(),Z) -> c_4():10 -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)):4 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)):3 6:S:plus#(s(X),Y) -> c_13(plus#(X,Y)) -->_1 plus#(0(),Y) -> c_12():15 -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):6 7:S:square#(X) -> c_14(times#(X,X)) -->_1 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):8 -->_1 times#(0(),Y) -> c_15():16 8:S:times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) -->_2 times#(0(),Y) -> c_15():16 -->_1 plus#(0(),Y) -> c_12():15 -->_2 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):8 -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):6 9:W:2ndsneg#(0(),Z) -> c_1() 10:W:2ndspos#(0(),Z) -> c_4() 11:W:activate#(X) -> c_7() 12:W:activate#(n__from(X)) -> c_8(from#(X)) -->_1 from#(X) -> c_10():14 -->_1 from#(X) -> c_9():13 13:W:from#(X) -> c_9() 14:W:from#(X) -> c_10() 15:W:plus#(0(),Y) -> c_12() 16:W:times#(0(),Y) -> c_15() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 16: times#(0(),Y) -> c_15() 15: plus#(0(),Y) -> c_12() 10: 2ndspos#(0(),Z) -> c_4() 9: 2ndsneg#(0(),Z) -> c_1() 11: activate#(X) -> c_7() 12: activate#(n__from(X)) -> c_8(from#(X)) 13: from#(X) -> c_9() 14: from#(X) -> c_10() ** Step 1.b:5: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)) pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) plus#(s(X),Y) -> c_13(plus#(X,Y)) square#(X) -> c_14(times#(X,X)) times#(s(X),Y) -> c_16(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,activate(Z))) 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z))) 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z))) activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(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,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/2,c_6/2,c_7/0,c_8/1,c_9/0,c_10/0,c_11/2,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)) -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)):2 2:S:2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)):4 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)):3 3:S:2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)):4 4:S:2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)) -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z)),activate#(Z)):2 -->_1 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z))),activate#(Z)):1 5:S:pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z)),activate#(Z)):4 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z))),activate#(Z)):3 6:S:plus#(s(X),Y) -> c_13(plus#(X,Y)) -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):6 7:S:square#(X) -> c_14(times#(X,X)) -->_1 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):8 8:S:times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) -->_2 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):8 -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):6 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) pi#(X) -> c_11(2ndspos#(X,from(0()))) ** Step 1.b:6: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(s(X),Y) -> c_13(plus#(X,Y)) square#(X) -> c_14(times#(X,X)) times#(s(X),Y) -> c_16(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,activate(Z))) 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z))) 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z))) activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(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,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,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_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(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#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(s(X),Y) -> c_13(plus#(X,Y)) square#(X) -> c_14(times#(X,X)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) ** Step 1.b:7: RemoveHeads WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(s(X),Y) -> c_13(plus#(X,Y)) square#(X) -> c_14(times#(X,X)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(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,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,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_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,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,activate(Z)))) -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))):2 2:S:2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))):3 3:S:2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4 4:S:2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))):2 -->_1 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))):1 5:S:pi#(X) -> c_11(2ndspos#(X,from(0()))) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))):3 6:S:plus#(s(X),Y) -> c_13(plus#(X,Y)) -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):6 7:S:square#(X) -> c_14(times#(X,X)) -->_1 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):8 8:S:times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) -->_2 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):8 -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):6 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). [(7,square#(X) -> c_14(times#(X,X)))] ** Step 1.b:8: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(s(X),Y) -> c_13(plus#(X,Y)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(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,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,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_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) pi#(X) -> c_11(2ndspos#(X,from(0()))) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) and a lower component plus#(s(X),Y) -> c_13(plus#(X,Y)) Further, following extension rules are added to the lower component. 