*** 1 Progress [(?,O(n^3))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: 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)) Weak DP Rules: Weak TRS Rules: 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: Full basic terms: {2ndsneg,2ndspos,activate,cons,from,pi,plus,square,times}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: DependencyPairs {dpKind_ = WIDP} Proof: We add the following weak dependency pairs: Strict DPs 2ndsneg#(0(),Z) -> c_1() 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,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#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_9(cons#(X,n__from(s(X)))) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) Weak DPs and mark the set of starting terms. *** 1.1 Progress [(?,O(n^3))] *** Considered Problem: Strict DP Rules: 2ndsneg#(0(),Z) -> c_1() 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,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#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_9(cons#(X,n__from(s(X)))) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) Strict TRS Rules: 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)) Weak DP Rules: Weak TRS Rules: 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: UsableRules Proof: We replace rewrite rules by usable rules: 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) plus(0(),Y) -> Y plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) 2ndsneg#(0(),Z) -> c_1() 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,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#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_9(cons#(X,n__from(s(X)))) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) *** 1.1.1 Progress [(?,O(n^3))] *** Considered Problem: Strict DP Rules: 2ndsneg#(0(),Z) -> c_1() 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,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#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_9(cons#(X,n__from(s(X)))) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) Strict TRS Rules: 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) 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)) Weak DP Rules: Weak TRS Rules: 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} Proof: We estimate the number of application of {1,3,15} by application of Pre({1,3,15}) = {2,4,5,8,10,11,12,14}. 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))) 3: 2ndspos#(0(),Z) -> c_3() 4: 2ndspos#(s(N) ,cons(X,n__cons(Y,Z))) -> c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) 5: activate#(X) -> c_5(X) 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(X1,X2) 9: from#(X) -> c_9(cons#(X ,n__from(s(X)))) 10: from#(X) -> c_10(X) 11: pi#(X) -> c_11(2ndspos#(X ,from(0()))) 12: plus#(0(),Y) -> c_12(Y) 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))) *** 1.1.1.1 Progress [(?,O(n^3))] *** Considered Problem: Strict DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_9(cons#(X,n__from(s(X)))) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) Strict TRS Rules: 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) 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)) Weak DP Rules: 2ndsneg#(0(),Z) -> c_1() 2ndspos#(0(),Z) -> c_3() times#(0(),Y) -> c_15() Weak TRS Rules: 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: PathAnalysis {onlyLinear = True} Proof: We employ 'linear path analysis' using the following approximated dependency graph: 1:S:2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) -->_1 activate#(n__from(X)) -> c_7(from#(X)):5 -->_1 activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)):4 -->_1 activate#(X) -> c_5(X):3 -->_2 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))):2 -->_2 2ndspos#(0(),Z) -> c_3():15 2:S:2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) -->_1 activate#(n__from(X)) -> c_7(from#(X)):5 -->_1 activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)):4 -->_1 activate#(X) -> c_5(X):3 -->_2 2ndsneg#(0(),Z) -> c_1():14 -->_2 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))):1 3:S:activate#(X) -> c_5(X) -->_1 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))):13 -->_1 square#(X) -> c_14(times#(X,X)):12 -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):11 -->_1 plus#(0(),Y) -> c_12(Y):10 -->_1 pi#(X) -> c_11(2ndspos#(X,from(0()))):9 -->_1 from#(X) -> c_10(X):8 -->_1 from#(X) -> c_9(cons#(X,n__from(s(X)))):7 -->_1 cons#(X1,X2) -> c_8(X1,X2):6 -->_1 activate#(n__from(X)) -> c_7(from#(X)):5 -->_1 activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)):4 -->_1 times#(0(),Y) -> c_15():16 -->_1 2ndspos#(0(),Z) -> c_3():15 -->_1 2ndsneg#(0(),Z) -> c_1():14 -->_1 activate#(X) -> c_5(X):3 -->_1 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))):2 -->_1 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))):1 4:S:activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) -->_1 cons#(X1,X2) -> c_8(X1,X2):6 5:S:activate#(n__from(X)) -> c_7(from#(X)) -->_1 from#(X) -> c_10(X):8 -->_1 from#(X) -> c_9(cons#(X,n__from(s(X)))):7 6:S:cons#(X1,X2) -> c_8(X1,X2) -->_2 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))):13 -->_1 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))):13 -->_2 square#(X) -> c_14(times#(X,X)):12 -->_1 square#(X) -> c_14(times#(X,X)):12 -->_2 plus#(s(X),Y) -> c_13(plus#(X,Y)):11 -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):11 -->_2 plus#(0(),Y) -> c_12(Y):10 -->_1 plus#(0(),Y) -> c_12(Y):10 -->_2 pi#(X) -> c_11(2ndspos#(X,from(0()))):9 -->_1 pi#(X) -> c_11(2ndspos#(X,from(0()))):9 -->_2 from#(X) -> c_10(X):8 -->_1 from#(X) -> c_10(X):8 -->_2 from#(X) -> c_9(cons#(X,n__from(s(X)))):7 -->_1 from#(X) -> c_9(cons#(X,n__from(s(X)))):7 -->_2 times#(0(),Y) -> c_15():16 -->_1 times#(0(),Y) -> c_15():16 -->_2 2ndspos#(0(),Z) -> c_3():15 -->_1 2ndspos#(0(),Z) -> c_3():15 -->_2 2ndsneg#(0(),Z) -> c_1():14 -->_1 2ndsneg#(0(),Z) -> c_1():14 -->_2 cons#(X1,X2) -> c_8(X1,X2):6 -->_1 cons#(X1,X2) -> c_8(X1,X2):6 -->_2 activate#(n__from(X)) -> c_7(from#(X)):5 -->_1 activate#(n__from(X)) -> c_7(from#(X)):5 -->_2 activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)):4 -->_1 activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)):4 -->_2 activate#(X) -> c_5(X):3 -->_1 activate#(X) -> c_5(X):3 -->_2 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))):2 -->_1 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))):2 -->_2 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))):1 -->_1 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))):1 7:S:from#(X) -> c_9(cons#(X,n__from(s(X)))) -->_1 cons#(X1,X2) -> c_8(X1,X2):6 8:S:from#(X) -> c_10(X) -->_1 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))):13 -->_1 square#(X) -> c_14(times#(X,X)):12 -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):11 -->_1 plus#(0(),Y) -> c_12(Y):10 -->_1 pi#(X) -> c_11(2ndspos#(X,from(0()))):9 -->_1 times#(0(),Y) -> c_15():16 -->_1 2ndspos#(0(),Z) -> c_3():15 -->_1 2ndsneg#(0(),Z) -> c_1():14 -->_1 from#(X) -> c_10(X):8 -->_1 from#(X) -> c_9(cons#(X,n__from(s(X)))):7 -->_1 cons#(X1,X2) -> c_8(X1,X2):6 -->_1 activate#(n__from(X)) -> c_7(from#(X)):5 -->_1 activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)):4 -->_1 activate#(X) -> c_5(X):3 -->_1 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))):2 -->_1 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))):1 9:S:pi#(X) -> c_11(2ndspos#(X,from(0()))) -->_1 2ndspos#(0(),Z) -> c_3():15 -->_1 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))):2 10:S:plus#(0(),Y) -> c_12(Y) -->_1 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))):13 -->_1 square#(X) -> c_14(times#(X,X)):12 -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):11 -->_1 times#(0(),Y) -> c_15():16 -->_1 2ndspos#(0(),Z) -> c_3():15 -->_1 2ndsneg#(0(),Z) -> c_1():14 -->_1 plus#(0(),Y) -> c_12(Y):10 -->_1 pi#(X) -> c_11(2ndspos#(X,from(0()))):9 -->_1 from#(X) -> c_10(X):8 -->_1 from#(X) -> c_9(cons#(X,n__from(s(X)))):7 -->_1 cons#(X1,X2) -> c_8(X1,X2):6 -->_1 activate#(n__from(X)) -> c_7(from#(X)):5 -->_1 activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)):4 -->_1 activate#(X) -> c_5(X):3 -->_1 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))):2 -->_1 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))):1 11:S:plus#(s(X),Y) -> c_13(plus#(X,Y)) -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):11 -->_1 plus#(0(),Y) -> c_12(Y):10 12:S:square#(X) -> c_14(times#(X,X)) -->_1 times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))):13 -->_1 times#(0(),Y) -> c_15():16 13:S:times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))) -->_1 plus#(s(X),Y) -> c_13(plus#(X,Y)):11 -->_1 plus#(0(),Y) -> c_12(Y):10 14:W:2ndsneg#(0(),Z) -> c_1() 15:W:2ndspos#(0(),Z) -> c_3() 16:W:times#(0(),Y) -> c_15() Obtaining following paths: 1.) Path: [1,4] 2.) Path: [1,3] 3.) Path: [1,2] *** 1.1.1.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_9(cons#(X,n__from(s(X)))) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) Strict TRS Rules: 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) 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)) Weak DP Rules: times#(0(),Y) -> c_15() Weak TRS Rules: 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [0] p(2ndspos) = [0] p(activate) = [1] x1 + [2] p(cons) = [1] x2 + [0] p(from) = [1] x1 + [0] p(n__cons) = [1] x2 + [0] p(n__from) = [1] x1 + [0] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [1] x2 + [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 + [0] p(2ndspos#) = [1] x2 + [0] p(activate#) = [0] p(cons#) = [0] p(from#) = [0] p(pi#) = [0] p(plus#) = [1] x2 + [0] p(square#) = [0] p(times#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [1] x1 + [0] Following rules are strictly oriented: activate(X) = [1] X + [2] > [1] X + [0] = X activate(n__cons(X1,X2)) = [1] X2 + [2] > [1] X2 + [0] = cons(X1,X2) activate(n__from(X)) = [1] X + [2] > [1] X + [0] = from(X) Following rules are (at-least) weakly oriented: 2ndsneg#(s(N) = [1] Z + [0] ,cons(X,n__cons(Y,Z))) >= [1] Z + [2] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(s(N) = [1] Z + [0] ,cons(X,n__cons(Y,Z))) >= [1] Z + [2] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = [0] >= [0] = c_5(X) activate#(n__cons(X1,X2)) = [0] >= [0] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [0] >= [0] = c_7(from#(X)) cons#(X1,X2) = [0] >= [0] = c_8(X1,X2) from#(X) = [0] >= [0] = c_9(cons#(X,n__from(s(X)))) from#(X) = [0] >= [0] = c_10(X) pi#(X) = [0] >= [0] = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = [1] Y + [0] >= [1] Y + [0] = c_12(Y) plus#(s(X),Y) = [1] Y + [0] >= [1] Y + [0] = c_13(plus#(X,Y)) square#(X) = [0] >= [0] = c_14(times#(X,X)) times#(0(),Y) = [0] >= [0] = c_15() times#(s(X),Y) = [0] >= [0] = c_16(plus#(Y,times(X,Y))) cons(X1,X2) = [1] X2 + [0] >= [1] X2 + [0] = n__cons(X1,X2) from(X) = [1] X + [0] >= [1] X + [0] = cons(X,n__from(s(X))) from(X) = [1] X + [0] >= [1] X + [0] = n__from(X) plus(0(),Y) = [1] Y + [0] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [0] >= [1] Y + [0] = s(plus(X,Y)) times(0(),Y) = [0] >= [0] = 0() times(s(X),Y) = [0] >= [0] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_9(cons#(X,n__from(s(X)))) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) Strict TRS Rules: cons(X1,X2) -> n__cons(X1,X2) 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)) Weak DP Rules: times#(0(),Y) -> c_15() Weak TRS Rules: activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [0] p(2ndspos) = [0] p(activate) = [1] x1 + [6] p(cons) = [1] x2 + [0] p(from) = [0] p(n__cons) = [1] x2 + [4] p(n__from) = [6] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [1] x2 + [3] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [1] p(square) = [0] p(times) = [5] p(2ndsneg#) = [1] x2 + [0] p(2ndspos#) = [1] x2 + [4] p(activate#) = [3] p(cons#) = [0] p(from#) = [0] p(pi#) = [0] p(plus#) = [4] x1 + [1] x2 + [0] p(square#) = [7] x1 + [0] p(times#) = [7] x2 + [6] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [4] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] p(c_9) = [1] x1 + [0] p(c_10) = [1] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [2] p(c_15) = [6] p(c_16) = [1] x1 + [0] Following rules are strictly oriented: activate#(X) = [3] > [0] = c_5(X) activate#(n__cons(X1,X2)) = [3] > [0] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [3] > [0] = c_7(from#(X)) plus#(s(X),Y) = [4] X + [1] Y + [4] > [4] X + [1] Y + [0] = c_13(plus#(X,Y)) times#(s(X),Y) = [7] Y + [6] > [4] Y + [5] = c_16(plus#(Y,times(X,Y))) plus(0(),Y) = [1] Y + [3] > [1] Y + [0] = Y times(0(),Y) = [5] > [0] = 0() Following rules are (at-least) weakly oriented: 2ndsneg#(s(N) = [1] Z + [4] ,cons(X,n__cons(Y,Z))) >= [1] Z + [13] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(s(N) = [1] Z + [8] ,cons(X,n__cons(Y,Z))) >= [1] Z + [13] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) cons#(X1,X2) = [0] >= [1] = c_8(X1,X2) from#(X) = [0] >= [0] = c_9(cons#(X,n__from(s(X)))) from#(X) = [0] >= [1] = c_10(X) pi#(X) = [0] >= [4] = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = [1] Y + [0] >= [1] Y + [0] = c_12(Y) square#(X) = [7] X + [0] >= [7] X + [8] = c_14(times#(X,X)) times#(0(),Y) = [7] Y + [6] >= [6] = c_15() activate(X) = [1] X + [6] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [1] X2 + [10] >= [1] X2 + [0] = cons(X1,X2) activate(n__from(X)) = [12] >= [0] = from(X) cons(X1,X2) = [1] X2 + [0] >= [1] X2 + [4] = n__cons(X1,X2) from(X) = [0] >= [6] = cons(X,n__from(s(X))) from(X) = [0] >= [6] = n__from(X) plus(s(X),Y) = [1] Y + [3] >= [1] Y + [4] = s(plus(X,Y)) times(s(X),Y) = [5] >= [8] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_9(cons#(X,n__from(s(X)))) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) square#(X) -> c_14(times#(X,X)) Strict TRS Rules: cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(s(X),Y) -> s(plus(X,Y)) times(s(X),Y) -> plus(Y,times(X,Y)) Weak DP Rules: activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) plus#(s(X),Y) -> c_13(plus#(X,Y)) times#(0(),Y) -> c_15() times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))) Weak TRS Rules: activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) plus(0(),Y) -> Y times(0(),Y) -> 0() 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [1] x1 + [0] p(2ndspos) = [1] x2 + [0] p(activate) = [1] x1 + [0] p(cons) = [1] x2 + [0] p(from) = [0] p(n__cons) = [1] x2 + [0] p(n__from) = [0] p(negrecip) = [0] p(pi) = [0] p(plus) = [1] x2 + [2] p(posrecip) = [1] p(rcons) = [1] x1 + [1] x2 + [1] p(rnil) = [0] p(s) = [1] x1 + [0] p(square) = [1] x1 + [2] p(times) = [4] x2 + [0] p(2ndsneg#) = [1] x2 + [4] p(2ndspos#) = [1] x2 + [2] p(activate#) = [3] p(cons#) = [2] p(from#) = [1] p(pi#) = [1] p(plus#) = [1] x2 + [4] p(square#) = [6] x1 + [1] p(times#) = [4] x2 + [6] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [1] p(c_5) = [1] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [3] p(c_10) = [0] p(c_11) = [1] x1 + [2] p(c_12) = [1] x1 + [1] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [1] p(c_15) = [0] p(c_16) = [1] x1 + [2] Following rules are strictly oriented: cons#(X1,X2) = [2] > [0] = c_8(X1,X2) from#(X) = [1] > [0] = c_10(X) plus#(0(),Y) = [1] Y + [4] > [1] Y + [1] = c_12(Y) Following rules are (at-least) weakly oriented: 2ndsneg#(s(N) = [1] Z + [4] ,cons(X,n__cons(Y,Z))) >= [1] Z + [5] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(s(N) = [1] Z + [2] ,cons(X,n__cons(Y,Z))) >= [1] Z + [8] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = [3] >= [1] = c_5(X) activate#(n__cons(X1,X2)) = [3] >= [2] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [3] >= [1] = c_7(from#(X)) from#(X) = [1] >= [5] = c_9(cons#(X,n__from(s(X)))) pi#(X) = [1] >= [4] = c_11(2ndspos#(X,from(0()))) plus#(s(X),Y) = [1] Y + [4] >= [1] Y + [4] = c_13(plus#(X,Y)) square#(X) = [6] X + [1] >= [4] X + [7] = c_14(times#(X,X)) times#(0(),Y) = [4] Y + [6] >= [0] = c_15() times#(s(X),Y) = [4] Y + [6] >= [4] Y + [6] = c_16(plus#(Y,times(X,Y))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [1] X2 + [0] >= [1] X2 + [0] = cons(X1,X2) activate(n__from(X)) = [0] >= [0] = from(X) cons(X1,X2) = [1] X2 + [0] >= [1] X2 + [0] = n__cons(X1,X2) from(X) = [0] >= [0] = cons(X,n__from(s(X))) from(X) = [0] >= [0] = n__from(X) plus(0(),Y) = [1] Y + [2] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [2] >= [1] Y + [2] = s(plus(X,Y)) times(0(),Y) = [4] Y + [0] >= [0] = 0() times(s(X),Y) = [4] Y + [0] >= [4] Y + [2] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) from#(X) -> c_9(cons#(X,n__from(s(X)))) pi#(X) -> c_11(2ndspos#(X,from(0()))) square#(X) -> c_14(times#(X,X)) Strict TRS Rules: cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(s(X),Y) -> s(plus(X,Y)) times(s(X),Y) -> plus(Y,times(X,Y)) Weak DP Rules: activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_10(X) plus#(0(),Y) -> c_12(Y) plus#(s(X),Y) -> c_13(plus#(X,Y)) times#(0(),Y) -> c_15() times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))) Weak TRS Rules: activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) plus(0(),Y) -> Y times(0(),Y) -> 0() 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [2] p(2ndsneg) = [0] p(2ndspos) = [0] p(activate) = [1] x1 + [5] p(cons) = [1] x2 + [0] p(from) = [1] x1 + [5] p(n__cons) = [1] x2 + [4] p(n__from) = [1] x1 + [0] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [1] x2 + [7] 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) = [4] p(2ndsneg#) = [1] x2 + [0] p(2ndspos#) = [1] x2 + [2] p(activate#) = [0] p(cons#) = [0] p(from#) = [0] p(pi#) = [0] p(plus#) = [1] x2 + [1] p(square#) = [5] x1 + [2] p(times#) = [5] x1 + [5] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [3] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [1] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [1] x1 + [0] Following rules are strictly oriented: from(X) = [1] X + [5] > [1] X + [0] = cons(X,n__from(s(X))) from(X) = [1] X + [5] > [1] X + [0] = n__from(X) Following rules are (at-least) weakly oriented: 2ndsneg#(s(N) = [1] Z + [4] ,cons(X,n__cons(Y,Z))) >= [1] Z + [7] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(s(N) = [1] Z + [6] ,cons(X,n__cons(Y,Z))) >= [1] Z + [8] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = [0] >= [0] = c_5(X) activate#(n__cons(X1,X2)) = [0] >= [0] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [0] >= [0] = c_7(from#(X)) cons#(X1,X2) = [0] >= [0] = c_8(X1,X2) from#(X) = [0] >= [0] = c_9(cons#(X,n__from(s(X)))) from#(X) = [0] >= [0] = c_10(X) pi#(X) = [0] >= [9] = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = [1] Y + [1] >= [1] Y + [1] = c_12(Y) plus#(s(X),Y) = [1] Y + [1] >= [1] Y + [1] = c_13(plus#(X,Y)) square#(X) = [5] X + [2] >= [5] X + [5] = c_14(times#(X,X)) times#(0(),Y) = [15] >= [0] = c_15() times#(s(X),Y) = [5] X + [5] >= [5] = c_16(plus#(Y,times(X,Y))) activate(X) = [1] X + [5] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [1] X2 + [9] >= [1] X2 + [0] = cons(X1,X2) activate(n__from(X)) = [1] X + [5] >= [1] X + [5] = from(X) cons(X1,X2) = [1] X2 + [0] >= [1] X2 + [4] = n__cons(X1,X2) plus(0(),Y) = [1] Y + [7] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [7] >= [1] Y + [7] = s(plus(X,Y)) times(0(),Y) = [4] >= [2] = 0() times(s(X),Y) = [4] >= [11] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) from#(X) -> c_9(cons#(X,n__from(s(X)))) pi#(X) -> c_11(2ndspos#(X,from(0()))) square#(X) -> c_14(times#(X,X)) Strict TRS Rules: cons(X1,X2) -> n__cons(X1,X2) plus(s(X),Y) -> s(plus(X,Y)) times(s(X),Y) -> plus(Y,times(X,Y)) Weak DP Rules: activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_10(X) plus#(0(),Y) -> c_12(Y) plus#(s(X),Y) -> c_13(plus#(X,Y)) times#(0(),Y) -> c_15() times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))) Weak TRS Rules: activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> Y times(0(),Y) -> 0() 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [1] p(2ndsneg) = [0] p(2ndspos) = [0] p(activate) = [4] x1 + [7] p(cons) = [4] x2 + [0] p(from) = [7] p(n__cons) = [1] x2 + [1] p(n__from) = [0] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [1] x2 + [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) = [4] x1 + [1] x2 + [4] p(2ndsneg#) = [1] x2 + [0] p(2ndspos#) = [1] x2 + [0] p(activate#) = [0] p(cons#) = [0] p(from#) = [0] p(pi#) = [0] p(plus#) = [2] x1 + [1] x2 + [6] p(square#) = [7] x1 + [0] p(times#) = [4] x1 + [3] x2 + [7] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [4] p(c_14) = [1] x1 + [0] p(c_15) = [1] p(c_16) = [1] x1 + [4] Following rules are strictly oriented: times(s(X),Y) = [4] X + [1] Y + [12] > [4] X + [1] Y + [4] = plus(Y,times(X,Y)) Following rules are (at-least) weakly oriented: 2ndsneg#(s(N) = [4] Z + [4] ,cons(X,n__cons(Y,Z))) >= [4] Z + [7] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(s(N) = [4] Z + [4] ,cons(X,n__cons(Y,Z))) >= [4] Z + [7] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = [0] >= [0] = c_5(X) activate#(n__cons(X1,X2)) = [0] >= [0] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [0] >= [0] = c_7(from#(X)) cons#(X1,X2) = [0] >= [0] = c_8(X1,X2) from#(X) = [0] >= [0] = c_9(cons#(X,n__from(s(X)))) from#(X) = [0] >= [0] = c_10(X) pi#(X) = [0] >= [7] = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = [1] Y + [8] >= [1] Y + [0] = c_12(Y) plus#(s(X),Y) = [2] X + [1] Y + [10] >= [2] X + [1] Y + [10] = c_13(plus#(X,Y)) square#(X) = [7] X + [0] >= [7] X + [7] = c_14(times#(X,X)) times#(0(),Y) = [3] Y + [11] >= [1] = c_15() times#(s(X),Y) = [4] X + [3] Y + [15] >= [4] X + [3] Y + [14] = c_16(plus#(Y,times(X,Y))) activate(X) = [4] X + [7] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [4] X2 + [11] >= [4] X2 + [0] = cons(X1,X2) activate(n__from(X)) = [7] >= [7] = from(X) cons(X1,X2) = [4] X2 + [0] >= [1] X2 + [1] = n__cons(X1,X2) from(X) = [7] >= [0] = cons(X,n__from(s(X))) from(X) = [7] >= [0] = n__from(X) plus(0(),Y) = [1] Y + [0] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [0] >= [1] Y + [2] = s(plus(X,Y)) times(0(),Y) = [1] Y + [8] >= [1] = 0() Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) from#(X) -> c_9(cons#(X,n__from(s(X)))) pi#(X) -> c_11(2ndspos#(X,from(0()))) square#(X) -> c_14(times#(X,X)) Strict TRS Rules: cons(X1,X2) -> n__cons(X1,X2) plus(s(X),Y) -> s(plus(X,Y)) Weak DP Rules: activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_10(X) plus#(0(),Y) -> c_12(Y) plus#(s(X),Y) -> c_13(plus#(X,Y)) times#(0(),Y) -> c_15() times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))) Weak TRS Rules: activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} 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(from) = [0] p(n__cons) = [1] x2 + [2] p(n__from) = [0] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [1] x2 + [0] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [4] p(square) = [0] p(times) = [0] p(2ndsneg#) = [1] x1 + [1] x2 + [0] p(2ndspos#) = [1] x1 + [1] x2 + [3] p(activate#) = [2] p(cons#) = [2] p(from#) = [2] p(pi#) = [4] x1 + [2] p(plus#) = [1] x1 + [1] x2 + [0] p(square#) = [1] x1 + [4] p(times#) = [1] x2 + [1] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [5] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [5] p(c_5) = [2] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [2] p(c_9) = [1] x1 + [0] p(c_10) = [1] p(c_11) = [1] x1 + [7] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [2] p(c_14) = [1] x1 + [0] p(c_15) = [1] p(c_16) = [1] x1 + [0] Following rules are strictly oriented: 2ndspos#(s(N) = [1] N + [1] Z + [9] ,cons(X,n__cons(Y,Z))) > [1] N + [1] Z + [7] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) square#(X) = [1] X + [4] > [1] X + [1] = c_14(times#(X,X)) Following rules are (at-least) weakly oriented: 2ndsneg#(s(N) = [1] N + [1] Z + [6] ,cons(X,n__cons(Y,Z))) >= [1] N + [1] Z + [10] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) activate#(X) = [2] >= [2] = c_5(X) activate#(n__cons(X1,X2)) = [2] >= [2] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [2] >= [2] = c_7(from#(X)) cons#(X1,X2) = [2] >= [2] = c_8(X1,X2) from#(X) = [2] >= [2] = c_9(cons#(X,n__from(s(X)))) from#(X) = [2] >= [1] = c_10(X) pi#(X) = [4] X + [2] >= [1] X + [10] = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = [1] Y + [0] >= [1] Y + [0] = c_12(Y) plus#(s(X),Y) = [1] X + [1] Y + [4] >= [1] X + [1] Y + [2] = c_13(plus#(X,Y)) times#(0(),Y) = [1] Y + [1] >= [1] = c_15() times#(s(X),Y) = [1] Y + [1] >= [1] Y + [0] = c_16(plus#(Y,times(X,Y))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [1] X2 + [2] >= [1] X2 + [0] = cons(X1,X2) activate(n__from(X)) = [0] >= [0] = from(X) cons(X1,X2) = [1] X2 + [0] >= [1] X2 + [2] = n__cons(X1,X2) from(X) = [0] >= [0] = cons(X,n__from(s(X))) from(X) = [0] >= [0] = n__from(X) plus(0(),Y) = [1] Y + [0] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [0] >= [1] Y + [4] = s(plus(X,Y)) times(0(),Y) = [0] >= [0] = 0() times(s(X),Y) = [0] >= [0] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) from#(X) -> c_9(cons#(X,n__from(s(X)))) pi#(X) -> c_11(2ndspos#(X,from(0()))) Strict TRS Rules: cons(X1,X2) -> n__cons(X1,X2) plus(s(X),Y) -> s(plus(X,Y)) Weak DP Rules: 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_10(X) plus#(0(),Y) -> c_12(Y) 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))) Weak TRS Rules: activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [0] p(2ndspos) = [0] p(activate) = [4] x1 + [0] p(cons) = [4] x2 + [0] p(from) = [4] p(n__cons) = [1] x2 + [0] p(n__from) = [1] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [1] x2 + [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 + [0] p(2ndspos#) = [1] x2 + [0] p(activate#) = [0] p(cons#) = [0] p(from#) = [0] p(pi#) = [5] p(plus#) = [1] x2 + [0] p(square#) = [0] p(times#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [1] x1 + [0] Following rules are strictly oriented: pi#(X) = [5] > [4] = c_11(2ndspos#(X,from(0()))) Following rules are (at-least) weakly oriented: 2ndsneg#(s(N) = [4] Z + [0] ,cons(X,n__cons(Y,Z))) >= [4] Z + [0] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(s(N) = [4] Z + [0] ,cons(X,n__cons(Y,Z))) >= [4] Z + [0] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = [0] >= [0] = c_5(X) activate#(n__cons(X1,X2)) = [0] >= [0] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [0] >= [0] = c_7(from#(X)) cons#(X1,X2) = [0] >= [0] = c_8(X1,X2) from#(X) = [0] >= [0] = c_9(cons#(X,n__from(s(X)))) from#(X) = [0] >= [0] = c_10(X) plus#(0(),Y) = [1] Y + [0] >= [1] Y + [0] = c_12(Y) plus#(s(X),Y) = [1] Y + [0] >= [1] Y + [0] = c_13(plus#(X,Y)) square#(X) = [0] >= [0] = c_14(times#(X,X)) times#(0(),Y) = [0] >= [0] = c_15() times#(s(X),Y) = [0] >= [0] = c_16(plus#(Y,times(X,Y))) activate(X) = [4] X + [0] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [4] X2 + [0] >= [4] X2 + [0] = cons(X1,X2) activate(n__from(X)) = [4] >= [4] = from(X) cons(X1,X2) = [4] X2 + [0] >= [1] X2 + [0] = n__cons(X1,X2) from(X) = [4] >= [4] = cons(X,n__from(s(X))) from(X) = [4] >= [1] = n__from(X) plus(0(),Y) = [1] Y + [0] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [0] >= [1] Y + [0] = s(plus(X,Y)) times(0(),Y) = [0] >= [0] = 0() times(s(X),Y) = [0] >= [0] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) from#(X) -> c_9(cons#(X,n__from(s(X)))) Strict TRS Rules: cons(X1,X2) -> n__cons(X1,X2) plus(s(X),Y) -> s(plus(X,Y)) Weak DP Rules: 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) Weak TRS Rules: activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [4] p(2ndsneg) = [0] p(2ndspos) = [0] p(activate) = [1] x1 + [4] p(cons) = [1] x2 + [3] p(from) = [4] p(n__cons) = [1] x2 + [7] p(n__from) = [0] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [1] x2 + [1] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [1] p(square) = [0] p(times) = [1] x1 + [4] p(2ndsneg#) = [1] x2 + [0] p(2ndspos#) = [1] x2 + [3] p(activate#) = [2] p(cons#) = [2] p(from#) = [1] p(pi#) = [1] x1 + [7] p(plus#) = [1] x2 + [0] p(square#) = [1] x1 + [7] p(times#) = [1] x1 + [7] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [5] p(c_5) = [2] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [4] p(c_10) = [1] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [1] p(c_16) = [1] x1 + [4] Following rules are strictly oriented: 2ndsneg#(s(N) = [1] Z + [10] ,cons(X,n__cons(Y,Z))) > [1] Z + [9] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) Following rules are (at-least) weakly oriented: 2ndspos#(s(N) = [1] Z + [13] ,cons(X,n__cons(Y,Z))) >= [1] Z + [11] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = [2] >= [2] = c_5(X) activate#(n__cons(X1,X2)) = [2] >= [2] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [2] >= [1] = c_7(from#(X)) cons#(X1,X2) = [2] >= [0] = c_8(X1,X2) from#(X) = [1] >= [6] = c_9(cons#(X,n__from(s(X)))) from#(X) = [1] >= [1] = c_10(X) pi#(X) = [1] X + [7] >= [7] = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = [1] Y + [0] >= [1] Y + [0] = c_12(Y) plus#(s(X),Y) = [1] Y + [0] >= [1] Y + [0] = c_13(plus#(X,Y)) square#(X) = [1] X + [7] >= [1] X + [7] = c_14(times#(X,X)) times#(0(),Y) = [11] >= [1] = c_15() times#(s(X),Y) = [1] X + [8] >= [1] X + [8] = c_16(plus#(Y,times(X,Y))) activate(X) = [1] X + [4] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [1] X2 + [11] >= [1] X2 + [3] = cons(X1,X2) activate(n__from(X)) = [4] >= [4] = from(X) cons(X1,X2) = [1] X2 + [3] >= [1] X2 + [7] = n__cons(X1,X2) from(X) = [4] >= [3] = cons(X,n__from(s(X))) from(X) = [4] >= [0] = n__from(X) plus(0(),Y) = [1] Y + [1] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [1] >= [1] Y + [2] = s(plus(X,Y)) times(0(),Y) = [8] >= [4] = 0() times(s(X),Y) = [1] X + [5] >= [1] X + [5] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: from#(X) -> c_9(cons#(X,n__from(s(X)))) Strict TRS Rules: cons(X1,X2) -> n__cons(X1,X2) plus(s(X),Y) -> s(plus(X,Y)) Weak DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) Weak TRS Rules: activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [4] x2 + [1] p(2ndspos) = [1] x1 + [0] p(activate) = [1] x1 + [2] p(cons) = [1] x2 + [1] p(from) = [1] p(n__cons) = [1] x2 + [7] p(n__from) = [0] p(negrecip) = [1] x1 + [0] p(pi) = [1] x1 + [0] p(plus) = [1] x2 + [0] p(posrecip) = [1] p(rcons) = [1] x2 + [4] p(rnil) = [0] p(s) = [1] x1 + [0] p(square) = [1] x1 + [0] p(times) = [1] x2 + [1] p(2ndsneg#) = [1] x2 + [2] p(2ndspos#) = [1] x2 + [4] p(activate#) = [3] p(cons#) = [0] p(from#) = [1] p(pi#) = [6] p(plus#) = [1] x2 + [0] p(square#) = [3] x1 + [5] p(times#) = [2] x1 + [1] x2 + [1] p(c_1) = [1] p(c_2) = [1] x1 + [1] x2 + [1] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [5] p(c_5) = [1] p(c_6) = [1] x1 + [3] p(c_7) = [1] x1 + [1] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [1] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [4] p(c_15) = [1] p(c_16) = [1] x1 + [0] Following rules are strictly oriented: from#(X) = [1] > [0] = c_9(cons#(X,n__from(s(X)))) Following rules are (at-least) weakly oriented: 2ndsneg#(s(N) = [1] Z + [10] ,cons(X,n__cons(Y,Z))) >= [1] Z + [10] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(s(N) = [1] Z + [12] ,cons(X,n__cons(Y,Z))) >= [1] Z + [12] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = [3] >= [1] = c_5(X) activate#(n__cons(X1,X2)) = [3] >= [3] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [3] >= [2] = c_7(from#(X)) cons#(X1,X2) = [0] >= [0] = c_8(X1,X2) from#(X) = [1] >= [0] = c_10(X) pi#(X) = [6] >= [6] = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = [1] Y + [0] >= [1] Y + [0] = c_12(Y) plus#(s(X),Y) = [1] Y + [0] >= [1] Y + [0] = c_13(plus#(X,Y)) square#(X) = [3] X + [5] >= [3] X + [5] = c_14(times#(X,X)) times#(0(),Y) = [1] Y + [1] >= [1] = c_15() times#(s(X),Y) = [2] X + [1] Y + [1] >= [1] Y + [1] = c_16(plus#(Y,times(X,Y))) activate(X) = [1] X + [2] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [1] X2 + [9] >= [1] X2 + [1] = cons(X1,X2) activate(n__from(X)) = [2] >= [1] = from(X) cons(X1,X2) = [1] X2 + [1] >= [1] X2 + [7] = n__cons(X1,X2) from(X) = [1] >= [1] = cons(X,n__from(s(X))) from(X) = [1] >= [0] = n__from(X) plus(0(),Y) = [1] Y + [0] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [0] >= [1] Y + [0] = s(plus(X,Y)) times(0(),Y) = [1] Y + [1] >= [0] = 0() times(s(X),Y) = [1] Y + [1] >= [1] Y + [1] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: cons(X1,X2) -> n__cons(X1,X2) plus(s(X),Y) -> s(plus(X,Y)) Weak DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_9(cons#(X,n__from(s(X)))) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) Weak TRS Rules: activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) plus(0(),Y) -> 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [4] p(2ndsneg) = [2] x1 + [4] p(2ndspos) = [4] p(activate) = [1] x1 + [2] p(cons) = [1] x2 + [2] p(from) = [2] p(n__cons) = [1] x2 + [0] p(n__from) = [0] p(negrecip) = [1] x1 + [0] p(pi) = [1] p(plus) = [1] x2 + [0] p(posrecip) = [1] p(rcons) = [0] p(rnil) = [0] p(s) = [1] x1 + [1] p(square) = [1] p(times) = [4] p(2ndsneg#) = [4] x1 + [1] x2 + [0] p(2ndspos#) = [4] x1 + [1] x2 + [4] p(activate#) = [0] p(cons#) = [0] p(from#) = [0] p(pi#) = [6] x1 + [7] p(plus#) = [1] x1 + [1] x2 + [0] p(square#) = [7] x1 + [5] p(times#) = [2] x1 + [4] x2 + [2] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [1] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [1] p(c_13) = [1] x1 + [1] p(c_14) = [1] x1 + [2] p(c_15) = [0] p(c_16) = [1] x1 + [0] Following rules are strictly oriented: cons(X1,X2) = [1] X2 + [2] > [1] X2 + [0] = n__cons(X1,X2) Following rules are (at-least) weakly oriented: 2ndsneg#(s(N) = [4] N + [1] Z + [6] ,cons(X,n__cons(Y,Z))) >= [4] N + [1] Z + [6] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(s(N) = [4] N + [1] Z + [10] ,cons(X,n__cons(Y,Z))) >= [4] N + [1] Z + [2] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = [0] >= [0] = c_5(X) activate#(n__cons(X1,X2)) = [0] >= [0] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [0] >= [0] = c_7(from#(X)) cons#(X1,X2) = [0] >= [0] = c_8(X1,X2) from#(X) = [0] >= [0] = c_9(cons#(X,n__from(s(X)))) from#(X) = [0] >= [0] = c_10(X) pi#(X) = [6] X + [7] >= [4] X + [6] = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = [1] Y + [4] >= [1] Y + [1] = c_12(Y) plus#(s(X),Y) = [1] X + [1] Y + [1] >= [1] X + [1] Y + [1] = c_13(plus#(X,Y)) square#(X) = [7] X + [5] >= [6] X + [4] = c_14(times#(X,X)) times#(0(),Y) = [4] Y + [10] >= [0] = c_15() times#(s(X),Y) = [2] X + [4] Y + [4] >= [1] Y + [4] = c_16(plus#(Y,times(X,Y))) activate(X) = [1] X + [2] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [1] X2 + [2] >= [1] X2 + [2] = cons(X1,X2) activate(n__from(X)) = [2] >= [2] = from(X) from(X) = [2] >= [2] = cons(X,n__from(s(X))) from(X) = [2] >= [0] = n__from(X) plus(0(),Y) = [1] Y + [0] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [0] >= [1] Y + [1] = s(plus(X,Y)) times(0(),Y) = [4] >= [4] = 0() times(s(X),Y) = [4] >= [4] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: plus(s(X),Y) -> s(plus(X,Y)) Weak DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_9(cons#(X,n__from(s(X)))) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) Weak TRS Rules: 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) plus(0(),Y) -> 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: NaturalPI {shape = Mixed 3, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any intersect of strict-rules and rewrite-rules, greedy = NoGreedy} Proof: We apply a polynomial interpretation of kind constructor-based(mixed(3)): The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = 0 p(2ndsneg) = 0 p(2ndspos) = 0 p(activate) = x1 p(cons) = x2 p(from) = 0 p(n__cons) = x2 p(n__from) = 0 p(negrecip) = 0 p(pi) = 0 p(plus) = x1 + x1^2 + x2 p(posrecip) = 0 p(rcons) = 0 p(rnil) = 0 p(s) = 1 + x1 p(square) = 0 p(times) = x1*x2 + x1*x2^2 p(2ndsneg#) = 1 + x1 + x1*x2 + x1^2*x2 + x2 p(2ndspos#) = 1 + x1 + x1^2*x2 + x2 p(activate#) = 1 p(cons#) = 0 p(from#) = 0 p(pi#) = 1 + x1 p(plus#) = x2 p(square#) = 1 + x1 + x1^2 + x1^3 p(times#) = 1 + x1 + x1*x2 + x1*x2^2 p(c_1) = 0 p(c_2) = x1 + x2 p(c_3) = 0 p(c_4) = x1 + x2 p(c_5) = 0 p(c_6) = x1 p(c_7) = 1 + x1 p(c_8) = 0 p(c_9) = x1 p(c_10) = 0 p(c_11) = x1 p(c_12) = x1 p(c_13) = x1 p(c_14) = x1 p(c_15) = 0 p(c_16) = x1 Following rules are strictly oriented: plus(s(X),Y) = 2 + 3*X + X^2 + Y > 1 + X + X^2 + Y = s(plus(X,Y)) Following rules are (at-least) weakly oriented: 2ndsneg#(s(N) = 2 + N + 3*N*Z + N^2*Z + 3*Z ,cons(X,n__cons(Y,Z))) >= 2 + N + N^2*Z + Z = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(s(N) = 2 + N + 2*N*Z + N^2*Z + 2*Z ,cons(X,n__cons(Y,Z))) >= 2 + N + N*Z + N^2*Z + Z = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = 1 >= 0 = c_5(X) activate#(n__cons(X1,X2)) = 1 >= 0 = c_6(cons#(X1,X2)) activate#(n__from(X)) = 1 >= 1 = c_7(from#(X)) cons#(X1,X2) = 0 >= 0 = c_8(X1,X2) from#(X) = 0 >= 0 = c_9(cons#(X,n__from(s(X)))) from#(X) = 0 >= 0 = c_10(X) pi#(X) = 1 + X >= 1 + X = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = Y >= Y = c_12(Y) plus#(s(X),Y) = Y >= Y = c_13(plus#(X,Y)) square#(X) = 1 + X + X^2 + X^3 >= 1 + X + X^2 + X^3 = c_14(times#(X,X)) times#(0(),Y) = 1 >= 0 = c_15() times#(s(X),Y) = 2 + X + X*Y + X*Y^2 + Y + Y^2 >= X*Y + X*Y^2 = c_16(plus#(Y,times(X,Y))) activate(X) = X >= X = X activate(n__cons(X1,X2)) = X2 >= X2 = cons(X1,X2) activate(n__from(X)) = 0 >= 0 = from(X) cons(X1,X2) = X2 >= X2 = n__cons(X1,X2) from(X) = 0 >= 0 = cons(X,n__from(s(X))) from(X) = 0 >= 0 = n__from(X) plus(0(),Y) = Y >= Y = Y times(0(),Y) = 0 >= 0 = 0() times(s(X),Y) = X*Y + X*Y^2 + Y + Y^2 >= X*Y + X*Y^2 + Y + Y^2 = plus(Y,times(X,Y)) *** 1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_9(cons#(X,n__from(s(X)))) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) Weak TRS Rules: 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) 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: EmptyProcessor Proof: The problem is already closed. The intended complexity is O(1). *** 1.1.1.1.2 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_9(cons#(X,n__from(s(X)))) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) Strict TRS Rules: 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) 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)) Weak DP Rules: 2ndspos#(0(),Z) -> c_3() Weak TRS Rules: 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [0] p(2ndspos) = [0] p(activate) = [1] x1 + [2] p(cons) = [1] x2 + [0] p(from) = [1] x1 + [0] p(n__cons) = [1] x2 + [0] p(n__from) = [1] x1 + [1] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [1] x2 + [0] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [3] p(square) = [0] p(times) = [3] x2 + [0] p(2ndsneg#) = [1] x2 + [6] p(2ndspos#) = [1] x2 + [0] p(activate#) = [0] p(cons#) = [7] p(from#) = [0] p(pi#) = [0] p(plus#) = [2] x1 + [1] x2 + [3] p(square#) = [5] x1 + [0] p(times#) = [5] x2 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [4] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [1] x1 + [0] Following rules are strictly oriented: 2ndsneg#(s(N) = [1] Z + [6] ,cons(X,n__cons(Y,Z))) > [1] Z + [2] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) cons#(X1,X2) = [7] > [0] = c_8(X1,X2) plus#(s(X),Y) = [2] X + [1] Y + [9] > [2] X + [1] Y + [3] = c_13(plus#(X,Y)) activate(X) = [1] X + [2] > [1] X + [0] = X activate(n__cons(X1,X2)) = [1] X2 + [2] > [1] X2 + [0] = cons(X1,X2) activate(n__from(X)) = [1] X + [3] > [1] X + [0] = from(X) Following rules are (at-least) weakly oriented: 2ndspos#(0(),Z) = [1] Z + [0] >= [0] = c_3() 2ndspos#(s(N) = [1] Z + [0] ,cons(X,n__cons(Y,Z))) >= [1] Z + [8] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = [0] >= [0] = c_5(X) activate#(n__cons(X1,X2)) = [0] >= [7] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [0] >= [0] = c_7(from#(X)) from#(X) = [0] >= [7] = c_9(cons#(X,n__from(s(X)))) from#(X) = [0] >= [0] = c_10(X) pi#(X) = [0] >= [0] = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = [1] Y + [3] >= [1] Y + [4] = c_12(Y) square#(X) = [5] X + [0] >= [5] X + [0] = c_14(times#(X,X)) times#(s(X),Y) = [5] Y + [0] >= [5] Y + [3] = c_16(plus#(Y,times(X,Y))) cons(X1,X2) = [1] X2 + [0] >= [1] X2 + [0] = n__cons(X1,X2) from(X) = [1] X + [0] >= [1] X + [4] = cons(X,n__from(s(X))) from(X) = [1] X + [0] >= [1] X + [1] = n__from(X) plus(0(),Y) = [1] Y + [0] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [0] >= [1] Y + [3] = s(plus(X,Y)) times(0(),Y) = [3] Y + [0] >= [0] = 0() times(s(X),Y) = [3] Y + [0] >= [3] Y + [0] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.2.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) activate#(X) -> c_5(X) 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)))) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) square#(X) -> c_14(times#(X,X)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))) Strict TRS Rules: cons(X1,X2) -> n__cons(X1,X2) 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)) Weak DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(0(),Z) -> c_3() cons#(X1,X2) -> c_8(X1,X2) plus#(s(X),Y) -> c_13(plus#(X,Y)) Weak TRS Rules: activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [5] p(2ndsneg) = [0] p(2ndspos) = [0] p(activate) = [1] x1 + [2] p(cons) = [1] x2 + [3] p(from) = [2] p(n__cons) = [1] x2 + [2] p(n__from) = [0] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [1] x2 + [2] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [4] p(square) = [1] x1 + [0] p(times) = [5] x2 + [3] p(2ndsneg#) = [2] x1 + [1] x2 + [0] p(2ndspos#) = [2] x1 + [1] x2 + [0] p(activate#) = [0] p(cons#) = [5] p(from#) = [0] p(pi#) = [4] x1 + [4] p(plus#) = [1] x2 + [1] p(square#) = [5] x1 + [0] p(times#) = [5] x2 + [2] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [4] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [4] p(c_8) = [4] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [1] x1 + [0] Following rules are strictly oriented: 2ndspos#(s(N) = [2] N + [1] Z + [13] ,cons(X,n__cons(Y,Z))) > [2] N + [1] Z + [2] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) pi#(X) = [4] X + [4] > [2] X + [2] = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = [1] Y + [1] > [1] Y + [0] = c_12(Y) cons(X1,X2) = [1] X2 + [3] > [1] X2 + [2] = n__cons(X1,X2) from(X) = [2] > [0] = n__from(X) plus(0(),Y) = [1] Y + [2] > [1] Y + [0] = Y Following rules are (at-least) weakly oriented: 2ndsneg#(s(N) = [2] N + [1] Z + [13] ,cons(X,n__cons(Y,Z))) >= [2] N + [1] Z + [6] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(0(),Z) = [1] Z + [10] >= [0] = c_3() activate#(X) = [0] >= [0] = c_5(X) activate#(n__cons(X1,X2)) = [0] >= [5] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [0] >= [4] = c_7(from#(X)) cons#(X1,X2) = [5] >= [4] = c_8(X1,X2) from#(X) = [0] >= [5] = c_9(cons#(X,n__from(s(X)))) from#(X) = [0] >= [0] = c_10(X) plus#(s(X),Y) = [1] Y + [1] >= [1] Y + [1] = c_13(plus#(X,Y)) square#(X) = [5] X + [0] >= [5] X + [2] = c_14(times#(X,X)) times#(s(X),Y) = [5] Y + [2] >= [5] Y + [4] = c_16(plus#(Y,times(X,Y))) activate(X) = [1] X + [2] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [1] X2 + [4] >= [1] X2 + [3] = cons(X1,X2) activate(n__from(X)) = [2] >= [2] = from(X) from(X) = [2] >= [3] = cons(X,n__from(s(X))) plus(s(X),Y) = [1] Y + [2] >= [1] Y + [6] = s(plus(X,Y)) times(0(),Y) = [5] Y + [3] >= [5] = 0() times(s(X),Y) = [5] Y + [3] >= [5] Y + [5] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.2.