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,n__cons(Y,Z))) -> rcons(negrecip(activate(Y)),2ndspos(N,activate(Z))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,n__cons(Y,Z))) -> rcons(posrecip(activate(Y)),2ndsneg(N,activate(Z))) activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) cons(X1,X2) -> n__cons(X1,X2) 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,cons/2,from/1,pi/1,plus/2,square/1,times/2} / {0/0,n__cons/2,n__from/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,square ,times} and constructors {0,n__cons,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,n__cons(Y,Z))) -> rcons(negrecip(activate(Y)),2ndspos(N,activate(Z))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,n__cons(Y,Z))) -> rcons(posrecip(activate(Y)),2ndsneg(N,activate(Z))) activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) cons(X1,X2) -> n__cons(X1,X2) 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,cons/2,from/1,pi/1,plus/2,square/1,times/2} / {0/0,n__cons/2,n__from/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,square ,times} and constructors {0,n__cons,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,n__cons(Y,Z))) -> rcons(negrecip(activate(Y)),2ndspos(N,activate(Z))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,n__cons(Y,Z))) -> rcons(posrecip(activate(Y)),2ndsneg(N,activate(Z))) activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) cons(X1,X2) -> n__cons(X1,X2) 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,cons/2,from/1,pi/1,plus/2,square/1,times/2} / {0/0,n__cons/2,n__from/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,square ,times} and constructors {0,n__cons,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,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)),activate#(Z)) 2ndspos#(0(),Z) -> c_3() 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)),activate#(Z)) activate#(X) -> c_5() activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8() from#(X) -> c_9(cons#(X,n__from(s(X)))) 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,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)),activate#(Z)) 2ndspos#(0(),Z) -> c_3() 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)),activate#(Z)) activate#(X) -> c_5() activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8() from#(X) -> c_9(cons#(X,n__from(s(X)))) 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,n__cons(Y,Z))) -> rcons(negrecip(activate(Y)),2ndspos(N,activate(Z))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,n__cons(Y,Z))) -> rcons(posrecip(activate(Y)),2ndsneg(N,activate(Z))) activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) cons(X1,X2) -> n__cons(X1,X2) 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,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/3,c_3/0,c_4/3,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,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#,cons#,from#,pi#,plus#,square# ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,4,5,8,10,12,15} by application of Pre({1,2,3,4,5,8,10,12,15}) = {6,7,9,11,13,14,16}. Here rules are labelled as follows: 1: 2ndsneg#(0(),Z) -> c_1() 2: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)),activate#(Z)) 3: 2ndspos#(0(),Z) -> c_3() 4: 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)),activate#(Z)) 5: activate#(X) -> c_5() 6: activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) 7: activate#(n__from(X)) -> c_7(from#(X)) 8: cons#(X1,X2) -> c_8() 9: from#(X) -> c_9(cons#(X,n__from(s(X)))) 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: activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) from#(X) -> c_9(cons#(X,n__from(s(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() 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)),activate#(Z)) 2ndspos#(0(),Z) -> c_3() 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)),activate#(Z)) activate#(X) -> c_5() cons#(X1,X2) -> c_8() 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,n__cons(Y,Z))) -> rcons(negrecip(activate(Y)),2ndspos(N,activate(Z))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,n__cons(Y,Z))) -> rcons(posrecip(activate(Y)),2ndsneg(N,activate(Z))) activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) cons(X1,X2) -> n__cons(X1,X2) 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,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/3,c_3/0,c_4/3,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,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#,cons#,from#,pi#,plus#,square# ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3} by application of Pre({1,3}) = {2,4}. Here rules are labelled as follows: 1: activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) 2: activate#(n__from(X)) -> c_7(from#(X)) 3: from#(X) -> c_9(cons#(X,n__from(s(X)))) 4: pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) 5: plus#(s(X),Y) -> c_13(plus#(X,Y)) 6: square#(X) -> c_14(times#(X,X)) 7: times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) 8: 2ndsneg#(0(),Z) -> c_1() 9: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)),activate#(Z)) 10: 2ndspos#(0(),Z) -> c_3() 11: 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)),activate#(Z)) 12: activate#(X) -> c_5() 13: cons#(X1,X2) -> c_8() 14: from#(X) -> c_10() 15: plus#(0(),Y) -> c_12() 16: times#(0(),Y) -> c_15() ** Step 1.b:4: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: activate#(n__from(X)) -> c_7(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() 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)),activate#(Z)) 2ndspos#(0(),Z) -> c_3() 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)),activate#(Z)) activate#(X) -> c_5() activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) cons#(X1,X2) -> c_8() from#(X) -> c_9(cons#(X,n__from(s(X)))) 