2ndsneg#(s(N),cons(X,Z)) -> 2ndsneg#(s(N),cons2(X,activate(Z))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> 2ndspos#(N,activate(Z)) 2ndspos#(s(N),cons(X,Z)) -> 2ndspos#(s(N),cons2(X,activate(Z))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> 2ndsneg#(N,activate(Z)) pi#(X) -> 2ndspos#(X,from(0())) times#(s(X),Y) -> plus#(Y,times(X,Y)) times#(s(X),Y) -> times#(X,Y) *** Step 1.b:8.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) pi#(X) -> c_11(2ndspos#(X,from(0()))) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(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,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,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_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))):2 2:S:2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))):3 3:S:2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4 4:S:2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))):2 -->_1 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))):1 5:S:pi#(X) -> c_11(2ndspos#(X,from(0()))) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))):4 -->_1 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))):3 6:S:times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) -->_2 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):6 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: times#(s(X),Y) -> c_16(times#(X,Y)) *** Step 1.b:8.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) pi#(X) -> c_11(2ndspos#(X,from(0()))) times#(s(X),Y) -> c_16(times#(X,Y)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(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,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,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_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) pi#(X) -> c_11(2ndspos#(X,from(0()))) times#(s(X),Y) -> c_16(times#(X,Y)) *** Step 1.b:8.a:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) pi#(X) -> c_11(2ndspos#(X,from(0()))) times#(s(X),Y) -> c_16(times#(X,Y)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,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_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(cons2) = {2}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_11) = {1}, uargs(c_16) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [0] p(2ndspos) = [0] p(activate) = [1] x1 + [0] p(cons) = [1] x2 + [0] p(cons2) = [1] x2 + [0] p(from) = [5] p(n__from) = [5] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [0] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [2] p(square) = [0] p(times) = [0] p(2ndsneg#) = [1] x2 + [0] p(2ndspos#) = [1] x2 + [4] p(activate#) = [1] p(from#) = [0] p(pi#) = [0] p(plus#) = [0] p(square#) = [0] p(times#) = [5] x1 + [1] x2 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [1] x1 + [0] Following rules are strictly oriented: 2ndspos#(s(N),cons2(X,cons(Y,Z))) = [1] Z + [4] > [1] Z + [0] = c_6(2ndsneg#(N,activate(Z))) times#(s(X),Y) = [5] X + [1] Y + [10] > [5] X + [1] Y + [0] = c_16(times#(X,Y)) Following rules are (at-least) weakly oriented: 2ndsneg#(s(N),cons(X,Z)) = [1] Z + [0] >= [1] Z + [0] = c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) = [1] Z + [0] >= [1] Z + [4] = c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) = [1] Z + [4] >= [1] Z + [4] = c_5(2ndspos#(s(N),cons2(X,activate(Z)))) pi#(X) = [0] >= [9] = c_11(2ndspos#(X,from(0()))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__from(X)) = [5] >= [5] = from(X) from(X) = [5] >= [5] = cons(X,n__from(s(X))) from(X) = [5] >= [5] = n__from(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:8.a:4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) pi#(X) -> c_11(2ndspos#(X,from(0()))) - Weak DPs: 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) times#(s(X),Y) -> c_16(times#(X,Y)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,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_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(cons2) = {2}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_11) = {1}, uargs(c_16) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [0] p(2ndspos) = [0] p(activate) = [1] x1 + [0] p(cons) = [1] x2 + [0] p(cons2) = [1] x2 + [7] p(from) = [0] p(n__from) = [0] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [0] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [0] p(square) = [0] p(times) = [0] p(2ndsneg#) = [1] x2 + [7] p(2ndspos#) = [1] x2 + [0] p(activate#) = [1] p(from#) = [0] p(pi#) = [1] p(plus#) = [0] p(square#) = [0] p(times#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [1] x1 + [0] Following rules are strictly oriented: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) = [1] Z + [14] > [1] Z + [0] = c_3(2ndspos#(N,activate(Z))) pi#(X) = [1] > [0] = c_11(2ndspos#(X,from(0()))) Following rules are (at-least) weakly oriented: 2ndsneg#(s(N),cons(X,Z)) = [1] Z + [7] >= [1] Z + [14] = c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons(X,Z)) = [1] Z + [0] >= [1] Z + [7] = c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) = [1] Z + [7] >= [1] Z + [7] = c_6(2ndsneg#(N,activate(Z))) times#(s(X),Y) = [0] >= [0] = c_16(times#(X,Y)) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__from(X)) = [0] >= [0] = from(X) from(X) = [0] >= [0] = cons(X,n__from(s(X))) from(X) = [0] >= [0] = n__from(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:8.a:5: MI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) - Weak DPs: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) pi#(X) -> c_11(2ndspos#(X,from(0()))) times#(s(X),Y) -> c_16(times#(X,Y)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,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_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: MI {miKind = Automaton Nothing, miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind Automaton Nothing: 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_11) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {activate,from,2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#,times#} TcT has computed the following interpretation: p(0) = [1] [0] p(2ndsneg) = [0 0] x_1 + [0 1] x_2 + [0] [0 2] [1 1] [0] p(2ndspos) = [0 0] x_2 + [0] [0 1] [4] p(activate) = [1 0] x_1 + [1] [0 1] [0] p(cons) = [0 1] x_2 + [1] [1 0] [0] p(cons2) = [0 1] x_2 + [0] [1 0] [0] p(from) = [1] [0] p(n__from) = [0] [0] p(negrecip) = [1] [1] p(pi) = [1 1] x_1 + [0] [1 1] [1] p(plus) = [0 0] x_2 + [2] [0 1] [0] p(posrecip) = [0] [1] p(rcons) = [1] [1] p(rnil) = [4] [1] p(s) = [1 0] x_1 + [4] [0 0] [2] p(square) = [0 0] x_1 + [0] [0 1] [1] p(times) = [4 0] x_1 + [0 0] x_2 + [4] [1 4] [0 4] [0] p(2ndsneg#) = [2 0] x_1 + [4 0] x_2 + [0] [2 0] [6 0] [1] p(2ndspos#) = [2 0] x_1 + [4 0] x_2 + [0] [0 3] [4 3] [0] p(activate#) = [0 0] x_1 + [4] [0 1] [1] p(from#) = [1] [0] p(pi#) = [3 0] x_1 + [4] [4 0] [2] p(plus#) = [4] [1] p(square#) = [0 0] x_1 + [0] [1 4] [0] p(times#) = [0 0] x_1 + [0] [1 0] [1] p(c_1) = [1] [1] p(c_2) = [1 0] x_1 + [3] [1 0] [2] p(c_3) = [1 0] x_1 + [4] [0 0] [3] p(c_4) = [1] [2] p(c_5) = [1 0] x_1 + [3] [0 0] [0] p(c_6) = [1 0] x_1 + [3] [0 0] [1] p(c_7) = [1] [0] p(c_8) = [0 0] x_1 + [1] [0 1] [0] p(c_9) = [1] [0] p(c_10) = [0] [0] p(c_11) = [1 0] x_1 + [0] [0 0] [2] p(c_12) = [1] [4] p(c_13) = [0 0] x_1 + [1] [0 1] [1] p(c_14) = [2 1] x_1 + [2] [0 0] [1] p(c_15) = [1] [4] p(c_16) = [2 0] x_1 + [0] [1 1] [3] Following rules are strictly oriented: 2ndsneg#(s(N),cons(X,Z)) = [2 0] N + [0 4] Z + [12] [2 0] [0 6] [15] > [2 0] N + [0 4] Z + [11] [2 0] [0 4] [10] = c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons(X,Z)) = [2 0] N + [0 4] Z + [12] [0 0] [3 4] [10] > [2 0] N + [0 4] Z + [11] [0 0] [0 0] [0] = c_5(2ndspos#(s(N),cons2(X,activate(Z)))) Following rules are (at-least) weakly oriented: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) = [2 0] N + [4 0] Z + [8] [2 0] [6 0] [9] >= [2 0] N + [4 0] Z + [8] [0 0] [0 0] [3] = c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) = [2 0] N + [4 0] Z + [8] [0 0] [4 3] [9] >= [2 0] N + [4 0] Z + [7] [0 0] [0 0] [1] = c_6(2ndsneg#(N,activate(Z))) pi#(X) = [3 0] X + [4] [4 0] [2] >= [2 0] X + [4] [0 0] [2] = c_11(2ndspos#(X,from(0()))) times#(s(X),Y) = [0 0] X + [0] [1 0] [5] >= [0 0] X + [0] [1 0] [4] = c_16(times#(X,Y)) activate(X) = [1 0] X + [1] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X activate(n__from(X)) = [1] [0] >= [1] [0] = from(X) from(X) = [1] [0] >= [1] [0] = cons(X,n__from(s(X))) from(X) = [1] [0] >= [0] [0] = n__from(X) *** Step 1.b:8.a:6: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: 2ndsneg#(s(N),cons(X,Z)) -> c_2(2ndsneg#(s(N),cons2(X,activate(Z)))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> c_3(2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,Z)) -> c_5(2ndspos#(s(N),cons2(X,activate(Z)))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> c_6(2ndsneg#(N,activate(Z))) pi#(X) -> c_11(2ndspos#(X,from(0()))) times#(s(X),Y) -> c_16(times#(X,Y)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {2ndsneg/2,2ndspos/2,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,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_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 1.b:8.