WORST_CASE(?,O(n^1)) * Step 1: Bounds WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: deeprev(C(x1,x2)) -> deeprevapp(C(x1,x2),N()) deeprev(N()) -> N() deeprev(V(n)) -> V(n) deeprevapp(C(x1,x2),rest) -> deeprevapp(x2,C(x1,rest)) deeprevapp(N(),rest) -> rest deeprevapp(V(n),rest) -> revconsapp(rest,V(n)) first(C(x1,x2)) -> x1 first(V(n)) -> N() goal(x) -> deeprev(x) isEmptyT(C(x1,x2)) -> False() isEmptyT(N()) -> True() isEmptyT(V(n)) -> False() isNotEmptyT(C(x1,x2)) -> True() isNotEmptyT(N()) -> False() isNotEmptyT(V(n)) -> False() isVal(C(x1,x2)) -> False() isVal(N()) -> False() isVal(V(n)) -> True() revconsapp(C(x1,x2),r) -> revconsapp(x2,C(x1,r)) revconsapp(N(),r) -> r revconsapp(V(n),r) -> r second(C(x1,x2)) -> x2 second(V(n)) -> N() - Signature: {deeprev/1,deeprevapp/2,first/1,goal/1,isEmptyT/1,isNotEmptyT/1,isVal/1,revconsapp/2,second/1} / {C/2 ,False/0,N/0,True/0,V/1} - Obligation: innermost runtime complexity wrt. defined symbols {deeprev,deeprevapp,first,goal,isEmptyT,isNotEmptyT,isVal ,revconsapp,second} and constructors {C,False,N,True,V} + Applied Processor: Bounds {initialAutomaton = minimal, enrichment = match} + Details: The problem is match-bounded by 3. The enriched problem is compatible with follwoing automaton. C_0(2,2) -> 1 C_0(2,2) -> 2 C_1(2,1) -> 1 C_1(2,2) -> 1 C_1(2,2) -> 3 C_1(2,3) -> 1 C_1(2,3) -> 3 C_1(2,5) -> 1 C_1(2,5) -> 3 C_1(2,6) -> 1 C_2(2,1) -> 1 C_2(2,1) -> 6 C_2(2,4) -> 1 C_2(2,4) -> 5 C_2(2,5) -> 1 C_2(2,5) -> 5 C_2(2,6) -> 1 C_2(2,6) -> 6 C_3(2,5) -> 1 C_3(2,5) -> 8 C_3(2,6) -> 1 C_3(2,6) -> 7 C_3(2,7) -> 1 C_3(2,7) -> 7 C_3(2,8) -> 1 C_3(2,8) -> 8 False_0() -> 1 False_0() -> 2 False_1() -> 1 N_0() -> 1 N_0() -> 2 N_1() -> 1 N_1() -> 4 True_0() -> 1 True_0() -> 2 True_1() -> 1 V_0(2) -> 1 V_0(2) -> 2 V_1(2) -> 1 deeprev_0(2) -> 1 deeprev_1(2) -> 1 deeprevapp_0(2,2) -> 1 deeprevapp_1(2,3) -> 1 deeprevapp_1(3,4) -> 1 deeprevapp_2(2,5) -> 1 deeprevapp_2(3,5) -> 1 deeprevapp_2(5,5) -> 1 deeprevapp_3(4,8) -> 1 deeprevapp_3(5,8) -> 1 first_0(2) -> 1 goal_0(2) -> 1 isEmptyT_0(2) -> 1 isNotEmptyT_0(2) -> 1 isVal_0(2) -> 1 revconsapp_0(2,2) -> 1 revconsapp_1(2,1) -> 1 revconsapp_1(3,1) -> 1 revconsapp_1(5,1) -> 1 revconsapp_2(2,6) -> 1 revconsapp_2(3,6) -> 1 revconsapp_2(4,6) -> 1 revconsapp_2(5,6) -> 1 revconsapp_3(4,7) -> 1 revconsapp_3(5,7) -> 1 second_0(2) -> 1 2 -> 1 3 -> 1 5 -> 1 6 -> 1 7 -> 1 8 -> 1 * Step 2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: deeprev(C(x1,x2)) -> deeprevapp(C(x1,x2),N()) deeprev(N()) -> N() deeprev(V(n)) -> V(n) deeprevapp(C(x1,x2),rest) -> deeprevapp(x2,C(x1,rest)) deeprevapp(N(),rest) -> rest deeprevapp(V(n),rest) -> revconsapp(rest,V(n)) first(C(x1,x2)) -> x1 first(V(n)) -> N() goal(x) -> deeprev(x) isEmptyT(C(x1,x2)) -> False() isEmptyT(N()) -> True() isEmptyT(V(n)) -> False() isNotEmptyT(C(x1,x2)) -> True() isNotEmptyT(N()) -> False() isNotEmptyT(V(n)) -> False() isVal(C(x1,x2)) -> False() isVal(N()) -> False() isVal(V(n)) -> True() revconsapp(C(x1,x2),r) -> revconsapp(x2,C(x1,r)) revconsapp(N(),r) -> r revconsapp(V(n),r) -> r second(C(x1,x2)) -> x2 second(V(n)) -> N() - Signature: {deeprev/1,deeprevapp/2,first/1,goal/1,isEmptyT/1,isNotEmptyT/1,isVal/1,revconsapp/2,second/1} / {C/2 ,False/0,N/0,True/0,V/1} - Obligation: innermost runtime complexity wrt. defined symbols {deeprev,deeprevapp,first,goal,isEmptyT,isNotEmptyT,isVal ,revconsapp,second} and constructors {C,False,N,True,V} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))