WORST_CASE(?,O(n^1)) * Step 1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if ,mark} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs a__add#(X1,X2) -> c_1() a__add#(0(),X) -> c_2(mark#(X)) a__add#(s(X),Y) -> c_3() a__and#(X1,X2) -> c_4() a__and#(false(),Y) -> c_5() a__and#(true(),X) -> c_6(mark#(X)) a__first#(X1,X2) -> c_7() a__first#(0(),X) -> c_8() a__first#(s(X),cons(Y,Z)) -> c_9() a__from#(X) -> c_10() a__from#(X) -> c_11() a__if#(X1,X2,X3) -> c_12() a__if#(false(),X,Y) -> c_13(mark#(Y)) a__if#(true(),X,Y) -> c_14(mark#(X)) mark#(0()) -> c_15() mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)) mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)) mark#(cons(X1,X2)) -> c_18() mark#(false()) -> c_19() mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(from(X)) -> c_21(a__from#(X)) mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(nil()) -> c_23() mark#(s(X)) -> c_24() mark#(true()) -> c_25() Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__add#(X1,X2) -> c_1() a__add#(0(),X) -> c_2(mark#(X)) a__add#(s(X),Y) -> c_3() a__and#(X1,X2) -> c_4() a__and#(false(),Y) -> c_5() a__and#(true(),X) -> c_6(mark#(X)) a__first#(X1,X2) -> c_7() a__first#(0(),X) -> c_8() a__first#(s(X),cons(Y,Z)) -> c_9() a__from#(X) -> c_10() a__from#(X) -> c_11() a__if#(X1,X2,X3) -> c_12() a__if#(false(),X,Y) -> c_13(mark#(Y)) a__if#(true(),X,Y) -> c_14(mark#(X)) mark#(0()) -> c_15() mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)) mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)) mark#(cons(X1,X2)) -> c_18() mark#(false()) -> c_19() mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(from(X)) -> c_21(a__from#(X)) mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(nil()) -> c_23() mark#(s(X)) -> c_24() mark#(true()) -> c_25() - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1,a__add#/2,a__and#/2,a__first#/2,a__from#/1,a__if#/3 ,mark#/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1,if/3,nil/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/0,c_16/2,c_17/2,c_18/0,c_19/0,c_20/3 ,c_21/1,c_22/2,c_23/0,c_24/0,c_25/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add#,a__and#,a__first#,a__from#,a__if# ,mark#} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,4,5,7,8,9,10,11,12,15,18,19,23,24,25} by application of Pre({1,3,4,5,7,8,9,10,11,12,15,18,19,23,24,25}) = {2,6,13,14,16,17,20,21,22}. Here rules are labelled as follows: 1: a__add#(X1,X2) -> c_1() 2: a__add#(0(),X) -> c_2(mark#(X)) 3: a__add#(s(X),Y) -> c_3() 4: a__and#(X1,X2) -> c_4() 5: a__and#(false(),Y) -> c_5() 6: a__and#(true(),X) -> c_6(mark#(X)) 7: a__first#(X1,X2) -> c_7() 8: a__first#(0(),X) -> c_8() 9: a__first#(s(X),cons(Y,Z)) -> c_9() 10: a__from#(X) -> c_10() 11: a__from#(X) -> c_11() 12: a__if#(X1,X2,X3) -> c_12() 13: a__if#(false(),X,Y) -> c_13(mark#(Y)) 14: a__if#(true(),X,Y) -> c_14(mark#(X)) 15: mark#(0()) -> c_15() 16: mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)) 17: mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)) 18: mark#(cons(X1,X2)) -> c_18() 19: mark#(false()) -> c_19() 20: mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) 21: mark#(from(X)) -> c_21(a__from#(X)) 22: mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)) 23: mark#(nil()) -> c_23() 24: mark#(s(X)) -> c_24() 25: mark#(true()) -> c_25() * Step 3: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__add#(0(),X) -> c_2(mark#(X)) a__and#(true(),X) -> c_6(mark#(X)) a__if#(false(),X,Y) -> c_13(mark#(Y)) a__if#(true(),X,Y) -> c_14(mark#(X)) mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)) mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)) mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(from(X)) -> c_21(a__from#(X)) mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)) - Weak DPs: a__add#(X1,X2) -> c_1() a__add#(s(X),Y) -> c_3() a__and#(X1,X2) -> c_4() a__and#(false(),Y) -> c_5() a__first#(X1,X2) -> c_7() a__first#(0(),X) -> c_8() a__first#(s(X),cons(Y,Z)) -> c_9() a__from#(X) -> c_10() a__from#(X) -> c_11() a__if#(X1,X2,X3) -> c_12() mark#(0()) -> c_15() mark#(cons(X1,X2)) -> c_18() mark#(false()) -> c_19() mark#(nil()) -> c_23() mark#(s(X)) -> c_24() mark#(true()) -> c_25() - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1,a__add#/2,a__and#/2,a__first#/2,a__from#/1,a__if#/3 ,mark#/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1,if/3,nil/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/0,c_16/2,c_17/2,c_18/0,c_19/0,c_20/3 ,c_21/1,c_22/2,c_23/0,c_24/0,c_25/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add#,a__and#,a__first#,a__from#,a__if# ,mark#} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {8} by application of Pre({8}) = {1,2,3,4,5,6,7,9}. Here rules are labelled as follows: 1: a__add#(0(),X) -> c_2(mark#(X)) 2: a__and#(true(),X) -> c_6(mark#(X)) 3: a__if#(false(),X,Y) -> c_13(mark#(Y)) 4: a__if#(true(),X,Y) -> c_14(mark#(X)) 5: mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)) 6: mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)) 7: mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) 8: mark#(from(X)) -> c_21(a__from#(X)) 9: mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)) 10: a__add#(X1,X2) -> c_1() 11: a__add#(s(X),Y) -> c_3() 12: a__and#(X1,X2) -> c_4() 13: a__and#(false(),Y) -> c_5() 14: a__first#(X1,X2) -> c_7() 15: a__first#(0(),X) -> c_8() 16: a__first#(s(X),cons(Y,Z)) -> c_9() 17: a__from#(X) -> c_10() 18: a__from#(X) -> c_11() 19: a__if#(X1,X2,X3) -> c_12() 20: mark#(0()) -> c_15() 21: mark#(cons(X1,X2)) -> c_18() 22: mark#(false()) -> c_19() 23: mark#(nil()) -> c_23() 24: mark#(s(X)) -> c_24() 25: mark#(true()) -> c_25() * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__add#(0(),X) -> c_2(mark#(X)) a__and#(true(),X) -> c_6(mark#(X)) a__if#(false(),X,Y) -> c_13(mark#(Y)) a__if#(true(),X,Y) -> c_14(mark#(X)) mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)) mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)) mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)) - Weak DPs: a__add#(X1,X2) -> c_1() a__add#(s(X),Y) -> c_3() a__and#(X1,X2) -> c_4() a__and#(false(),Y) -> c_5() a__first#(X1,X2) -> c_7() a__first#(0(),X) -> c_8() a__first#(s(X),cons(Y,Z)) -> c_9() a__from#(X) -> c_10() a__from#(X) -> c_11() a__if#(X1,X2,X3) -> c_12() mark#(0()) -> c_15() mark#(cons(X1,X2)) -> c_18() mark#(false()) -> c_19() mark#(from(X)) -> c_21(a__from#(X)) mark#(nil()) -> c_23() mark#(s(X)) -> c_24() mark#(true()) -> c_25() - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1,a__add#/2,a__and#/2,a__first#/2,a__from#/1,a__if#/3 ,mark#/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1,if/3,nil/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/0,c_16/2,c_17/2,c_18/0,c_19/0,c_20/3 ,c_21/1,c_22/2,c_23/0,c_24/0,c_25/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add#,a__and#,a__first#,a__from#,a__if# ,mark#} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:a__add#(0(),X) -> c_2(mark#(X)) -->_1 mark#(from(X)) -> c_21(a__from#(X)):22 -->_1 mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_1 mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):7 -->_1 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):6 -->_1 mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)):5 -->_1 mark#(true()) -> c_25():25 -->_1 mark#(s(X)) -> c_24():24 -->_1 mark#(nil()) -> c_23():23 -->_1 mark#(false()) -> c_19():21 -->_1 mark#(cons(X1,X2)) -> c_18():20 -->_1 mark#(0()) -> c_15():19 2:S:a__and#(true(),X) -> c_6(mark#(X)) -->_1 mark#(from(X)) -> c_21(a__from#(X)):22 -->_1 mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_1 mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):7 -->_1 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):6 -->_1 mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)):5 -->_1 mark#(true()) -> c_25():25 -->_1 mark#(s(X)) -> c_24():24 -->_1 mark#(nil()) -> c_23():23 -->_1 mark#(false()) -> c_19():21 -->_1 mark#(cons(X1,X2)) -> c_18():20 -->_1 mark#(0()) -> c_15():19 3:S:a__if#(false(),X,Y) -> c_13(mark#(Y)) -->_1 mark#(from(X)) -> c_21(a__from#(X)):22 -->_1 mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_1 mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):7 -->_1 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):6 -->_1 mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)):5 -->_1 mark#(true()) -> c_25():25 -->_1 mark#(s(X)) -> c_24():24 -->_1 mark#(nil()) -> c_23():23 -->_1 mark#(false()) -> c_19():21 -->_1 mark#(cons(X1,X2)) -> c_18():20 -->_1 mark#(0()) -> c_15():19 4:S:a__if#(true(),X,Y) -> c_14(mark#(X)) -->_1 mark#(from(X)) -> c_21(a__from#(X)):22 -->_1 mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_1 mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):7 -->_1 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):6 -->_1 mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)):5 -->_1 mark#(true()) -> c_25():25 -->_1 mark#(s(X)) -> c_24():24 -->_1 mark#(nil()) -> c_23():23 -->_1 mark#(false()) -> c_19():21 -->_1 mark#(cons(X1,X2)) -> c_18():20 -->_1 mark#(0()) -> c_15():19 5:S:mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)) -->_2 mark#(from(X)) -> c_21(a__from#(X)):22 -->_2 mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_2 mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):7 -->_2 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):6 -->_2 mark#(true()) -> c_25():25 -->_2 mark#(s(X)) -> c_24():24 -->_2 mark#(nil()) -> c_23():23 -->_2 mark#(false()) -> c_19():21 -->_2 mark#(cons(X1,X2)) -> c_18():20 -->_2 mark#(0()) -> c_15():19 -->_1 a__add#(s(X),Y) -> c_3():10 -->_1 a__add#(X1,X2) -> c_1():9 -->_2 mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)):5 -->_1 a__add#(0(),X) -> c_2(mark#(X)):1 6:S:mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)) -->_2 mark#(from(X)) -> c_21(a__from#(X)):22 -->_2 mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_2 mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):7 -->_2 mark#(true()) -> c_25():25 -->_2 mark#(s(X)) -> c_24():24 -->_2 mark#(nil()) -> c_23():23 -->_2 mark#(false()) -> c_19():21 -->_2 mark#(cons(X1,X2)) -> c_18():20 -->_2 mark#(0()) -> c_15():19 -->_1 a__and#(false(),Y) -> c_5():12 -->_1 a__and#(X1,X2) -> c_4():11 -->_2 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):6 -->_2 mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)):5 -->_1 a__and#(true(),X) -> c_6(mark#(X)):2 7:S:mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) -->_3 mark#(from(X)) -> c_21(a__from#(X)):22 -->_2 mark#(from(X)) -> c_21(a__from#(X)):22 -->_3 mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_2 mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_3 mark#(true()) -> c_25():25 -->_2 mark#(true()) -> c_25():25 -->_3 mark#(s(X)) -> c_24():24 -->_2 mark#(s(X)) -> c_24():24 -->_3 mark#(nil()) -> c_23():23 -->_2 mark#(nil()) -> c_23():23 -->_3 mark#(false()) -> c_19():21 -->_2 mark#(false()) -> c_19():21 -->_3 mark#(cons(X1,X2)) -> c_18():20 -->_2 mark#(cons(X1,X2)) -> c_18():20 -->_3 mark#(0()) -> c_15():19 -->_2 mark#(0()) -> c_15():19 -->_1 a__first#(s(X),cons(Y,Z)) -> c_9():15 -->_1 a__first#(0(),X) -> c_8():14 -->_1 a__first#(X1,X2) -> c_7():13 -->_3 mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):7 -->_2 mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):7 -->_3 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):6 -->_2 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):6 -->_3 mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)):5 -->_2 mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)):5 8:S:mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)) -->_2 mark#(from(X)) -> c_21(a__from#(X)):22 -->_2 mark#(true()) -> c_25():25 -->_2 mark#(s(X)) -> c_24():24 -->_2 mark#(nil()) -> c_23():23 -->_2 mark#(false()) -> c_19():21 -->_2 mark#(cons(X1,X2)) -> c_18():20 -->_2 mark#(0()) -> c_15():19 -->_1 a__if#(X1,X2,X3) -> c_12():18 -->_2 mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_2 mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):7 -->_2 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):6 -->_2 mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)):5 -->_1 a__if#(true(),X,Y) -> c_14(mark#(X)):4 -->_1 a__if#(false(),X,Y) -> c_13(mark#(Y)):3 9:W:a__add#(X1,X2) -> c_1() 10:W:a__add#(s(X),Y) -> c_3() 11:W:a__and#(X1,X2) -> c_4() 12:W:a__and#(false(),Y) -> c_5() 13:W:a__first#(X1,X2) -> c_7() 14:W:a__first#(0(),X) -> c_8() 15:W:a__first#(s(X),cons(Y,Z)) -> c_9() 16:W:a__from#(X) -> c_10() 17:W:a__from#(X) -> c_11() 18:W:a__if#(X1,X2,X3) -> c_12() 19:W:mark#(0()) -> c_15() 