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: activate#(X) -> c_5(X) 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)))) from#(X) -> c_10(X) square#(X) -> c_14(times#(X,X)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))) Strict TRS Rules: from(X) -> cons(X,n__from(s(X))) plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) Weak DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(0(),Z) -> c_3() 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) cons#(X1,X2) -> c_8(X1,X2) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) plus#(s(X),Y) -> c_13(plus#(X,Y)) Weak TRS Rules: 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) -> n__from(X) plus(0(),Y) -> 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [3] p(2ndsneg) = [0] p(2ndspos) = [0] p(activate) = [1] x1 + [1] p(cons) = [1] x2 + [5] p(from) = [0] p(n__cons) = [1] x2 + [4] p(n__from) = [0] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [1] x2 + [5] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [3] p(square) = [0] p(times) = [1] x2 + [0] p(2ndsneg#) = [1] x2 + [0] p(2ndspos#) = [1] x2 + [0] p(activate#) = [4] p(cons#) = [2] p(from#) = [1] p(pi#) = [1] p(plus#) = [3] x1 + [1] x2 + [1] p(square#) = [7] x1 + [0] p(times#) = [3] x1 + [4] x2 + [5] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [4] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [4] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [5] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [1] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [3] p(c_14) = [1] x1 + [0] p(c_15) = [2] p(c_16) = [1] x1 + [0] Following rules are strictly oriented: activate#(X) = [4] > [0] = c_5(X) activate#(n__cons(X1,X2)) = [4] > [2] = c_6(cons#(X1,X2)) from#(X) = [1] > [0] = c_10(X) times#(s(X),Y) = [3] X + [4] Y + [14] > [4] Y + [1] = c_16(plus#(Y,times(X,Y))) Following rules are (at-least) weakly oriented: 2ndsneg#(s(N) = [1] Z + [9] ,cons(X,n__cons(Y,Z))) >= [1] Z + [9] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(0(),Z) = [1] Z + [0] >= [0] = c_3() 2ndspos#(s(N) = [1] Z + [9] ,cons(X,n__cons(Y,Z))) >= [1] Z + [9] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(n__from(X)) = [4] >= [6] = c_7(from#(X)) cons#(X1,X2) = [2] >= [0] = c_8(X1,X2) from#(X) = [1] >= [2] = c_9(cons#(X,n__from(s(X)))) pi#(X) = [1] >= [1] = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = [1] Y + [10] >= [1] Y + [0] = c_12(Y) plus#(s(X),Y) = [3] X + [1] Y + [10] >= [3] X + [1] Y + [4] = c_13(plus#(X,Y)) square#(X) = [7] X + [0] >= [7] X + [5] = c_14(times#(X,X)) activate(X) = [1] X + [1] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [1] X2 + [5] >= [1] X2 + [5] = cons(X1,X2) activate(n__from(X)) = [1] >= [0] = from(X) cons(X1,X2) = [1] X2 + [5] >= [1] X2 + [4] = n__cons(X1,X2) from(X) = [0] >= [5] = cons(X,n__from(s(X))) from(X) = [0] >= [0] = n__from(X) plus(0(),Y) = [1] Y + [5] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [5] >= [1] Y + [8] = s(plus(X,Y)) times(0(),Y) = [1] Y + [0] >= [3] = 0() times(s(X),Y) = [1] Y + [0] >= [1] Y + [5] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.2.1.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: activate#(n__from(X)) -> c_7(from#(X)) from#(X) -> c_9(cons#(X,n__from(s(X)))) square#(X) -> c_14(times#(X,X)) Strict TRS Rules: from(X) -> cons(X,n__from(s(X))) plus(s(X),Y) -> s(plus(X,Y)) times(0(),Y) -> 0() times(s(X),Y) -> plus(Y,times(X,Y)) Weak DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,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#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) plus#(s(X),Y) -> c_13(plus#(X,Y)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))) Weak TRS Rules: 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) -> n__from(X) plus(0(),Y) -> 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [2] p(2ndsneg) = [1] x1 + [4] p(2ndspos) = [4] x1 + [0] p(activate) = [3] x1 + [0] p(cons) = [3] x2 + [0] p(from) = [3] p(n__cons) = [1] x2 + [0] p(n__from) = [1] p(negrecip) = [1] x1 + [0] p(pi) = [4] x1 + [1] p(plus) = [1] x2 + [4] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x1 + [0] p(rnil) = [1] p(s) = [1] x1 + [4] p(square) = [2] p(times) = [2] x1 + [1] x2 + [4] p(2ndsneg#) = [1] x2 + [1] p(2ndspos#) = [1] x2 + [1] p(activate#) = [0] p(cons#) = [0] p(from#) = [0] p(pi#) = [4] p(plus#) = [1] x1 + [1] x2 + [4] p(square#) = [4] x1 + [1] p(times#) = [2] x1 + [2] x2 + [0] p(c_1) = [4] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [1] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [2] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [1] p(c_13) = [1] x1 + [2] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [1] x1 + [0] Following rules are strictly oriented: square#(X) = [4] X + [1] > [4] X + [0] = c_14(times#(X,X)) times(0(),Y) = [1] Y + [8] > [2] = 0() times(s(X),Y) = [2] X + [1] Y + [12] > [2] X + [1] Y + [8] = plus(Y,times(X,Y)) Following rules are (at-least) weakly oriented: 2ndsneg#(s(N) = [3] Z + [1] ,cons(X,n__cons(Y,Z))) >= [3] Z + [1] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(0(),Z) = [1] Z + [1] >= [1] = c_3() 2ndspos#(s(N) = [3] Z + [1] ,cons(X,n__cons(Y,Z))) >= [3] Z + [1] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = [0] >= [0] = c_5(X) activate#(n__cons(X1,X2)) = [0] >= [0] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [0] >= [0] = c_7(from#(X)) cons#(X1,X2) = [0] >= [0] = c_8(X1,X2) from#(X) = [0] >= [2] = c_9(cons#(X,n__from(s(X)))) from#(X) = [0] >= [0] = c_10(X) pi#(X) = [4] >= [4] = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = [1] Y + [6] >= [1] Y + [1] = c_12(Y) plus#(s(X),Y) = [1] X + [1] Y + [8] >= [1] X + [1] Y + [6] = c_13(plus#(X,Y)) times#(s(X),Y) = [2] X + [2] Y + [8] >= [2] X + [2] Y + [8] = c_16(plus#(Y,times(X,Y))) activate(X) = [3] X + [0] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [3] X2 + [0] >= [3] X2 + [0] = cons(X1,X2) activate(n__from(X)) = [3] >= [3] = from(X) cons(X1,X2) = [3] X2 + [0] >= [1] X2 + [0] = n__cons(X1,X2) from(X) = [3] >= [3] = cons(X,n__from(s(X))) from(X) = [3] >= [1] = n__from(X) plus(0(),Y) = [1] Y + [4] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [4] >= [1] Y + [8] = s(plus(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.2.1.1.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: activate#(n__from(X)) -> c_7(from#(X)) from#(X) -> c_9(cons#(X,n__from(s(X)))) Strict TRS Rules: from(X) -> cons(X,n__from(s(X))) plus(s(X),Y) -> s(plus(X,Y)) Weak DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,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#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) Weak TRS Rules: 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) -> n__from(X) plus(0(),Y) -> 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [1] p(2ndsneg) = [0] p(2ndspos) = [0] p(activate) = [1] x1 + [0] p(cons) = [1] x2 + [4] p(from) = [0] p(n__cons) = [1] x2 + [4] p(n__from) = [0] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [1] x2 + [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) = [2] x2 + [1] p(2ndsneg#) = [1] x2 + [0] p(2ndspos#) = [1] x2 + [4] p(activate#) = [3] p(cons#) = [0] p(from#) = [0] p(pi#) = [2] x1 + [5] p(plus#) = [4] x1 + [1] x2 + [1] p(square#) = [7] x1 + [6] p(times#) = [6] x2 + [6] p(c_1) = [1] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [3] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [1] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [1] x1 + [4] Following rules are strictly oriented: activate#(n__from(X)) = [3] > [0] = c_7(from#(X)) Following rules are (at-least) weakly oriented: 2ndsneg#(s(N) = [1] Z + [8] ,cons(X,n__cons(Y,Z))) >= [1] Z + [7] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(0(),Z) = [1] Z + [4] >= [0] = c_3() 2ndspos#(s(N) = [1] Z + [12] ,cons(X,n__cons(Y,Z))) >= [1] Z + [3] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = [3] >= [3] = c_5(X) activate#(n__cons(X1,X2)) = [3] >= [0] = c_6(cons#(X1,X2)) cons#(X1,X2) = [0] >= [0] = c_8(X1,X2) from#(X) = [0] >= [0] = c_9(cons#(X,n__from(s(X)))) from#(X) = [0] >= [0] = c_10(X) pi#(X) = [2] X + [5] >= [4] = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = [1] Y + [5] >= [1] Y + [1] = c_12(Y) plus#(s(X),Y) = [4] X + [1] Y + [1] >= [4] X + [1] Y + [1] = c_13(plus#(X,Y)) square#(X) = [7] X + [6] >= [6] X + [6] = c_14(times#(X,X)) times#(s(X),Y) = [6] Y + [6] >= [6] Y + [6] = c_16(plus#(Y,times(X,Y))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [1] X2 + [4] >= [1] X2 + [4] = cons(X1,X2) activate(n__from(X)) = [0] >= [0] = from(X) cons(X1,X2) = [1] X2 + [4] >= [1] X2 + [4] = n__cons(X1,X2) from(X) = [0] >= [4] = cons(X,n__from(s(X))) from(X) = [0] >= [0] = n__from(X) plus(0(),Y) = [1] Y + [0] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [0] >= [1] Y + [0] = s(plus(X,Y)) times(0(),Y) = [2] Y + [1] >= [1] = 0() times(s(X),Y) = [2] Y + [1] >= [2] Y + [1] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.2.1.1.1.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: from#(X) -> c_9(cons#(X,n__from(s(X)))) Strict TRS Rules: from(X) -> cons(X,n__from(s(X))) plus(s(X),Y) -> s(plus(X,Y)) Weak DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,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#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) Weak TRS Rules: 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) -> n__from(X) plus(0(),Y) -> 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [4] p(2ndspos) = [1] x1 + [1] x2 + [4] p(activate) = [1] x1 + [0] p(cons) = [1] x2 + [0] p(from) = [1] x1 + [0] p(n__cons) = [1] x2 + [0] p(n__from) = [1] x1 + [0] p(negrecip) = [0] p(pi) = [1] p(plus) = [1] x2 + [4] p(posrecip) = [1] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [4] p(square) = [0] p(times) = [2] x1 + [4] p(2ndsneg#) = [2] x1 + [1] x2 + [1] p(2ndspos#) = [2] x1 + [1] x2 + [1] p(activate#) = [2] p(cons#) = [0] p(from#) = [1] p(pi#) = [4] x1 + [6] p(plus#) = [1] x2 + [4] p(square#) = [5] x1 + [7] p(times#) = [2] x1 + [2] x2 + [7] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [6] p(c_3) = [1] p(c_4) = [1] x1 + [1] x2 + [6] p(c_5) = [2] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [5] p(c_12) = [1] x1 + [1] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [1] p(c_16) = [1] x1 + [3] Following rules are strictly oriented: from#(X) = [1] > [0] = c_9(cons#(X,n__from(s(X)))) Following rules are (at-least) weakly oriented: 2ndsneg#(s(N) = [2] N + [1] Z + [9] ,cons(X,n__cons(Y,Z))) >= [2] N + [1] Z + [9] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(0(),Z) = [1] Z + [1] >= [1] = c_3() 2ndspos#(s(N) = [2] N + [1] Z + [9] ,cons(X,n__cons(Y,Z))) >= [2] N + [1] Z + [9] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = [2] >= [2] = c_5(X) activate#(n__cons(X1,X2)) = [2] >= [0] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [2] >= [1] = c_7(from#(X)) cons#(X1,X2) = [0] >= [0] = c_8(X1,X2) from#(X) = [1] >= [0] = c_10(X) pi#(X) = [4] X + [6] >= [2] X + [6] = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = [1] Y + [4] >= [1] Y + [1] = c_12(Y) plus#(s(X),Y) = [1] Y + [4] >= [1] Y + [4] = c_13(plus#(X,Y)) square#(X) = [5] X + [7] >= [4] X + [7] = c_14(times#(X,X)) times#(s(X),Y) = [2] X + [2] Y + [15] >= [2] X + [11] = c_16(plus#(Y,times(X,Y))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [1] X2 + [0] >= [1] X2 + [0] = cons(X1,X2) activate(n__from(X)) = [1] X + [0] >= [1] X + [0] = from(X) cons(X1,X2) = [1] X2 + [0] >= [1] X2 + [0] = n__cons(X1,X2) from(X) = [1] X + [0] >= [1] X + [4] = cons(X,n__from(s(X))) from(X) = [1] X + [0] >= [1] X + [0] = n__from(X) plus(0(),Y) = [1] Y + [4] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [4] >= [1] Y + [8] = s(plus(X,Y)) times(0(),Y) = [4] >= [0] = 0() times(s(X),Y) = [2] X + [12] >= [2] X + [8] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.2.1.1.1.1.