WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum WORST_CASE(Omega(n^1),O(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(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__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)) s(X) -> n__s(X) 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,s/1,square/1,times/2} / {0/0,n__cons/2,n__from/1 ,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square ,times} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + 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(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__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)) s(X) -> n__s(X) 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,s/1,square/1,times/2} / {0/0,n__cons/2,n__from/1 ,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square ,times} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: activate(x){x -> n__cons(x,y)} = activate(n__cons(x,y)) ->^+ cons(activate(x),y) = C[activate(x) = activate(x){}] ** Step 1.b:1: DependencyPairs WORST_CASE(?,O(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(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__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)) s(X) -> n__s(X) 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,s/1,square/1,times/2} / {0/0,n__cons/2,n__from/1 ,n__s/1,negrecip/1,posrecip/1,rcons/2,rnil/0} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,activate,cons,from,pi,plus,s,square ,times} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + 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#(activate(X1),X2),activate#(X1)) activate#(n__from(X)) -> c_7(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_8(s#(activate(X)),activate#(X)) cons#(X1,X2) -> c_9() from#(X) -> c_10(cons#(X,n__from(n__s(X)))) from#(X) -> c_11() pi#(X) -> c_12(2ndspos#(X,from(0())),from#(0())) plus#(0(),Y) -> c_13() plus#(s(X),Y) -> c_14(s#(plus(X,Y)),plus#(X,Y)) s#(X) -> c_15() square#(X) -> c_16(times#(X,X)) times#(0(),Y) -> c_17() times#(s(X),Y) -> c_18(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^1)) + 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#(activate(X1),X2),activate#(X1)) activate#(n__from(X)) -> c_7(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_8(s#(activate(X)),activate#(X)) cons#(X1,X2) -> c_9() from#(X) -> c_10(cons#(X,n__from(n__s(X)))) from#(X) -> c_11() pi#(X) -> c_12(2ndspos#(X,from(0())),from#(0())) plus#(0(),Y) -> c_13() plus#(s(X),Y) -> c_14(s#(plus(X,Y)),plus#(X,Y)) s#(X) -> c_15() square#(X) -> c_16(times#(X,X)) times#(0(),Y) -> c_17() times#(s(X),Y) -> c_18(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(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__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)) s(X) -> n__s(X) 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,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/3,c_3/0,c_4/3,c_5/0,c_6/2,c_7/2,c_8/2,c_9/0,c_10/1,c_11/0 ,c_12/2,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,4,5,9,11,13,14,15,17,18} by application of Pre({1,2,3,4,5,9,11,13,14,15,17,18}) = {6,7,8,10,12,16}. 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#(activate(X1),X2),activate#(X1)) 7: activate#(n__from(X)) -> c_7(from#(activate(X)),activate#(X)) 8: activate#(n__s(X)) -> c_8(s#(activate(X)),activate#(X)) 9: cons#(X1,X2) -> c_9() 10: from#(X) -> c_10(cons#(X,n__from(n__s(X)))) 11: from#(X) -> c_11() 12: pi#(X) -> c_12(2ndspos#(X,from(0())),from#(0())) 13: plus#(0(),Y) -> c_13() 14: plus#(s(X),Y) -> c_14(s#(plus(X,Y)),plus#(X,Y)) 15: s#(X) -> c_15() 16: square#(X) -> c_16(times#(X,X)) 17: times#(0(),Y) -> c_17() 18: times#(s(X),Y) -> c_18(plus#(Y,times(X,Y)),times#(X,Y)) ** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_6(cons#(activate(X1),X2),activate#(X1)) activate#(n__from(X)) -> c_7(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_8(s#(activate(X)),activate#(X)) from#(X) -> c_10(cons#(X,n__from(n__s(X)))) pi#(X) -> c_12(2ndspos#(X,from(0())),from#(0())) square#(X) -> c_16(times#(X,X)) - 