20:W:mark#(cons(X1,X2)) -> c_18() 21:W:mark#(false()) -> c_19() 22:W:mark#(from(X)) -> c_21(a__from#(X)) -->_1 a__from#(X) -> c_11():17 -->_1 a__from#(X) -> c_10():16 23:W:mark#(nil()) -> c_23() 24:W:mark#(s(X)) -> c_24() 25:W:mark#(true()) -> c_25() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: a__add#(X1,X2) -> c_1() 10: a__add#(s(X),Y) -> c_3() 11: a__and#(X1,X2) -> c_4() 12: a__and#(false(),Y) -> c_5() 13: a__first#(X1,X2) -> c_7() 14: a__first#(0(),X) -> c_8() 15: a__first#(s(X),cons(Y,Z)) -> c_9() 18: a__if#(X1,X2,X3) -> c_12() 19: mark#(0()) -> c_15() 20: mark#(cons(X1,X2)) -> c_18() 21: mark#(false()) -> c_19() 23: mark#(nil()) -> c_23() 24: mark#(s(X)) -> c_24() 25: mark#(true()) -> c_25() 22: mark#(from(X)) -> c_21(a__from#(X)) 16: a__from#(X) -> c_10() 17: a__from#(X) -> c_11() * Step 5: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__add#(0(),X) -> c_2(mark#(X)) a__and#(true(),X) -> c_6(mark#(X)) a__if#(false(),X,Y) -> c_13(mark#(Y)) a__if#(true(),X,Y) -> c_14(mark#(X)) mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)) mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)) mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1,a__add#/2,a__and#/2,a__first#/2,a__from#/1,a__if#/3 ,mark#/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1,if/3,nil/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/0,c_16/2,c_17/2,c_18/0,c_19/0,c_20/3 ,c_21/1,c_22/2,c_23/0,c_24/0,c_25/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add#,a__and#,a__first#,a__from#,a__if# ,mark#} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:a__add#(0(),X) -> c_2(mark#(X)) -->_1 mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_1 mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):7 -->_1 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):6 -->_1 mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)):5 2:S:a__and#(true(),X) -> c_6(mark#(X)) -->_1 mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_1 mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):7 -->_1 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):6 -->_1 mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)):5 3:S:a__if#(false(),X,Y) -> c_13(mark#(Y)) -->_1 mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_1 mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):7 -->_1 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):6 -->_1 mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)):5 4:S:a__if#(true(),X,Y) -> c_14(mark#(X)) -->_1 mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_1 mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):7 -->_1 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):6 -->_1 mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)):5 5:S:mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)) -->_2 mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_2 mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):7 -->_2 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):6 -->_2 mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)):5 -->_1 a__add#(0(),X) -> c_2(mark#(X)):1 6:S:mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)) -->_2 mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_2 mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):7 -->_2 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):6 -->_2 mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)):5 -->_1 a__and#(true(),X) -> c_6(mark#(X)):2 7:S:mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) -->_3 mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_2 mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_3 mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):7 -->_2 mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):7 -->_3 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):6 -->_2 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):6 -->_3 mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)):5 -->_2 mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)):5 8:S:mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)) -->_2 mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_2 mark#(first(X1,X2)) -> c_20(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):7 -->_2 mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)):6 -->_2 mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)):5 -->_1 a__if#(true(),X,Y) -> c_14(mark#(X)):4 -->_1 a__if#(false(),X,Y) -> c_13(mark#(Y)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: mark#(first(X1,X2)) -> c_20(mark#(X1),mark#(X2)) * Step 6: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__add#(0(),X) -> c_2(mark#(X)) a__and#(true(),X) -> c_6(mark#(X)) a__if#(false(),X,Y) -> c_13(mark#(Y)) a__if#(true(),X,Y) -> c_14(mark#(X)) mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)) mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)) mark#(first(X1,X2)) -> c_20(mark#(X1),mark#(X2)) mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1,a__add#/2,a__and#/2,a__first#/2,a__from#/1,a__if#/3 ,mark#/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1,if/3,nil/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/0,c_16/2,c_17/2,c_18/0,c_19/0,c_20/2 ,c_21/1,c_22/2,c_23/0,c_24/0,c_25/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add#,a__and#,a__first#,a__from#,a__if# ,mark#} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__add) = {1}, uargs(a__and) = {1}, uargs(a__first) = {1,2}, uargs(a__if) = {1}, uargs(a__add#) = {1}, uargs(a__and#) = {1}, uargs(a__if#) = {1}, uargs(c_2) = {1}, uargs(c_6) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1,2}, uargs(c_17) = {1,2}, uargs(c_20) = {1,2}, uargs(c_22) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__add) = [1] x1 + [0] p(a__and) = [1] x1 + [0] p(a__first) = [1] x1 + [1] x2 + [0] p(a__from) = [0] p(a__if) = [1] x1 + [0] p(add) = [0] p(and) = [0] p(cons) = [0] p(false) = [0] p(first) = [0] p(from) = [0] p(if) = [0] p(mark) = [0] p(nil) = [0] p(s) = [0] p(true) = [0] p(a__add#) = [1] x1 + [0] p(a__and#) = [1] x1 + [0] p(a__first#) = [0] p(a__from#) = [0] p(a__if#) = [1] x1 + [5] p(mark#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [1] x1 + [1] x2 + [0] p(c_17) = [1] x1 + [1] x2 + [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [1] x1 + [1] x2 + [0] p(c_21) = [0] p(c_22) = [1] x1 + [1] x2 + [0] p(c_23) = [0] p(c_24) = [0] p(c_25) = [0] Following rules are strictly oriented: a__if#(false(),X,Y) = [5] > [0] = c_13(mark#(Y)) a__if#(true(),X,Y) = [5] > [0] = c_14(mark#(X)) Following rules are (at-least) weakly oriented: a__add#(0(),X) = [0] >= [0] = c_2(mark#(X)) a__and#(true(),X) = [0] >= [0] = c_6(mark#(X)) mark#(add(X1,X2)) = [0] >= [0] = c_16(a__add#(mark(X1),X2),mark#(X1)) mark#(and(X1,X2)) = [0] >= [0] = c_17(a__and#(mark(X1),X2),mark#(X1)) mark#(first(X1,X2)) = [0] >= [0] = c_20(mark#(X1),mark#(X2)) mark#(if(X1,X2,X3)) = [0] >= [5] = c_22(a__if#(mark(X1),X2,X3),mark#(X1)) a__add(X1,X2) = [1] X1 + [0] >= [0] = add(X1,X2) a__add(0(),X) = [0] >= [0] = mark(X) a__add(s(X),Y) = [0] >= [0] = s(add(X,Y)) a__and(X1,X2) = [1] X1 + [0] >= [0] = and(X1,X2) a__and(false(),Y) = [0] >= [0] = false() a__and(true(),X) = [0] >= [0] = mark(X) a__first(X1,X2) = [1] X1 + [1] X2 + [0] >= [0] = first(X1,X2) a__first(0(),X) = [1] X + [0] >= [0] = nil() a__first(s(X),cons(Y,Z)) = [0] >= [0] = cons(Y,first(X,Z)) a__from(X) = [0] >= [0] = cons(X,from(s(X))) a__from(X) = [0] >= [0] = from(X) a__if(X1,X2,X3) = [1] X1 + [0] >= [0] = if(X1,X2,X3) a__if(false(),X,Y) = [0] >= [0] = mark(Y) a__if(true(),X,Y) = [0] >= [0] = mark(X) mark(0()) = [0] >= [0] = 0() mark(add(X1,X2)) = [0] >= [0] = a__add(mark(X1),X2) mark(and(X1,X2)) = [0] >= [0] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [0] = cons(X1,X2) mark(false()) = [0] >= [0] = false() mark(first(X1,X2)) = [0] >= [0] = a__first(mark(X1),mark(X2)) mark(from(X)) = [0] >= [0] = a__from(X) mark(if(X1,X2,X3)) = [0] >= [0] = a__if(mark(X1),X2,X3) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [0] = s(X) mark(true()) = [0] >= [0] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__add#(0(),X) -> c_2(mark#(X)) a__and#(true(),X) -> c_6(mark#(X)) mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)) mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)) mark#(first(X1,X2)) -> c_20(mark#(X1),mark#(X2)) mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)) - Weak DPs: a__if#(false(),X,Y) -> c_13(mark#(Y)) a__if#(true(),X,Y) -> c_14(mark#(X)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1,a__add#/2,a__and#/2,a__first#/2,a__from#/1,a__if#/3 ,mark#/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1,if/3,nil/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/0,c_16/2,c_17/2,c_18/0,c_19/0,c_20/2 ,c_21/1,c_22/2,c_23/0,c_24/0,c_25/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add#,a__and#,a__first#,a__from#,a__if# ,mark#} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__add) = {1}, uargs(a__and) = {1}, uargs(a__first) = {1,2}, uargs(a__if) = {1}, uargs(a__add#) = {1}, uargs(a__and#) = {1}, uargs(a__if#) = {1}, uargs(c_2) = {1}, uargs(c_6) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1,2}, uargs(c_17) = {1,2}, uargs(c_20) = {1,2}, uargs(c_22) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__add) = [1] x1 + [0] p(a__and) = [1] x1 + [0] p(a__first) = [1] x1 + [1] x2 + [0] p(a__from) = [0] p(a__if) = [1] x1 + [0] p(add) = [1] x1 + [0] p(and) = [0] p(cons) = [0] p(false) = [0] p(first) = [0] p(from) = [0] p(if) = [0] p(mark) = [0] p(nil) = [0] p(s) = [0] p(true) = [0] p(a__add#) = [1] x1 + [3] p(a__and#) = [1] x1 + [3] p(a__first#) = [0] p(a__from#) = [0] p(a__if#) = [1] x1 + [0] p(mark#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [1] x1 + [1] x2 + [0] p(c_17) = [1] x1 + [1] x2 + [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [1] x1 + [1] x2 + [0] p(c_21) = [0] p(c_22) = [1] x1 + [1] x2 + [0] p(c_23) = [0] p(c_24) = [0] p(c_25) = [0] Following rules are strictly oriented: a__add#(0(),X) = [3] > [0] = c_2(mark#(X)) a__and#(true(),X) = [3] > [0] = c_6(mark#(X)) Following rules are (at-least) weakly oriented: a__if#(false(),X,Y) = [0] >= [0] = c_13(mark#(Y)) a__if#(true(),X,Y) = [0] >= [0] = c_14(mark#(X)) mark#(add(X1,X2)) = [0] >= [3] = c_16(a__add#(mark(X1),X2),mark#(X1)) mark#(and(X1,X2)) = [0] >= [3] = c_17(a__and#(mark(X1),X2),mark#(X1)) mark#(first(X1,X2)) = [0] >= [0] = c_20(mark#(X1),mark#(X2)) mark#(if(X1,X2,X3)) = [0] >= [0] = c_22(a__if#(mark(X1),X2,X3),mark#(X1)) a__add(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = add(X1,X2) a__add(0(),X) = [0] >= [0] = mark(X) a__add(s(X),Y) = [0] >= [0] = s(add(X,Y)) a__and(X1,X2) = [1] X1 + [0] >= [0] = and(X1,X2) a__and(false(),Y) = [0] >= [0] = false() a__and(true(),X) = [0] >= [0] = mark(X) a__first(X1,X2) = [1] X1 + [1] X2 + [0] >= [0] = first(X1,X2) a__first(0(),X) = [1] X + [0] >= [0] = nil() a__first(s(X),cons(Y,Z)) = [0] >= [0] = cons(Y,first(X,Z)) a__from(X) = [0] >= [0] = cons(X,from(s(X))) a__from(X) = [0] >= [0] = from(X) a__if(X1,X2,X3) = [1] X1 + [0] >= [0] = if(X1,X2,X3) a__if(false(),X,Y) = [0] >= [0] = mark(Y) a__if(true(),X,Y) = [0] >= [0] = mark(X) mark(0()) = [0] >= [0] = 0() mark(add(X1,X2)) = [0] >= [0] = a__add(mark(X1),X2) mark(and(X1,X2)) = [0] >= [0] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [0] = cons(X1,X2) mark(false()) = [0] >= [0] = false() mark(first(X1,X2)) = [0] >= [0] = a__first(mark(X1),mark(X2)) mark(from(X)) = [0] >= [0] = a__from(X) mark(if(X1,X2,X3)) = [0] >= [0] = a__if(mark(X1),X2,X3) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [0] = s(X) mark(true()) = [0] >= [0] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 8: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)) mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)) mark#(first(X1,X2)) -> c_20(mark#(X1),mark#(X2)) mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)) - Weak DPs: a__add#(0(),X) -> c_2(mark#(X)) a__and#(true(),X) -> c_6(mark#(X)) a__if#(false(),X,Y) -> c_13(mark#(Y)) a__if#(true(),X,Y) -> c_14(mark#(X)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1,a__add#/2,a__and#/2,a__first#/2,a__from#/1,a__if#/3 ,mark#/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1,if/3,nil/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/0,c_16/2,c_17/2,c_18/0,c_19/0,c_20/2 ,c_21/1,c_22/2,c_23/0,c_24/0,c_25/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add#,a__and#,a__first#,a__from#,a__if# ,mark#} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_6) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1,2}, uargs(c_17) = {1,2}, uargs(c_20) = {1,2}, uargs(c_22) = {1,2} Following