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: plus#(s(X),Y) -> c_13(plus#(X,Y)) - Weak DPs: 2ndsneg#(s(N),cons(X,Z)) -> 2ndsneg#(s(N),cons2(X,activate(Z))) 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> 2ndspos#(N,activate(Z)) 2ndspos#(s(N),cons(X,Z)) -> 2ndspos#(s(N),cons2(X,activate(Z))) 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> 2ndsneg#(N,activate(Z)) pi#(X) -> 2ndspos#(X,from(0())) times#(s(X),Y) -> plus#(Y,times(X,Y)) times#(s(X),Y) -> times#(X,Y) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(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,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,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_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:plus#(s(X),Y) -> c_13(plus#(X,Y)) -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):1 2:W:2ndsneg#(s(N),cons(X,Z)) -> 2ndsneg#(s(N),cons2(X,activate(Z))) -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> 2ndspos#(N,activate(Z)):3 3:W:2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> 2ndspos#(N,activate(Z)) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> 2ndsneg#(N,activate(Z)):5 -->_1 2ndspos#(s(N),cons(X,Z)) -> 2ndspos#(s(N),cons2(X,activate(Z))):4 4:W:2ndspos#(s(N),cons(X,Z)) -> 2ndspos#(s(N),cons2(X,activate(Z))) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> 2ndsneg#(N,activate(Z)):5 5:W:2ndspos#(s(N),cons2(X,cons(Y,Z))) -> 2ndsneg#(N,activate(Z)) -->_1 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> 2ndspos#(N,activate(Z)):3 -->_1 2ndsneg#(s(N),cons(X,Z)) -> 2ndsneg#(s(N),cons2(X,activate(Z))):2 6:W:pi#(X) -> 2ndspos#(X,from(0())) -->_1 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> 2ndsneg#(N,activate(Z)):5 -->_1 2ndspos#(s(N),cons(X,Z)) -> 2ndspos#(s(N),cons2(X,activate(Z))):4 7:W:times#(s(X),Y) -> plus#(Y,times(X,Y)) -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):1 8:W:times#(s(X),Y) -> times#(X,Y) -->_1 times#(s(X),Y) -> times#(X,Y):8 -->_1 times#(s(X),Y) -> plus#(Y,times(X,Y)):7 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: pi#(X) -> 2ndspos#(X,from(0())) 2: 2ndsneg#(s(N),cons(X,Z)) -> 2ndsneg#(s(N),cons2(X,activate(Z))) 5: 2ndspos#(s(N),cons2(X,cons(Y,Z))) -> 2ndsneg#(N,activate(Z)) 4: 2ndspos#(s(N),cons(X,Z)) -> 2ndspos#(s(N),cons2(X,activate(Z))) 3: 2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> 2ndspos#(N,activate(Z)) *** Step 1.b:8.b:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: plus#(s(X),Y) -> c_13(plus#(X,Y)) - Weak DPs: times#(s(X),Y) -> plus#(Y,times(X,Y)) times#(s(X),Y) -> times#(X,Y) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(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,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,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_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: 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)) plus#(s(X),Y) -> c_13(plus#(X,Y)) times#(s(X),Y) -> plus#(Y,times(X,Y)) times#(s(X),Y) -> times#(X,Y) *** Step 1.b:8.b:3: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: plus#(s(X),Y) -> c_13(plus#(X,Y)) - Weak DPs: times#(s(X),Y) -> plus#(Y,times(X,Y)) times#(s(X),Y) -> times#(X,Y) - Weak TRS: 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,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,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_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(c_13) = {1} Following symbols are considered usable: {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square#,times#} TcT has computed the following interpretation: p(0) = 0 p(2ndsneg) = 2*x1 p(2ndspos) = 8 + x2 p(activate) = 4 p(cons) = 0 p(cons2) = 1 + x1 p(from) = 0 p(n__from) = x1 p(negrecip) = x1 p(pi) = 2*x1 p(plus) = 1 + x2 p(posrecip) = 1 + x1 p(rcons) = 1 p(rnil) = 1 p(s) = 4 + x1 p(square) = 1 p(times) = 8 + x1 + 4*x2 p(2ndsneg#) = 8 + x2 p(2ndspos#) = 1 + x2 p(activate#) = 2 + x1 p(from#) = 2 + 2*x1 p(pi#) = 2 + x1 p(plus#) = 2 + 4*x1 p(square#) = 1 p(times#) = 6 + 5*x2 p(c_1) = 0 p(c_2) = 4 p(c_3) = 2 p(c_4) = 0 p(c_5) = 1 p(c_6) = 0 p(c_7) = 1 p(c_8) = 1 p(c_9) = 8 p(c_10) = 2 p(c_11) = 2 + x1 p(c_12) = 0 p(c_13) = 6 + x1 p(c_14) = 2 p(c_15) = 2 p(c_16) = x1 Following rules are strictly oriented: plus#(s(X),Y) = 18 + 4*X > 8 + 4*X = c_13(plus#(X,Y)) Following rules are (at-least) weakly oriented: times#(s(X),Y) = 6 + 5*Y >= 2 + 4*Y = plus#(Y,times(X,Y)) times#(s(X),Y) = 6 + 5*Y >= 6 + 5*Y = times#(X,Y) *** Step 1.b:8.b:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: plus#(s(X),Y) -> c_13(plus#(X,Y)) times#(s(X),Y) -> plus#(Y,times(X,Y)) times#(s(X),Y) -> times#(X,Y) - Weak TRS: 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,activate/1,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,cons/2,cons2/2,n__from/1,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_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0 ,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,from#,pi#,plus#,square# ,times#} and constructors {0,cons,cons2,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^2))