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_9() from#(X) -> c_11() plus#(0(),Y) -> c_13() plus#(s(X),Y) -> c_14(s#(plus(X,Y)),plus#(X,Y)) s#(X) -> c_15() times#(0(),Y) -> c_17() times#(s(X),Y) -> c_18(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(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__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)) s(X) -> n__s(X) 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,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/3,c_3/0,c_4/3,c_5/0,c_6/2,c_7/2,c_8/2,c_9/0,c_10/1,c_11/0 ,c_12/2,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {4,6} by application of Pre({4,6}) = {2,5}. Here rules are labelled as follows: 1: activate#(n__cons(X1,X2)) -> c_6(cons#(activate(X1),X2),activate#(X1)) 2: activate#(n__from(X)) -> c_7(from#(activate(X)),activate#(X)) 3: activate#(n__s(X)) -> c_8(s#(activate(X)),activate#(X)) 4: from#(X) -> c_10(cons#(X,n__from(n__s(X)))) 5: pi#(X) -> c_12(2ndspos#(X,from(0())),from#(0())) 6: square#(X) -> c_16(times#(X,X)) 7: 2ndsneg#(0(),Z) -> c_1() 8: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)),activate#(Z)) 9: 2ndspos#(0(),Z) -> c_3() 10: 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)),activate#(Z)) 11: activate#(X) -> c_5() 12: cons#(X1,X2) -> c_9() 13: from#(X) -> c_11() 14: plus#(0(),Y) -> c_13() 15: plus#(s(X),Y) -> c_14(s#(plus(X,Y)),plus#(X,Y)) 16: s#(X) -> c_15() 17: times#(0(),Y) -> c_17() 18: times#(s(X),Y) -> c_18(plus#(Y,times(X,Y)),times#(X,Y)) ** Step 1.b:4: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_6(cons#(activate(X1),X2),activate#(X1)) activate#(n__from(X)) -> c_7(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_8(s#(activate(X)),activate#(X)) pi#(X) -> c_12(2ndspos#(X,from(0())),from#(0())) - 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_9() from#(X) -> c_10(cons#(X,n__from(n__s(X)))) from#(X) -> c_11() plus#(0(),Y) -> c_13() plus#(s(X),Y) -> c_14(s#(plus(X,Y)),plus#(X,Y)) s#(X) -> c_15() square#(X) -> c_16(times#(X,X)) times#(0(),Y) -> c_17() times#(s(X),Y) -> c_18(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(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__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)) s(X) -> n__s(X) 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,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/3,c_3/0,c_4/3,c_5/0,c_6/2,c_7/2,c_8/2,c_9/0,c_10/1,c_11/0 ,c_12/2,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {4} by application of Pre({4}) = {}. Here rules are labelled as follows: 1: activate#(n__cons(X1,X2)) -> c_6(cons#(activate(X1),X2),activate#(X1)) 2: activate#(n__from(X)) -> c_7(from#(activate(X)),activate#(X)) 3: activate#(n__s(X)) -> c_8(s#(activate(X)),activate#(X)) 4: pi#(X) -> c_12(2ndspos#(X,from(0())),from#(0())) 5: 2ndsneg#(0(),Z) -> c_1() 6: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z)),activate#(Z)) 7: 2ndspos#(0(),Z) -> c_3() 8: 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z)),activate#(Z)) 9: activate#(X) -> c_5() 10: cons#(X1,X2) -> c_9() 11: from#(X) -> c_10(cons#(X,n__from(n__s(X)))) 12: from#(X) -> c_11() 13: plus#(0(),Y) -> c_13() 14: plus#(s(X),Y) -> c_14(s#(plus(X,Y)),plus#(X,Y)) 15: s#(X) -> c_15() 16: square#(X) -> c_16(times#(X,X)) 17: times#(0(),Y) -> c_17() 18: times#(s(X),Y) -> c_18(plus#(Y,times(X,Y)),times#(X,Y)) ** Step 1.b:5: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_6(cons#(activate(X1),X2),activate#(X1)) activate#(n__from(X)) -> c_7(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_8(s#(activate(X)),activate#(X)) - 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_9() from#(X) -> c_10(cons#(X,n__from(n__s(X)))) from#(X) -> c_11() pi#(X) -> c_12(2ndspos#(X,from(0())),from#(0())) plus#(0(),Y) -> c_13() plus#(s(X),Y) -> c_14(s#(plus(X,Y)),plus#(X,Y)) s#(X) -> c_15() square#(X) -> c_16(times#(X,X)) times#(0(),Y) -> c_17() times#(s(X),Y) -> c_18(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(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__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)) s(X) -> n__s(X) 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,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/3,c_3/0,c_4/3,c_5/0,c_6/2,c_7/2,c_8/2,c_9/0,c_10/1,c_11/0 ,c_12/2,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__cons(X1,X2)) -> c_6(cons#(activate(X1),X2),activate#(X1)) -->_2 activate#(n__s(X)) -> c_8(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_7(from#(activate(X)),activate#(X)):2 -->_1 cons#(X1,X2) -> c_9():9 -->_2 activate#(X) -> c_5():8 -->_2 activate#(n__cons(X1,X2)) -> c_6(cons#(activate(X1),X2),activate#(X1)):1 2:S:activate#(n__from(X)) -> c_7(from#(activate(X)),activate#(X)) -->_1 from#(X) -> c_10(cons#(X,n__from(n__s(X)))):10 -->_2 activate#(n__s(X)) -> c_8(s#(activate(X)),activate#(X)):3 -->_1 from#(X) -> c_11():11 -->_2 activate#(X) -> c_5():8 -->_2 activate#(n__from(X)) -> c_7(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__cons(X1,X2)) -> c_6(cons#(activate(X1),X2),activate#(X1)):1 3:S:activate#(n__s(X)) -> c_8(s#(activate(X)),activate#(X)) -->_1 s#(X) -> c_15():15 -->_2 activate#(X) -> c_5():8 -->_2 activate#(n__s(X)) -> c_8(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_7(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__cons(X1,X2)) -> c_6(cons#(activate(X1),X2),activate#(X1)):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:cons#(X1,X2) -> c_9() 10:W:from#(X) -> c_10(cons#(X,n__from(n__s(X)))) -->_1 cons#(X1,X2) -> c_9():9 11:W:from#(X) -> c_11() 12:W:pi#(X) -> c_12(2ndspos#(X,from(0())),from#(0())) -->_2 from#(X) -> c_11():11 -->_2 from#(X) -> c_10(cons#(X,n__from(n__s(X)))):10 -->_1 2ndspos#(0(),Z) -> c_3():6 13:W:plus#(0(),Y) -> c_13() 14:W:plus#(s(X),Y) -> c_14(s#(plus(X,Y)),plus#(X,Y)) 15:W:s#(X) -> c_15() 16:W:square#(X) -> c_16(times#(X,X)) -->_1 times#(0(),Y) -> c_17():17 17:W:times#(0(),Y) -> c_17() 18:W:times#(s(X),Y) -> c_18(plus#(Y,times(X,Y)),times#(X,Y)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 18: times#(s(X),Y) -> c_18(plus#(Y,times(X,Y)),times#(X,Y)) 16: square#(X) -> c_16(times#(X,X)) 17: times#(0(),Y) -> c_17() 14: plus#(s(X),Y) -> c_14(s#(plus(X,Y)),plus#(X,Y)) 13: plus#(0(),Y) -> c_13() 12: pi#(X) -> c_12(2ndspos#(X,from(0())),from#(0())) 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() 11: from#(X) -> c_11() 10: from#(X) -> c_10(cons#(X,n__from(n__s(X)))) 9: cons#(X1,X2) -> c_9() 8: activate#(X) -> c_5() 15: s#(X) -> c_15() ** Step 1.b:6: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_6(cons#(activate(X1),X2),activate#(X1)) activate#(n__from(X)) -> c_7(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_8(s#(activate(X)),activate#(X)) - 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(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__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)) s(X) -> n__s(X) 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,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/3,c_3/0,c_4/3,c_5/0,c_6/2,c_7/2,c_8/2,c_9/0,c_10/1,c_11/0 ,c_12/2,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:activate#(n__cons(X1,X2)) -> c_6(cons#(activate(X1),X2),activate#(X1)) -->_2 activate#(n__s(X)) -> c_8(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_7(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__cons(X1,X2)) -> c_6(cons#(activate(X1),X2),activate#(X1)):1 2:S:activate#(n__from(X)) -> c_7(from#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_8(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_7(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__cons(X1,X2)) -> c_6(cons#(activate(X1),X2),activate#(X1)):1 3:S:activate#(n__s(X)) -> c_8(s#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_8(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_7(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__cons(X1,X2)) -> c_6(cons#(activate(X1),X2),activate#(X1)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: activate#(n__cons(X1,X2)) -> c_6(activate#(X1)) activate#(n__from(X)) -> c_7(activate#(X)) activate#(n__s(X)) -> c_8(activate#(X)) ** Step 1.b:7: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_6(activate#(X1)) activate#(n__from(X)) -> c_7(activate#(X)) activate#(n__s(X)) -> c_8(activate#(X)) - 