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,n__cons(Y,Z))) -> rcons(negrecip(activate(Y)),2ndspos(N,activate(Z))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,n__cons(Y,Z))) -> rcons(posrecip(activate(Y)),2ndsneg(N,activate(Z))) activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) cons(X1,X2) -> n__cons(X1,X2) 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,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/3,c_3/0,c_4/3,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,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#,cons#,from#,pi#,plus#,square# ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2} by application of Pre({1,2}) = {}. Here rules are labelled as follows: 1: activate#(n__from(X)) -> c_7(from#(X)) 2: pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) 3: plus#(s(X),Y) -> c_13(plus#(X,Y)) 4: square#(X) -> c_14(times#(X,X)) 5: times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) 6: 2ndsneg#(0(),Z) -> c_1() 7: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)),activate#(Z)) 8: 2ndspos#(0(),Z) -> c_3() 9: 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)),activate#(Z)) 10: activate#(X) -> c_5() 11: activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) 12: cons#(X1,X2) -> c_8() 13: from#(X) -> c_9(cons#(X,n__from(s(X)))) 14: from#(X) -> c_10() 15: plus#(0(),Y) -> c_12() 16: times#(0(),Y) -> c_15() ** Step 1.b:5: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 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() 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)),activate#(Z)) 2ndspos#(0(),Z) -> c_3() 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)),activate#(Z)) activate#(X) -> c_5() activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8() from#(X) -> c_9(cons#(X,n__from(s(X)))) from#(X) -> c_10() pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) plus#(0(),Y) -> c_12() times#(0(),Y) -> c_15() - Weak TRS: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,n__cons(Y,Z))) -> rcons(negrecip(activate(Y)),2ndspos(N,activate(Z))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,n__cons(Y,Z))) -> rcons(posrecip(activate(Y)),2ndsneg(N,activate(Z))) activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) cons(X1,X2) -> n__cons(X1,X2) 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,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/3,c_3/0,c_4/3,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,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#,cons#,from#,pi#,plus#,square# ,times#} and constructors {0,n__cons,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#(0(),Y) -> c_12():15 -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):1 2:S:square#(X) -> c_14(times#(X,X)) -->_1 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):3 -->_1 times#(0(),Y) -> c_15():16 3: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)):3 -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):1 4:W:2ndsneg#(0(),Z) -> c_1() 5:W:2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)),activate#(Z)) 6:W:2ndspos#(0(),Z) -> c_3() 7:W:2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)),activate#(Z)) 8:W:activate#(X) -> c_5() 9:W:activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) -->_1 cons#(X1,X2) -> c_8():11 10:W:activate#(n__from(X)) -> c_7(from#(X)) -->_1 from#(X) -> c_9(cons#(X,n__from(s(X)))):12 -->_1 from#(X) -> c_10():13 11:W:cons#(X1,X2) -> c_8() 12:W:from#(X) -> c_9(cons#(X,n__from(s(X)))) -->_1 cons#(X1,X2) -> c_8():11 13:W:from#(X) -> c_10() 14:W:pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) -->_2 from#(X) -> c_10():13 -->_2 from#(X) -> c_9(cons#(X,n__from(s(X)))):12 -->_1 2ndspos#(0(),Z) -> c_3():6 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. 14: pi#(X) -> c_11(2ndspos#(X,from(0())),from#(0())) 10: activate#(n__from(X)) -> c_7(from#(X)) 13: from#(X) -> c_10() 12: from#(X) -> c_9(cons#(X,n__from(s(X)))) 9: activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) 11: cons#(X1,X2) -> c_8() 8: activate#(X) -> c_5() 7: 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)),activate#(Z)) 6: 2ndspos#(0(),Z) -> c_3() 5: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)),activate#(Z)) 4: 2ndsneg#(0(),Z) -> c_1() 16: times#(0(),Y) -> c_15() 15: plus#(0(),Y) -> c_12() ** Step 1.b:6: RemoveHeads WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 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,n__cons(Y,Z))) -> rcons(negrecip(activate(Y)),2ndspos(N,activate(Z))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,n__cons(Y,Z))) -> rcons(posrecip(activate(Y)),2ndsneg(N,activate(Z))) activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) cons(X1,X2) -> n__cons(X1,X2) 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,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/3,c_3/0,c_4/3,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,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#,cons#,from#,pi#,plus#,square# ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: RemoveHeads + 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:S:square#(X) -> c_14(times#(X,X)) -->_1 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)):3 3: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)):3 -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):1 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). [(2,square#(X) -> c_14(times#(X,X)))] ** Step 1.b:7: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 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: 2ndsneg(0(),Z) -> rnil() 2ndsneg(s(N),cons(X,n__cons(Y,Z))) -> rcons(negrecip(activate(Y)),2ndspos(N,activate(Z))) 