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: from(X) -> cons(X,n__from(s(X))) plus(s(X),Y) -> s(plus(X,Y)) Weak DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,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#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_9(cons#(X,n__from(s(X)))) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) Weak TRS Rules: 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) -> n__from(X) plus(0(),Y) -> 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [4] x1 + [1] p(2ndspos) = [4] x1 + [0] p(activate) = [7] x1 + [3] p(cons) = [7] x2 + [2] p(from) = [3] p(n__cons) = [1] x2 + [1] p(n__from) = [0] p(negrecip) = [1] x1 + [2] p(pi) = [1] x1 + [1] p(plus) = [1] x2 + [0] p(posrecip) = [1] x1 + [1] p(rcons) = [1] x2 + [0] p(rnil) = [4] p(s) = [1] x1 + [1] p(square) = [2] p(times) = [2] x2 + [0] p(2ndsneg#) = [2] x1 + [1] x2 + [4] p(2ndspos#) = [2] x1 + [1] x2 + [0] p(activate#) = [3] p(cons#) = [2] p(from#) = [2] p(pi#) = [2] x1 + [3] p(plus#) = [1] x2 + [0] p(square#) = [6] x1 + [7] p(times#) = [1] x1 + [5] x2 + [5] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [5] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [3] p(c_6) = [1] x1 + [1] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [2] p(c_15) = [1] p(c_16) = [1] x1 + [2] Following rules are strictly oriented: from(X) = [3] > [2] = cons(X,n__from(s(X))) Following rules are (at-least) weakly oriented: 2ndsneg#(s(N) = [2] N + [7] Z + [15] ,cons(X,n__cons(Y,Z))) >= [2] N + [7] Z + [11] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(0(),Z) = [1] Z + [0] >= [0] = c_3() 2ndspos#(s(N) = [2] N + [7] Z + [11] ,cons(X,n__cons(Y,Z))) >= [2] N + [7] Z + [10] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = [3] >= [3] = c_5(X) activate#(n__cons(X1,X2)) = [3] >= [3] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [3] >= [2] = c_7(from#(X)) cons#(X1,X2) = [2] >= [0] = c_8(X1,X2) from#(X) = [2] >= [2] = c_9(cons#(X,n__from(s(X)))) from#(X) = [2] >= [0] = c_10(X) pi#(X) = [2] X + [3] >= [2] X + [3] = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = [1] Y + [0] >= [1] Y + [0] = c_12(Y) plus#(s(X),Y) = [1] Y + [0] >= [1] Y + [0] = c_13(plus#(X,Y)) square#(X) = [6] X + [7] >= [6] X + [7] = c_14(times#(X,X)) times#(s(X),Y) = [1] X + [5] Y + [6] >= [2] Y + [2] = c_16(plus#(Y,times(X,Y))) activate(X) = [7] X + [3] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [7] X2 + [10] >= [7] X2 + [2] = cons(X1,X2) activate(n__from(X)) = [3] >= [3] = from(X) cons(X1,X2) = [7] X2 + [2] >= [1] X2 + [1] = n__cons(X1,X2) from(X) = [3] >= [0] = n__from(X) plus(0(),Y) = [1] Y + [0] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [0] >= [1] Y + [1] = s(plus(X,Y)) times(0(),Y) = [2] Y + [0] >= [0] = 0() times(s(X),Y) = [2] Y + [0] >= [2] Y + [0] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.2.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: plus(s(X),Y) -> s(plus(X,Y)) Weak DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,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#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_9(cons#(X,n__from(s(X)))) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) Weak TRS Rules: 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) plus(0(),Y) -> 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: NaturalPI {shape = Mixed 3, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any intersect of strict-rules and rewrite-rules, greedy = NoGreedy} Proof: We apply a polynomial interpretation of kind constructor-based(mixed(3)): The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = 0 p(2ndsneg) = 0 p(2ndspos) = 0 p(activate) = 1 + x1 p(cons) = 1 + x2 p(from) = 1 p(n__cons) = x2 p(n__from) = 0 p(negrecip) = 0 p(pi) = 0 p(plus) = x1 + x1^2 + x2 p(posrecip) = 0 p(rcons) = 0 p(rnil) = 0 p(s) = 1 + x1 p(square) = 0 p(times) = x1*x2 + x1*x2^2 p(2ndsneg#) = x1*x2^2 + x1^2 + x2 p(2ndspos#) = x1*x2^2 + x1^2 + x2 p(activate#) = 1 p(cons#) = 0 p(from#) = 0 p(pi#) = 1 + x1 + x1^2 + x1^3 p(plus#) = x1 + x1^2 + x2 p(square#) = x1^2 + x1^3 p(times#) = x1*x2 + x1*x2^2 p(c_1) = 0 p(c_2) = x1 + x2 p(c_3) = 0 p(c_4) = 1 + x1 + x2 p(c_5) = 1 p(c_6) = 1 + x1 p(c_7) = x1 p(c_8) = 0 p(c_9) = x1 p(c_10) = 0 p(c_11) = x1 p(c_12) = x1 p(c_13) = x1 p(c_14) = x1 p(c_15) = 0 p(c_16) = x1 Following rules are strictly oriented: plus(s(X),Y) = 2 + 3*X + X^2 + Y > 1 + X + X^2 + Y = s(plus(X,Y)) Following rules are (at-least) weakly oriented: 2ndsneg#(s(N) = 3 + 3*N + 2*N*Z + N*Z^2 + N^2 + 3*Z + Z^2 ,cons(X,n__cons(Y,Z))) >= 2 + N + 2*N*Z + N*Z^2 + N^2 + Z = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(0(),Z) = Z >= 0 = c_3() 2ndspos#(s(N) = 3 + 3*N + 2*N*Z + N*Z^2 + N^2 + 3*Z + Z^2 ,cons(X,n__cons(Y,Z))) >= 3 + N + 2*N*Z + N*Z^2 + N^2 + Z = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = 1 >= 1 = c_5(X) activate#(n__cons(X1,X2)) = 1 >= 1 = c_6(cons#(X1,X2)) activate#(n__from(X)) = 1 >= 0 = c_7(from#(X)) cons#(X1,X2) = 0 >= 0 = c_8(X1,X2) from#(X) = 0 >= 0 = c_9(cons#(X,n__from(s(X)))) from#(X) = 0 >= 0 = c_10(X) pi#(X) = 1 + X + X^2 + X^3 >= 1 + X + X^2 = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = Y >= Y = c_12(Y) plus#(s(X),Y) = 2 + 3*X + X^2 + Y >= X + X^2 + Y = c_13(plus#(X,Y)) square#(X) = X^2 + X^3 >= X^2 + X^3 = c_14(times#(X,X)) times#(s(X),Y) = X*Y + X*Y^2 + Y + Y^2 >= X*Y + X*Y^2 + Y + Y^2 = c_16(plus#(Y,times(X,Y))) activate(X) = 1 + X >= X = X activate(n__cons(X1,X2)) = 1 + X2 >= 1 + X2 = cons(X1,X2) activate(n__from(X)) = 1 >= 1 = from(X) cons(X1,X2) = 1 + X2 >= X2 = n__cons(X1,X2) from(X) = 1 >= 1 = cons(X,n__from(s(X))) from(X) = 1 >= 0 = n__from(X) plus(0(),Y) = Y >= Y = Y times(0(),Y) = 0 >= 0 = 0() times(s(X),Y) = X*Y + X*Y^2 + Y + Y^2 >= X*Y + X*Y^2 + Y + Y^2 = plus(Y,times(X,Y)) *** 1.1.1.1.2.1.1.1.1.1.1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,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#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_9(cons#(X,n__from(s(X)))) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) Weak TRS Rules: 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) 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: EmptyProcessor Proof: The problem is already closed. The intended complexity is O(1). *** 1.1.1.1.3 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_9(cons#(X,n__from(s(X)))) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) Strict TRS Rules: 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) 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)) Weak DP Rules: 2ndsneg#(0(),Z) -> c_1() Weak TRS Rules: 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [0] p(2ndspos) = [0] p(activate) = [1] x1 + [2] p(cons) = [1] x2 + [0] p(from) = [1] x1 + [0] p(n__cons) = [1] x2 + [7] p(n__from) = [1] x1 + [0] p(negrecip) = [1] x1 + [0] p(pi) = [1] x1 + [0] p(plus) = [1] x2 + [0] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [3] p(square) = [0] p(times) = [1] x2 + [0] p(2ndsneg#) = [1] x2 + [0] p(2ndspos#) = [1] x2 + [0] p(activate#) = [3] p(cons#) = [0] p(from#) = [0] p(pi#) = [0] p(plus#) = [4] x1 + [1] x2 + [0] p(square#) = [5] x1 + [0] p(times#) = [5] x2 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [1] x1 + [0] Following rules are strictly oriented: 2ndsneg#(s(N) = [1] Z + [7] ,cons(X,n__cons(Y,Z))) > [1] Z + [5] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(s(N) = [1] Z + [7] ,cons(X,n__cons(Y,Z))) > [1] Z + [5] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = [3] > [0] = c_5(X) activate#(n__cons(X1,X2)) = [3] > [0] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [3] > [0] = c_7(from#(X)) plus#(s(X),Y) = [4] X + [1] Y + [12] > [4] X + [1] Y + [0] = c_13(plus#(X,Y)) activate(X) = [1] X + [2] > [1] X + [0] = X activate(n__cons(X1,X2)) = [1] X2 + [9] > [1] X2 + [0] = cons(X1,X2) activate(n__from(X)) = [1] X + [2] > [1] X + [0] = from(X) Following rules are (at-least) weakly oriented: 2ndsneg#(0(),Z) = [1] Z + [0] >= [0] = c_1() cons#(X1,X2) = [0] >= [0] = c_8(X1,X2) from#(X) = [0] >= [0] = c_9(cons#(X,n__from(s(X)))) from#(X) = [0] >= [0] = c_10(X) pi#(X) = [0] >= [0] = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = [1] Y + [0] >= [1] Y + [0] = c_12(Y) square#(X) = [5] X + [0] >= [5] X + [0] = c_14(times#(X,X)) times#(s(X),Y) = [5] Y + [0] >= [5] Y + [0] = c_16(plus#(Y,times(X,Y))) cons(X1,X2) = [1] X2 + [0] >= [1] X2 + [7] = n__cons(X1,X2) from(X) = [1] X + [0] >= [1] X + [3] = cons(X,n__from(s(X))) from(X) = [1] X + [0] >= [1] X + [0] = n__from(X) plus(0(),Y) = [1] Y + [0] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [0] >= [1] Y + [3] = s(plus(X,Y)) times(0(),Y) = [1] Y + [0] >= [0] = 0() times(s(X),Y) = [1] Y + [0] >= [1] Y + [0] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.3.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_9(cons#(X,n__from(s(X)))) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) square#(X) -> c_14(times#(X,X)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))) Strict TRS Rules: cons(X1,X2) -> n__cons(X1,X2) 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)) Weak DP Rules: 2ndsneg#(0(),Z) -> c_1() 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) plus#(s(X),Y) -> c_13(plus#(X,Y)) Weak TRS Rules: activate(X) -> X activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__from(X)) -> from(X) 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [0] p(2ndspos) = [1] x2 + [0] p(activate) = [1] x1 + [5] p(cons) = [1] x2 + [4] p(from) = [3] p(n__cons) = [1] x2 + [2] p(n__from) = [2] p(negrecip) = [1] x1 + [1] p(pi) = [2] p(plus) = [1] x2 + [4] p(posrecip) = [1] x1 + [1] p(rcons) = [1] x2 + [4] p(rnil) = [1] p(s) = [1] x1 + [0] p(square) = [0] p(times) = [2] p(2ndsneg#) = [1] x2 + [0] p(2ndspos#) = [1] x2 + [0] p(activate#) = [1] p(cons#) = [1] p(from#) = [1] p(pi#) = [0] p(plus#) = [1] x2 + [0] p(square#) = [1] x1 + [0] p(times#) = [1] x2 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [1] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [1] x1 + [0] Following rules are strictly oriented: cons#(X1,X2) = [1] > [0] = c_8(X1,X2) from#(X) = [1] > [0] = c_10(X) cons(X1,X2) = [1] X2 + [4] > [1] X2 + [2] = n__cons(X1,X2) from(X) = [3] > [2] = n__from(X) plus(0(),Y) = [1] Y + [4] > [1] Y + [0] = Y times(0(),Y) = [2] > [0] = 0() Following rules are (at-least) weakly oriented: 2ndsneg#(0(),Z) = [1] Z + [0] >= [0] = c_1() 2ndsneg#(s(N) = [1] Z + [6] ,cons(X,n__cons(Y,Z))) >= [1] Z + [6] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(s(N) = [1] Z + [6] ,cons(X,n__cons(Y,Z))) >= [1] Z + [6] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = [1] >= [1] = c_5(X) activate#(n__cons(X1,X2)) = [1] >= [1] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [1] >= [1] = c_7(from#(X)) from#(X) = [1] >= [1] = c_9(cons#(X,n__from(s(X)))) pi#(X) = [0] >= [3] = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = [1] Y + [0] >= [1] Y + [0] = c_12(Y) plus#(s(X),Y) = [1] Y + [0] >= [1] Y + [0] = c_13(plus#(X,Y)) square#(X) = [1] X + [0] >= [1] X + [0] = c_14(times#(X,X)) times#(s(X),Y) = [1] Y + [0] >= [2] = c_16(plus#(Y,times(X,Y))) activate(X) = [1] X + [5] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [1] X2 + [7] >= [1] X2 + [4] = cons(X1,X2) activate(n__from(X)) = [7] >= [3] = from(X) from(X) = [3] >= [6] = cons(X,n__from(s(X))) plus(s(X),Y) = [1] Y + [4] >= [1] Y + [4] = s(plus(X,Y)) times(s(X),Y) = [2] >= [6] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.3.