symbols are considered usable: {a__add#,a__and#,a__first#,a__from#,a__if#,mark#} TcT has computed the following interpretation: p(0) = [0] p(a__add) = [0] p(a__and) = [1] x1 + [0] p(a__first) = [6] x2 + [0] p(a__from) = [2] x1 + [0] p(a__if) = [2] x1 + [1] x2 + [2] p(add) = [1] x1 + [1] x2 + [2] p(and) = [1] x1 + [1] x2 + [2] p(cons) = [1] x1 + [1] x2 + [2] p(false) = [0] p(first) = [1] x1 + [1] x2 + [3] p(from) = [1] x1 + [4] p(if) = [1] x1 + [1] x2 + [1] x3 + [2] p(mark) = [0] p(nil) = [1] p(s) = [1] x1 + [6] p(true) = [5] p(a__add#) = [4] x2 + [4] p(a__and#) = [4] x2 + [7] p(a__first#) = [4] x1 + [1] p(a__from#) = [1] p(a__if#) = [4] x2 + [4] x3 + [6] p(mark#) = [4] x1 + [3] p(c_1) = [0] p(c_2) = [1] x1 + [1] p(c_3) = [2] p(c_4) = [1] p(c_5) = [0] p(c_6) = [1] x1 + [4] p(c_7) = [4] p(c_8) = [4] p(c_9) = [0] p(c_10) = [1] p(c_11) = [4] p(c_12) = [2] p(c_13) = [1] x1 + [3] p(c_14) = [1] x1 + [1] p(c_15) = [0] p(c_16) = [1] x1 + [1] x2 + [4] p(c_17) = [1] x1 + [1] x2 + [1] p(c_18) = [0] p(c_19) = [4] p(c_20) = [1] x1 + [1] x2 + [2] p(c_21) = [1] x1 + [0] p(c_22) = [1] x1 + [1] x2 + [0] p(c_23) = [1] p(c_24) = [0] p(c_25) = [4] Following rules are strictly oriented: mark#(first(X1,X2)) = [4] X1 + [4] X2 + [15] > [4] X1 + [4] X2 + [8] = c_20(mark#(X1),mark#(X2)) mark#(if(X1,X2,X3)) = [4] X1 + [4] X2 + [4] X3 + [11] > [4] X1 + [4] X2 + [4] X3 + [9] = c_22(a__if#(mark(X1),X2,X3),mark#(X1)) Following rules are (at-least) weakly oriented: a__add#(0(),X) = [4] X + [4] >= [4] X + [4] = c_2(mark#(X)) a__and#(true(),X) = [4] X + [7] >= [4] X + [7] = c_6(mark#(X)) a__if#(false(),X,Y) = [4] X + [4] Y + [6] >= [4] Y + [6] = c_13(mark#(Y)) a__if#(true(),X,Y) = [4] X + [4] Y + [6] >= [4] X + [4] = c_14(mark#(X)) mark#(add(X1,X2)) = [4] X1 + [4] X2 + [11] >= [4] X1 + [4] X2 + [11] = c_16(a__add#(mark(X1),X2),mark#(X1)) mark#(and(X1,X2)) = [4] X1 + [4] X2 + [11] >= [4] X1 + [4] X2 + [11] = c_17(a__and#(mark(X1),X2),mark#(X1)) * Step 9: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)) mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)) - Weak DPs: a__add#(0(),X) -> c_2(mark#(X)) a__and#(true(),X) -> c_6(mark#(X)) a__if#(false(),X,Y) -> c_13(mark#(Y)) a__if#(true(),X,Y) -> c_14(mark#(X)) mark#(first(X1,X2)) -> c_20(mark#(X1),mark#(X2)) mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1,a__add#/2,a__and#/2,a__first#/2,a__from#/1,a__if#/3 ,mark#/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1,if/3,nil/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/0,c_16/2,c_17/2,c_18/0,c_19/0,c_20/2 ,c_21/1,c_22/2,c_23/0,c_24/0,c_25/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add#,a__and#,a__first#,a__from#,a__if# ,mark#} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_6) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1,2}, uargs(c_17) = {1,2}, uargs(c_20) = {1,2}, uargs(c_22) = {1,2} Following symbols are considered usable: {a__add#,a__and#,a__first#,a__from#,a__if#,mark#} TcT has computed the following interpretation: p(0) = [1] p(a__add) = [6] x1 + [0] p(a__and) = [1] x1 + [7] p(a__first) = [4] x1 + [2] x2 + [4] p(a__from) = [4] x1 + [4] p(a__if) = [4] x1 + [1] x3 + [0] p(add) = [1] x1 + [1] x2 + [0] p(and) = [1] x1 + [1] x2 + [2] p(cons) = [1] x1 + [4] p(false) = [1] p(first) = [1] x1 + [1] x2 + [0] p(from) = [1] x1 + [5] p(if) = [1] x1 + [1] x2 + [1] x3 + [3] p(mark) = [1] x1 + [1] p(nil) = [5] p(s) = [0] p(true) = [0] p(a__add#) = [4] x2 + [0] p(a__and#) = [4] x2 + [0] p(a__first#) = [2] x2 + [1] p(a__from#) = [2] x1 + [0] p(a__if#) = [4] x2 + [4] x3 + [1] p(mark#) = [4] x1 + [0] p(c_1) = [1] p(c_2) = [1] x1 + [0] p(c_3) = [4] p(c_4) = [1] p(c_5) = [1] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [1] p(c_16) = [1] x1 + [1] x2 + [0] p(c_17) = [1] x1 + [1] x2 + [6] p(c_18) = [4] p(c_19) = [1] p(c_20) = [1] x1 + [1] x2 + [0] p(c_21) = [1] p(c_22) = [1] x1 + [1] x2 + [6] p(c_23) = [0] p(c_24) = [0] p(c_25) = [1] Following rules are strictly oriented: mark#(and(X1,X2)) = [4] X1 + [4] X2 + [8] > [4] X1 + [4] X2 + [6] = c_17(a__and#(mark(X1),X2),mark#(X1)) Following rules are (at-least) weakly oriented: a__add#(0(),X) = [4] X + [0] >= [4] X + [0] = c_2(mark#(X)) a__and#(true(),X) = [4] X + [0] >= [4] X + [0] = c_6(mark#(X)) a__if#(false(),X,Y) = [4] X + [4] Y + [1] >= [4] Y + [0] = c_13(mark#(Y)) a__if#(true(),X,Y) = [4] X + [4] Y + [1] >= [4] X + [0] = c_14(mark#(X)) mark#(add(X1,X2)) = [4] X1 + [4] X2 + [0] >= [4] X1 + [4] X2 + [0] = c_16(a__add#(mark(X1),X2),mark#(X1)) mark#(first(X1,X2)) = [4] X1 + [4] X2 + [0] >= [4] X1 + [4] X2 + [0] = c_20(mark#(X1),mark#(X2)) mark#(if(X1,X2,X3)) = [4] X1 + [4] X2 + [4] X3 + [12] >= [4] X1 + [4] X2 + [4] X3 + [7] = c_22(a__if#(mark(X1),X2,X3),mark#(X1)) * Step 10: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)) - Weak DPs: a__add#(0(),X) -> c_2(mark#(X)) a__and#(true(),X) -> c_6(mark#(X)) a__if#(false(),X,Y) -> c_13(mark#(Y)) a__if#(true(),X,Y) -> c_14(mark#(X)) mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)) mark#(first(X1,X2)) -> c_20(mark#(X1),mark#(X2)) mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1,a__add#/2,a__and#/2,a__first#/2,a__from#/1,a__if#/3 ,mark#/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1,if/3,nil/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/0,c_16/2,c_17/2,c_18/0,c_19/0,c_20/2 ,c_21/1,c_22/2,c_23/0,c_24/0,c_25/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add#,a__and#,a__first#,a__from#,a__if# ,mark#} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_6) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1,2}, uargs(c_17) = {1,2}, uargs(c_20) = {1,2}, uargs(c_22) = {1,2} Following symbols are considered usable: {a__add#,a__and#,a__first#,a__from#,a__if#,mark#} TcT has computed the following interpretation: p(0) = [0] p(a__add) = [0] p(a__and) = [2] p(a__first) = [5] x1 + [4] p(a__from) = [1] x1 + [4] p(a__if) = [1] x1 + [0] p(add) = [1] x1 + [1] x2 + [3] p(and) = [1] x1 + [1] x2 + [2] p(cons) = [1] p(false) = [0] p(first) = [1] x1 + [1] x2 + [1] p(from) = [4] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [0] p(nil) = [1] p(s) = [1] x1 + [1] p(true) = [0] p(a__add#) = [4] x2 + [1] p(a__and#) = [4] x2 + [0] p(a__first#) = [1] x1 + [0] p(a__from#) = [0] p(a__if#) = [4] x2 + [4] x3 + [0] p(mark#) = [4] x1 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [1] p(c_3) = [1] p(c_4) = [1] p(c_5) = [1] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [1] p(c_16) = [1] x1 + [1] x2 + [7] p(c_17) = [1] x1 + [1] x2 + [5] p(c_18) = [1] p(c_19) = [1] p(c_20) = [1] x1 + [1] x2 + [2] p(c_21) = [1] x1 + [4] p(c_22) = [1] x1 + [1] x2 + [0] p(c_23) = [0] p(c_24) = [1] p(c_25) = [1] Following rules are strictly oriented: mark#(add(X1,X2)) = [4] X1 + [4] X2 + [12] > [4] X1 + [4] X2 + [8] = c_16(a__add#(mark(X1),X2),mark#(X1)) Following rules are (at-least) weakly oriented: a__add#(0(),X) = [4] X + [1] >= [4] X + [1] = c_2(mark#(X)) a__and#(true(),X) = [4] X + [0] >= [4] X + [0] = c_6(mark#(X)) a__if#(false(),X,Y) = [4] X + [4] Y + [0] >= [4] Y + [0] = c_13(mark#(Y)) a__if#(true(),X,Y) = [4] X + [4] Y + [0] >= [4] X + [0] = c_14(mark#(X)) mark#(and(X1,X2)) = [4] X1 + [4] X2 + [8] >= [4] X1 + [4] X2 + [5] = c_17(a__and#(mark(X1),X2),mark#(X1)) mark#(first(X1,X2)) = [4] X1 + [4] X2 + [4] >= [4] X1 + [4] X2 + [2] = c_20(mark#(X1),mark#(X2)) mark#(if(X1,X2,X3)) = [4] X1 + [4] X2 + [4] X3 + [0] >= [4] X1 + [4] X2 + [4] X3 + [0] = c_22(a__if#(mark(X1),X2,X3),mark#(X1)) * Step 11: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: a__add#(0(),X) -> c_2(mark#(X)) a__and#(true(),X) -> c_6(mark#(X)) a__if#(false(),X,Y) -> c_13(mark#(Y)) a__if#(true(),X,Y) -> c_14(mark#(X)) mark#(add(X1,X2)) -> c_16(a__add#(mark(X1),X2),mark#(X1)) mark#(and(X1,X2)) -> c_17(a__and#(mark(X1),X2),mark#(X1)) mark#(first(X1,X2)) -> c_20(mark#(X1),mark#(X2)) mark#(if(X1,X2,X3)) -> c_22(a__if#(mark(X1),X2,X3),mark#(X1)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1,a__add#/2,a__and#/2,a__first#/2,a__from#/1,a__if#/3 ,mark#/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1,if/3,nil/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/0,c_16/2,c_17/2,c_18/0,c_19/0,c_20/2 ,c_21/1,c_22/2,c_23/0,c_24/0,c_25/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add#,a__and#,a__first#,a__from#,a__if# ,mark#} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))