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(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__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)) s(X) -> n__s(X) 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,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/3,c_3/0,c_4/3,c_5/0,c_6/1,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0 ,c_12/2,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate#(n__cons(X1,X2)) -> c_6(activate#(X1)) activate#(n__from(X)) -> c_7(activate#(X)) activate#(n__s(X)) -> c_8(activate#(X)) ** Step 1.b:8: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_6(activate#(X1)) activate#(n__from(X)) -> c_7(activate#(X)) activate#(n__s(X)) -> c_8(activate#(X)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/3,c_3/0,c_4/3,c_5/0,c_6/1,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0 ,c_12/2,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + 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(c_6) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [0] p(2ndspos) = [0] p(activate) = [2] x1 + [1] p(cons) = [8] x2 + [2] p(from) = [0] p(n__cons) = [1] x1 + [1] x2 + [0] p(n__from) = [1] x1 + [1] p(n__s) = [1] x1 + [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) = [0] p(square) = [0] p(times) = [0] p(2ndsneg#) = [0] p(2ndspos#) = [0] p(activate#) = [10] x1 + [0] p(cons#) = [0] p(from#) = [0] p(pi#) = [0] p(plus#) = [0] p(s#) = [0] p(square#) = [0] p(times#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] Following rules are strictly oriented: activate#(n__from(X)) = [10] X + [10] > [10] X + [0] = c_7(activate#(X)) Following rules are (at-least) weakly oriented: activate#(n__cons(X1,X2)) = [10] X1 + [10] X2 + [0] >= [10] X1 + [0] = c_6(activate#(X1)) activate#(n__s(X)) = [10] X + [0] >= [10] X + [0] = c_8(activate#(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:9: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_6(activate#(X1)) activate#(n__s(X)) -> c_8(activate#(X)) - Weak DPs: activate#(n__from(X)) -> c_7(activate#(X)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/3,c_3/0,c_4/3,c_5/0,c_6/1,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0 ,c_12/2,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + 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(c_6) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [2] x1 + [2] p(2ndspos) = [1] p(activate) = [1] x1 + [2] p(cons) = [1] x1 + [8] x2 + [4] p(from) = [2] p(n__cons) = [1] x1 + [1] x2 + [6] p(n__from) = [1] x1 + [4] p(n__s) = [1] x1 + [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) = [0] p(square) = [0] p(times) = [2] p(2ndsneg#) = [0] p(2ndspos#) = [8] x1 + [0] p(activate#) = [4] x1 + [4] p(cons#) = [0] p(from#) = [0] p(pi#) = [0] p(plus#) = [0] p(s#) = [0] p(square#) = [0] p(times#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [1] x2 + [0] p(c_5) = [1] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [2] p(c_12) = [1] x1 + [2] x2 + [2] p(c_13) = [0] p(c_14) = [2] x1 + [2] x2 + [2] p(c_15) = [0] p(c_16) = [2] p(c_17) = [1] p(c_18) = [2] x1 + [1] x2 + [0] Following rules are strictly oriented: activate#(n__cons(X1,X2)) = [4] X1 + [4] X2 + [28] > [4] X1 + [4] = c_6(activate#(X1)) activate#(n__s(X)) = [4] X + [24] > [4] X + [4] = c_8(activate#(X)) Following rules are (at-least) weakly oriented: activate#(n__from(X)) = [4] X + [20] >= [4] X + [4] = c_7(activate#(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:10: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__cons(X1,X2)) -> c_6(activate#(X1)) activate#(n__from(X)) -> c_7(activate#(X)) activate#(n__s(X)) -> c_8(activate#(X)) - Signature: {2ndsneg/2,2ndspos/2,activate/1,cons/2,from/1,pi/1,plus/2,s/1,square/1,times/2,2ndsneg#/2,2ndspos#/2 ,activate#/1,cons#/2,from#/1,pi#/1,plus#/2,s#/1,square#/1,times#/2} / {0/0,n__cons/2,n__from/1,n__s/1 ,negrecip/1,posrecip/1,rcons/2,rnil/0,c_1/0,c_2/3,c_3/0,c_4/3,c_5/0,c_6/1,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0 ,c_12/2,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,s# ,square#,times#} and constructors {0,n__cons,n__from,n__s,negrecip,posrecip,rcons,rnil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))