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: from#(X) -> c_9(cons#(X,n__from(s(X)))) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) square#(X) -> c_14(times#(X,X)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))) Strict TRS Rules: from(X) -> cons(X,n__from(s(X))) plus(s(X),Y) -> s(plus(X,Y)) times(s(X),Y) -> plus(Y,times(X,Y)) Weak DP Rules: 2ndsneg#(0(),Z) -> c_1() 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_10(X) plus#(s(X),Y) -> c_13(plus#(X,Y)) Weak TRS Rules: 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) -> n__from(X) plus(0(),Y) -> Y times(0(),Y) -> 0() 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [0] p(2ndspos) = [0] p(activate) = [1] x1 + [2] p(cons) = [1] x2 + [1] p(from) = [5] p(n__cons) = [1] x2 + [0] p(n__from) = [4] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [1] x2 + [0] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [5] p(square) = [0] p(times) = [0] p(2ndsneg#) = [1] x1 + [1] x2 + [5] p(2ndspos#) = [1] x1 + [1] x2 + [3] p(activate#) = [0] p(cons#) = [0] p(from#) = [0] p(pi#) = [4] x1 + [0] p(plus#) = [1] x2 + [1] p(square#) = [1] p(times#) = [5] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [6] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [1] x1 + [0] Following rules are strictly oriented: plus#(0(),Y) = [1] Y + [1] > [1] Y + [0] = c_12(Y) times#(s(X),Y) = [5] > [1] = c_16(plus#(Y,times(X,Y))) Following rules are (at-least) weakly oriented: 2ndsneg#(0(),Z) = [1] Z + [5] >= [0] = c_1() 2ndsneg#(s(N) = [1] N + [1] Z + [11] ,cons(X,n__cons(Y,Z))) >= [1] N + [1] Z + [11] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(s(N) = [1] N + [1] Z + [9] ,cons(X,n__cons(Y,Z))) >= [1] N + [1] Z + [7] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = [0] >= [0] = c_5(X) activate#(n__cons(X1,X2)) = [0] >= [0] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [0] >= [0] = c_7(from#(X)) cons#(X1,X2) = [0] >= [0] = c_8(X1,X2) from#(X) = [0] >= [0] = c_9(cons#(X,n__from(s(X)))) from#(X) = [0] >= [0] = c_10(X) pi#(X) = [4] X + [0] >= [1] X + [8] = c_11(2ndspos#(X,from(0()))) plus#(s(X),Y) = [1] Y + [1] >= [1] Y + [1] = c_13(plus#(X,Y)) square#(X) = [1] >= [5] = c_14(times#(X,X)) activate(X) = [1] X + [2] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [1] X2 + [2] >= [1] X2 + [1] = cons(X1,X2) activate(n__from(X)) = [6] >= [5] = from(X) cons(X1,X2) = [1] X2 + [1] >= [1] X2 + [0] = n__cons(X1,X2) from(X) = [5] >= [5] = cons(X,n__from(s(X))) from(X) = [5] >= [4] = n__from(X) plus(0(),Y) = [1] Y + [0] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [0] >= [1] Y + [5] = s(plus(X,Y)) times(0(),Y) = [0] >= [0] = 0() times(s(X),Y) = [0] >= [0] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.3.1.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: from#(X) -> c_9(cons#(X,n__from(s(X)))) pi#(X) -> c_11(2ndspos#(X,from(0()))) square#(X) -> c_14(times#(X,X)) Strict TRS Rules: from(X) -> cons(X,n__from(s(X))) plus(s(X),Y) -> s(plus(X,Y)) times(s(X),Y) -> plus(Y,times(X,Y)) Weak DP Rules: 2ndsneg#(0(),Z) -> c_1() 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_10(X) plus#(0(),Y) -> c_12(Y) plus#(s(X),Y) -> c_13(plus#(X,Y)) times#(s(X),Y) -> c_16(plus#(Y,times(X,Y))) Weak TRS Rules: 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) -> n__from(X) plus(0(),Y) -> Y times(0(),Y) -> 0() 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [0] p(2ndspos) = [0] p(activate) = [3] x1 + [0] p(cons) = [3] x2 + [2] p(from) = [6] p(n__cons) = [1] x2 + [2] p(n__from) = [2] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [1] x2 + [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) = [1] x2 + [0] p(2ndsneg#) = [1] x2 + [2] p(2ndspos#) = [1] x2 + [0] p(activate#) = [0] p(cons#) = [0] p(from#) = [0] p(pi#) = [0] p(plus#) = [1] x1 + [1] x2 + [0] p(square#) = [3] x1 + [5] p(times#) = [1] x1 + [2] x2 + [4] p(c_1) = [2] p(c_2) = [1] x1 + [1] x2 + [4] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [6] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [4] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [1] x1 + [0] Following rules are strictly oriented: square#(X) = [3] X + [5] > [3] X + [4] = c_14(times#(X,X)) Following rules are (at-least) weakly oriented: 2ndsneg#(0(),Z) = [1] Z + [2] >= [2] = c_1() 2ndsneg#(s(N) = [3] Z + [10] ,cons(X,n__cons(Y,Z))) >= [3] Z + [4] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(s(N) = [3] Z + [8] ,cons(X,n__cons(Y,Z))) >= [3] Z + [8] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = [0] >= [0] = c_5(X) activate#(n__cons(X1,X2)) = [0] >= [0] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [0] >= [0] = c_7(from#(X)) cons#(X1,X2) = [0] >= [0] = c_8(X1,X2) from#(X) = [0] >= [0] = c_9(cons#(X,n__from(s(X)))) from#(X) = [0] >= [0] = c_10(X) pi#(X) = [0] >= [10] = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = [1] Y + [0] >= [1] Y + [0] = c_12(Y) plus#(s(X),Y) = [1] X + [1] Y + [0] >= [1] X + [1] Y + [0] = c_13(plus#(X,Y)) times#(s(X),Y) = [1] X + [2] Y + [4] >= [2] Y + [0] = c_16(plus#(Y,times(X,Y))) activate(X) = [3] X + [0] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [3] X2 + [6] >= [3] X2 + [2] = cons(X1,X2) activate(n__from(X)) = [6] >= [6] = from(X) cons(X1,X2) = [3] X2 + [2] >= [1] X2 + [2] = n__cons(X1,X2) from(X) = [6] >= [8] = cons(X,n__from(s(X))) from(X) = [6] >= [2] = n__from(X) plus(0(),Y) = [1] Y + [0] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [0] >= [1] Y + [0] = s(plus(X,Y)) times(0(),Y) = [1] Y + [0] >= [0] = 0() times(s(X),Y) = [1] Y + [0] >= [1] Y + [0] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.3.1.1.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: from#(X) -> c_9(cons#(X,n__from(s(X)))) pi#(X) -> c_11(2ndspos#(X,from(0()))) Strict TRS Rules: from(X) -> cons(X,n__from(s(X))) plus(s(X),Y) -> s(plus(X,Y)) times(s(X),Y) -> plus(Y,times(X,Y)) Weak DP Rules: 2ndsneg#(0(),Z) -> c_1() 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_10(X) plus#(0(),Y) -> c_12(Y) 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))) Weak TRS Rules: 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) -> n__from(X) plus(0(),Y) -> Y times(0(),Y) -> 0() 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [0] p(2ndspos) = [0] p(activate) = [1] x1 + [1] p(cons) = [1] x2 + [0] p(from) = [1] p(n__cons) = [1] x2 + [0] p(n__from) = [0] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [1] x2 + [0] p(posrecip) = [1] p(rcons) = [1] x1 + [4] p(rnil) = [0] p(s) = [1] x1 + [1] p(square) = [4] x1 + [0] p(times) = [1] x2 + [0] p(2ndsneg#) = [1] x1 + [1] x2 + [0] p(2ndspos#) = [1] x1 + [1] x2 + [0] p(activate#) = [0] p(cons#) = [0] p(from#) = [0] p(pi#) = [1] x1 + [1] p(plus#) = [5] x1 + [1] x2 + [0] p(square#) = [7] x1 + [0] p(times#) = [7] x2 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [2] p(c_10) = [0] p(c_11) = [1] x1 + [1] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [5] p(c_14) = [1] x1 + [0] p(c_15) = [4] p(c_16) = [1] x1 + [0] Following rules are strictly oriented: from(X) = [1] > [0] = cons(X,n__from(s(X))) Following rules are (at-least) weakly oriented: 2ndsneg#(0(),Z) = [1] Z + [0] >= [0] = c_1() 2ndsneg#(s(N) = [1] N + [1] Z + [1] ,cons(X,n__cons(Y,Z))) >= [1] N + [1] Z + [1] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(s(N) = [1] N + [1] Z + [1] ,cons(X,n__cons(Y,Z))) >= [1] N + [1] Z + [1] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = [0] >= [0] = c_5(X) activate#(n__cons(X1,X2)) = [0] >= [0] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [0] >= [0] = c_7(from#(X)) cons#(X1,X2) = [0] >= [0] = c_8(X1,X2) from#(X) = [0] >= [2] = c_9(cons#(X,n__from(s(X)))) from#(X) = [0] >= [0] = c_10(X) pi#(X) = [1] X + [1] >= [1] X + [2] = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = [1] Y + [0] >= [1] Y + [0] = c_12(Y) plus#(s(X),Y) = [5] X + [1] Y + [5] >= [5] X + [1] Y + [5] = c_13(plus#(X,Y)) square#(X) = [7] X + [0] >= [7] X + [0] = c_14(times#(X,X)) times#(s(X),Y) = [7] Y + [0] >= [6] Y + [0] = c_16(plus#(Y,times(X,Y))) activate(X) = [1] X + [1] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [1] X2 + [1] >= [1] X2 + [0] = cons(X1,X2) activate(n__from(X)) = [1] >= [1] = from(X) cons(X1,X2) = [1] X2 + [0] >= [1] X2 + [0] = n__cons(X1,X2) from(X) = [1] >= [0] = n__from(X) plus(0(),Y) = [1] Y + [0] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [0] >= [1] Y + [1] = s(plus(X,Y)) times(0(),Y) = [1] Y + [0] >= [0] = 0() times(s(X),Y) = [1] Y + [0] >= [1] Y + [0] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.3.1.1.1.