2ndspos(0(),Z) -> rnil() 2ndspos(s(N),cons(X,n__cons(Y,Z))) -> rcons(posrecip(activate(Y)),2ndsneg(N,activate(Z))) activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) cons(X1,X2) -> n__cons(X1,X2) 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,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/3,c_3/0,c_4/3,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,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#,cons#,from#,pi#,plus#,square# ,times#} and constructors {0,n__cons,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) -> c_16(plus#(Y,times(X,Y)),times#(X,Y)) ** Step 1.b:8: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 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: 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,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/3,c_3/0,c_4/3,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,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#,cons#,from#,pi#,plus#,square# ,times#} and constructors {0,n__cons,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 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. 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: times#(s(X),Y) -> c_16(plus#(Y,times(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,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/3,c_3/0,c_4/3,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,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#,cons#,from#,pi#,plus#,square# ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1: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)):1 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: times#(s(X),Y) -> c_16(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,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/3,c_3/0,c_4/3,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2,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#,cons#,from#,pi#,plus#,square# ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: times#(s(X),Y) -> c_16(times#(X,Y)) *** Step 1.b:8.a:3: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: times#(s(X),Y) -> c_16(times#(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/3,c_3/0,c_4/3,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2,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#,cons#,from#,pi#,plus#,square# ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_16) = {1} Following symbols are considered usable: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#} TcT has computed the following interpretation: p(0) = [1] p(2ndsneg) = [1] x1 + [1] p(2ndspos) = [1] x1 + [2] x2 + [0] p(activate) = [0] p(cons) = [8] x2 + [0] p(from) = [1] x1 + [2] p(n__cons) = [1] x1 + [1] x2 + [0] p(n__from) = [8] p(negrecip) = [1] x1 + [1] 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#) = [0] p(2ndspos#) = [0] p(activate#) = [0] p(cons#) = [0] p(from#) = [0] p(pi#) = [0] p(plus#) = [0] p(square#) = [0] p(times#) = [2] x1 + [2] p(c_1) = [0] p(c_2) = [8] p(c_3) = [1] p(c_4) = [1] x2 + [1] p(c_5) = [2] p(c_6) = [1] x1 + [2] p(c_7) = [1] x1 + [8] p(c_8) = [1] p(c_9) = [1] p(c_10) = [1] p(c_11) = [1] x1 + [2] x2 + [1] p(c_12) = [8] p(c_13) = [0] p(c_14) = [2] x1 + [1] p(c_15) = [0] p(c_16) = [1] x1 + [0] Following rules are strictly oriented: times#(s(X),Y) = [2] X + [6] > [2] X + [2] = c_16(times#(X,Y)) Following rules are (at-least) weakly oriented: *** Step 1.b:8.a:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: times#(s(X),Y) -> c_16(times#(X,Y)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/3,c_3/0,c_4/3,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/0,c_11/2,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#,cons#,from#,pi#,plus#,square# ,times#} and constructors {0,n__cons,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: NaturalMI 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,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/3,c_3/0,c_4/3,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,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#,cons#,from#,pi#,plus#,square# ,times#} and constructors {0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_13) = {1} Following symbols are considered usable: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#} TcT has computed the following interpretation: p(0) = [3] p(2ndsneg) = [0] p(2ndspos) = [0] p(activate) = [0] p(cons) = [0] p(from) = [0] p(n__cons) = [1] x1 + [1] x2 + [0] p(n__from) = [1] x1 + [0] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [3] x1 + [12] p(posrecip) = [0] p(rcons) = [0] p(rnil) = [0] p(s) = [1] x1 + [1] p(square) = [0] p(times) = [0] p(2ndsneg#) = [1] x1 + [0] p(2ndspos#) = [0] p(activate#) = [0] p(cons#) = [1] x1 + [0] p(from#) = [0] p(pi#) = [0] p(plus#) = [4] x1 + [8] p(square#) = [0] p(times#) = [4] x2 + [8] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [1] x1 + [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] Following rules are strictly oriented: plus#(s(X),Y) = [4] X + [12] > [4] X + [8] = c_13(plus#(X,Y)) Following rules are (at-least) weakly oriented: times#(s(X),Y) = [4] Y + [8] >= [4] Y + [8] = plus#(Y,times(X,Y)) times#(s(X),Y) = [4] Y + [8] >= [4] Y + [8] = times#(X,Y) *** Step 1.b:8.b:2: 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,cons/2,from/1,pi/1,plus/2,square/1,times/2,2ndsneg#/2,2ndspos#/2,activate#/1 ,cons#/2,from#/1,pi#/1,plus#/2,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,negrecip/1,posrecip/1,rcons/2 ,rnil/0,s/1,c_1/0,c_2/3,c_3/0,c_4/3,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,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#,cons#,from#,pi#,plus#,square# ,times#} and constructors {0,n__cons,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))