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: from#(X) -> c_9(cons#(X,n__from(s(X)))) pi#(X) -> c_11(2ndspos#(X,from(0()))) Strict TRS Rules: plus(s(X),Y) -> s(plus(X,Y)) times(s(X),Y) -> plus(Y,times(X,Y)) Weak DP Rules: 2ndsneg#(0(),Z) -> c_1() 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_10(X) plus#(0(),Y) -> c_12(Y) 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))) Weak TRS Rules: 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) plus(0(),Y) -> Y times(0(),Y) -> 0() 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [1] x1 + [1] p(2ndspos) = [1] p(activate) = [1] x1 + [0] p(cons) = [1] x2 + [0] p(from) = [0] p(n__cons) = [1] x2 + [0] p(n__from) = [0] p(negrecip) = [1] p(pi) = [1] p(plus) = [1] x2 + [0] p(posrecip) = [1] x1 + [2] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [2] p(s) = [1] x1 + [0] p(square) = [1] x1 + [1] p(times) = [0] p(2ndsneg#) = [1] x2 + [2] p(2ndspos#) = [1] x2 + [2] p(activate#) = [0] p(cons#) = [0] p(from#) = [0] p(pi#) = [4] x1 + [6] p(plus#) = [3] x1 + [1] x2 + [0] p(square#) = [3] x1 + [4] p(times#) = [3] x2 + [4] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [1] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [3] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [1] p(c_16) = [1] x1 + [4] Following rules are strictly oriented: pi#(X) = [4] X + [6] > [5] = c_11(2ndspos#(X,from(0()))) Following rules are (at-least) weakly oriented: 2ndsneg#(0(),Z) = [1] Z + [2] >= [0] = c_1() 2ndsneg#(s(N) = [1] Z + [2] ,cons(X,n__cons(Y,Z))) >= [1] Z + [2] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(s(N) = [1] Z + [2] ,cons(X,n__cons(Y,Z))) >= [1] Z + [2] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = [0] >= [0] = c_5(X) activate#(n__cons(X1,X2)) = [0] >= [0] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [0] >= [0] = c_7(from#(X)) cons#(X1,X2) = [0] >= [0] = c_8(X1,X2) from#(X) = [0] >= [0] = c_9(cons#(X,n__from(s(X)))) from#(X) = [0] >= [0] = c_10(X) plus#(0(),Y) = [1] Y + [0] >= [1] Y + [0] = c_12(Y) plus#(s(X),Y) = [3] X + [1] Y + [0] >= [3] X + [1] Y + [0] = c_13(plus#(X,Y)) square#(X) = [3] X + [4] >= [3] X + [4] = c_14(times#(X,X)) times#(s(X),Y) = [3] Y + [4] >= [3] Y + [4] = c_16(plus#(Y,times(X,Y))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [1] X2 + [0] >= [1] X2 + [0] = cons(X1,X2) activate(n__from(X)) = [0] >= [0] = from(X) cons(X1,X2) = [1] X2 + [0] >= [1] X2 + [0] = n__cons(X1,X2) from(X) = [0] >= [0] = cons(X,n__from(s(X))) from(X) = [0] >= [0] = n__from(X) plus(0(),Y) = [1] Y + [0] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [0] >= [1] Y + [0] = s(plus(X,Y)) times(0(),Y) = [0] >= [0] = 0() times(s(X),Y) = [0] >= [0] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.3.1.1.1.1.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: from#(X) -> c_9(cons#(X,n__from(s(X)))) Strict TRS Rules: plus(s(X),Y) -> s(plus(X,Y)) times(s(X),Y) -> plus(Y,times(X,Y)) Weak DP Rules: 2ndsneg#(0(),Z) -> c_1() 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) Weak TRS Rules: 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) plus(0(),Y) -> Y times(0(),Y) -> 0() 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [0] p(2ndsneg) = [0] p(2ndspos) = [0] p(activate) = [4] x1 + [3] p(cons) = [4] x2 + [0] p(from) = [1] p(n__cons) = [1] x2 + [0] p(n__from) = [0] p(negrecip) = [1] x1 + [0] p(pi) = [0] p(plus) = [1] x2 + [3] p(posrecip) = [1] x1 + [0] p(rcons) = [1] x1 + [1] x2 + [0] p(rnil) = [2] p(s) = [1] x1 + [1] p(square) = [0] p(times) = [2] x1 + [1] x2 + [0] p(2ndsneg#) = [4] x1 + [1] x2 + [1] p(2ndspos#) = [4] x1 + [1] x2 + [1] p(activate#) = [1] p(cons#) = [0] p(from#) = [1] p(pi#) = [4] x1 + [2] p(plus#) = [1] x2 + [4] p(square#) = [3] x1 + [4] p(times#) = [2] x1 + [1] x2 + [3] p(c_1) = [1] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [1] p(c_6) = [1] x1 + [1] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [1] p(c_15) = [0] p(c_16) = [1] x1 + [1] Following rules are strictly oriented: from#(X) = [1] > [0] = c_9(cons#(X,n__from(s(X)))) Following rules are (at-least) weakly oriented: 2ndsneg#(0(),Z) = [1] Z + [1] >= [1] = c_1() 2ndsneg#(s(N) = [4] N + [4] Z + [5] ,cons(X,n__cons(Y,Z))) >= [4] N + [4] Z + [5] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(s(N) = [4] N + [4] Z + [5] ,cons(X,n__cons(Y,Z))) >= [4] N + [4] Z + [5] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = [1] >= [1] = c_5(X) activate#(n__cons(X1,X2)) = [1] >= [1] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [1] >= [1] = c_7(from#(X)) cons#(X1,X2) = [0] >= [0] = c_8(X1,X2) from#(X) = [1] >= [1] = c_10(X) pi#(X) = [4] X + [2] >= [4] X + [2] = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = [1] Y + [4] >= [1] Y + [0] = c_12(Y) plus#(s(X),Y) = [1] Y + [4] >= [1] Y + [4] = c_13(plus#(X,Y)) square#(X) = [3] X + [4] >= [3] X + [4] = c_14(times#(X,X)) times#(s(X),Y) = [2] X + [1] Y + [5] >= [2] X + [1] Y + [5] = c_16(plus#(Y,times(X,Y))) activate(X) = [4] X + [3] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [4] X2 + [3] >= [4] X2 + [0] = cons(X1,X2) activate(n__from(X)) = [3] >= [1] = from(X) cons(X1,X2) = [4] X2 + [0] >= [1] X2 + [0] = n__cons(X1,X2) from(X) = [1] >= [0] = cons(X,n__from(s(X))) from(X) = [1] >= [0] = n__from(X) plus(0(),Y) = [1] Y + [3] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [3] >= [1] Y + [4] = s(plus(X,Y)) times(0(),Y) = [1] Y + [0] >= [0] = 0() times(s(X),Y) = [2] X + [1] Y + [2] >= [2] X + [1] Y + [3] = plus(Y,times(X,Y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.3.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: plus(s(X),Y) -> s(plus(X,Y)) times(s(X),Y) -> plus(Y,times(X,Y)) Weak DP Rules: 2ndsneg#(0(),Z) -> c_1() 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_9(cons#(X,n__from(s(X)))) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) Weak TRS Rules: 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) plus(0(),Y) -> Y times(0(),Y) -> 0() 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} Proof: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = [4] p(2ndsneg) = [4] x2 + [2] p(2ndspos) = [2] x2 + [2] p(activate) = [4] x1 + [0] p(cons) = [4] x2 + [0] p(from) = [0] p(n__cons) = [1] x2 + [0] p(n__from) = [0] p(negrecip) = [0] p(pi) = [2] x1 + [1] p(plus) = [1] x2 + [0] p(posrecip) = [1] p(rcons) = [1] x2 + [0] p(rnil) = [0] p(s) = [1] x1 + [1] p(square) = [4] x1 + [0] p(times) = [1] x1 + [4] p(2ndsneg#) = [1] x1 + [1] x2 + [2] p(2ndspos#) = [1] x1 + [1] x2 + [2] p(activate#) = [0] p(cons#) = [0] p(from#) = [0] p(pi#) = [4] x1 + [4] p(plus#) = [1] x2 + [0] p(square#) = [6] x1 + [0] p(times#) = [4] x1 + [1] x2 + [0] p(c_1) = [1] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [1] p(c_4) = [1] x1 + [1] x2 + [1] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [1] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [1] x1 + [0] Following rules are strictly oriented: times(s(X),Y) = [1] X + [5] > [1] X + [4] = plus(Y,times(X,Y)) Following rules are (at-least) weakly oriented: 2ndsneg#(0(),Z) = [1] Z + [6] >= [1] = c_1() 2ndsneg#(s(N) = [1] N + [4] Z + [3] ,cons(X,n__cons(Y,Z))) >= [1] N + [4] Z + [2] = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(s(N) = [1] N + [4] Z + [3] ,cons(X,n__cons(Y,Z))) >= [1] N + [4] Z + [3] = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = [0] >= [0] = c_5(X) activate#(n__cons(X1,X2)) = [0] >= [0] = c_6(cons#(X1,X2)) activate#(n__from(X)) = [0] >= [0] = c_7(from#(X)) cons#(X1,X2) = [0] >= [0] = c_8(X1,X2) from#(X) = [0] >= [0] = c_9(cons#(X,n__from(s(X)))) from#(X) = [0] >= [0] = c_10(X) pi#(X) = [4] X + [4] >= [1] X + [3] = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = [1] Y + [0] >= [1] Y + [0] = c_12(Y) plus#(s(X),Y) = [1] Y + [0] >= [1] Y + [0] = c_13(plus#(X,Y)) square#(X) = [6] X + [0] >= [5] X + [0] = c_14(times#(X,X)) times#(s(X),Y) = [4] X + [1] Y + [4] >= [1] X + [4] = c_16(plus#(Y,times(X,Y))) activate(X) = [4] X + [0] >= [1] X + [0] = X activate(n__cons(X1,X2)) = [4] X2 + [0] >= [4] X2 + [0] = cons(X1,X2) activate(n__from(X)) = [0] >= [0] = from(X) cons(X1,X2) = [4] X2 + [0] >= [1] X2 + [0] = n__cons(X1,X2) from(X) = [0] >= [0] = cons(X,n__from(s(X))) from(X) = [0] >= [0] = n__from(X) plus(0(),Y) = [1] Y + [0] >= [1] Y + [0] = Y plus(s(X),Y) = [1] Y + [0] >= [1] Y + [1] = s(plus(X,Y)) times(0(),Y) = [8] >= [4] = 0() Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** 1.1.1.1.3.1.1.1.1.1.1.1.1 Progress [(O(1),O(n^3))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: plus(s(X),Y) -> s(plus(X,Y)) Weak DP Rules: 2ndsneg#(0(),Z) -> c_1() 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_9(cons#(X,n__from(s(X)))) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) Weak TRS Rules: 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) plus(0(),Y) -> 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: NaturalPI {shape = Mixed 3, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any intersect of strict-rules and rewrite-rules, greedy = NoGreedy} Proof: We apply a polynomial interpretation of kind constructor-based(mixed(3)): The following argument positions are considered usable: uargs(plus) = {2}, uargs(s) = {1}, uargs(2ndsneg#) = {2}, uargs(2ndspos#) = {2}, uargs(plus#) = {2}, uargs(c_2) = {1,2}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {} TcT has computed the following interpretation: p(0) = 0 p(2ndsneg) = 0 p(2ndspos) = 0 p(activate) = x1 p(cons) = x2 p(from) = 0 p(n__cons) = x2 p(n__from) = 0 p(negrecip) = 0 p(pi) = 0 p(plus) = x1 + x1^2 + x2 p(posrecip) = 0 p(rcons) = 0 p(rnil) = 0 p(s) = 1 + x1 p(square) = 0 p(times) = x1*x2 + x1*x2^2 p(2ndsneg#) = x2 p(2ndspos#) = x2 p(activate#) = 0 p(cons#) = 0 p(from#) = 0 p(pi#) = 0 p(plus#) = x1 + x2 p(square#) = x1^2 + x1^3 p(times#) = x1*x2 + x1*x2^2 p(c_1) = 0 p(c_2) = x1 + x2 p(c_3) = 0 p(c_4) = x1 + x2 p(c_5) = 0 p(c_6) = x1 p(c_7) = x1 p(c_8) = 0 p(c_9) = x1 p(c_10) = 0 p(c_11) = x1 p(c_12) = x1 p(c_13) = x1 p(c_14) = x1 p(c_15) = 0 p(c_16) = x1 Following rules are strictly oriented: plus(s(X),Y) = 2 + 3*X + X^2 + Y > 1 + X + X^2 + Y = s(plus(X,Y)) Following rules are (at-least) weakly oriented: 2ndsneg#(0(),Z) = Z >= 0 = c_1() 2ndsneg#(s(N) = Z ,cons(X,n__cons(Y,Z))) >= Z = c_2(activate#(Y) ,2ndspos#(N,activate(Z))) 2ndspos#(s(N) = Z ,cons(X,n__cons(Y,Z))) >= Z = c_4(activate#(Y) ,2ndsneg#(N,activate(Z))) activate#(X) = 0 >= 0 = c_5(X) activate#(n__cons(X1,X2)) = 0 >= 0 = c_6(cons#(X1,X2)) activate#(n__from(X)) = 0 >= 0 = c_7(from#(X)) cons#(X1,X2) = 0 >= 0 = c_8(X1,X2) from#(X) = 0 >= 0 = c_9(cons#(X,n__from(s(X)))) from#(X) = 0 >= 0 = c_10(X) pi#(X) = 0 >= 0 = c_11(2ndspos#(X,from(0()))) plus#(0(),Y) = Y >= Y = c_12(Y) plus#(s(X),Y) = 1 + X + Y >= X + Y = c_13(plus#(X,Y)) square#(X) = X^2 + X^3 >= X^2 + X^3 = c_14(times#(X,X)) times#(s(X),Y) = X*Y + X*Y^2 + Y + Y^2 >= X*Y + X*Y^2 + Y = c_16(plus#(Y,times(X,Y))) activate(X) = X >= X = X activate(n__cons(X1,X2)) = X2 >= X2 = cons(X1,X2) activate(n__from(X)) = 0 >= 0 = from(X) cons(X1,X2) = X2 >= X2 = n__cons(X1,X2) from(X) = 0 >= 0 = cons(X,n__from(s(X))) from(X) = 0 >= 0 = n__from(X) plus(0(),Y) = Y >= Y = Y times(0(),Y) = 0 >= 0 = 0() times(s(X),Y) = X*Y + X*Y^2 + Y + Y^2 >= X*Y + X*Y^2 + Y + Y^2 = plus(Y,times(X,Y)) *** 1.1.1.1.3.1.1.1.1.1.1.1.1.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP Rules: 2ndsneg#(0(),Z) -> c_1() 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) -> c_2(activate#(Y),2ndspos#(N,activate(Z))) 2ndspos#(s(N),cons(X,n__cons(Y,Z))) -> c_4(activate#(Y),2ndsneg#(N,activate(Z))) activate#(X) -> c_5(X) activate#(n__cons(X1,X2)) -> c_6(cons#(X1,X2)) activate#(n__from(X)) -> c_7(from#(X)) cons#(X1,X2) -> c_8(X1,X2) from#(X) -> c_9(cons#(X,n__from(s(X)))) from#(X) -> c_10(X) pi#(X) -> c_11(2ndspos#(X,from(0()))) plus#(0(),Y) -> c_12(Y) 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))) Weak TRS Rules: 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) 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_2/2,c_3/0,c_4/2,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/1,c_15/0,c_16/1} Obligation: Full basic terms: {2ndsneg#,2ndspos#,activate#,cons#,from#,pi#,plus#,square#,times#}/{0,n__cons,n__from,negrecip,posrecip,rcons,rnil,s} Applied Processor: EmptyProcessor Proof: The problem is already closed. The intended complexity is O(1).