WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: REVAPP(Cons(z0,z1),z2) -> c5(REVAPP(z1,Cons(z0,z2))) REVAPP(Nil(),z0) -> c6() SELECT(Cons(z0,z1)) -> c3(SELECTS(z0,Nil(),z1)) SELECT(Nil()) -> c4() SELECTS(z0,z1,Cons(z2,z3)) -> c(REVAPP(z1,Cons(z2,z3))) SELECTS(z0,z1,Cons(z2,z3)) -> c1(SELECTS(z2,Cons(z0,z1),z3)) SELECTS(z0,z1,Nil()) -> c2(REVAPP(z1,Nil())) - Weak TRS: revapp(Cons(z0,z1),z2) -> revapp(z1,Cons(z0,z2)) revapp(Nil(),z0) -> z0 select(Cons(z0,z1)) -> selects(z0,Nil(),z1) select(Nil()) -> Nil() selects(z0,z1,Cons(z2,z3)) -> Cons(Cons(z0,revapp(z1,Cons(z2,z3))),selects(z2,Cons(z0,z1),z3)) selects(z0,z1,Nil()) -> Cons(Cons(z0,revapp(z1,Nil())),Nil()) - Signature: {REVAPP/2,SELECT/1,SELECTS/3,revapp/2,select/1,selects/3} / {Cons/2,Nil/0,c/1,c1/1,c2/1,c3/1,c4/0,c5/1,c6/0} - Obligation: innermost runtime complexity wrt. defined symbols {REVAPP,SELECT,SELECTS,revapp,select ,selects} and constructors {Cons,Nil,c,c1,c2,c3,c4,c5,c6} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: REVAPP(Cons(z0,z1),z2) -> c5(REVAPP(z1,Cons(z0,z2))) REVAPP(Nil(),z0) -> c6() SELECT(Cons(z0,z1)) -> c3(SELECTS(z0,Nil(),z1)) SELECT(Nil()) -> c4() SELECTS(z0,z1,Cons(z2,z3)) -> c(REVAPP(z1,Cons(z2,z3))) SELECTS(z0,z1,Cons(z2,z3)) -> c1(SELECTS(z2,Cons(z0,z1),z3)) SELECTS(z0,z1,Nil()) -> c2(REVAPP(z1,Nil())) - Weak TRS: revapp(Cons(z0,z1),z2) -> revapp(z1,Cons(z0,z2)) revapp(Nil(),z0) -> z0 select(Cons(z0,z1)) -> selects(z0,Nil(),z1) select(Nil()) -> Nil() selects(z0,z1,Cons(z2,z3)) -> Cons(Cons(z0,revapp(z1,Cons(z2,z3))),selects(z2,Cons(z0,z1),z3)) selects(z0,z1,Nil()) -> Cons(Cons(z0,revapp(z1,Nil())),Nil()) - Signature: {REVAPP/2,SELECT/1,SELECTS/3,revapp/2,select/1,selects/3} / {Cons/2,Nil/0,c/1,c1/1,c2/1,c3/1,c4/0,c5/1,c6/0} - Obligation: innermost runtime complexity wrt. defined symbols {REVAPP,SELECT,SELECTS,revapp,select ,selects} and constructors {Cons,Nil,c,c1,c2,c3,c4,c5,c6} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: REVAPP(Cons(z0,z1),z2) -> c5(REVAPP(z1,Cons(z0,z2))) REVAPP(Nil(),z0) -> c6() SELECT(Cons(z0,z1)) -> c3(SELECTS(z0,Nil(),z1)) SELECT(Nil()) -> c4() SELECTS(z0,z1,Cons(z2,z3)) -> c(REVAPP(z1,Cons(z2,z3))) SELECTS(z0,z1,Cons(z2,z3)) -> c1(SELECTS(z2,Cons(z0,z1),z3)) SELECTS(z0,z1,Nil()) -> c2(REVAPP(z1,Nil())) - Weak TRS: revapp(Cons(z0,z1),z2) -> revapp(z1,Cons(z0,z2)) revapp(Nil(),z0) -> z0 select(Cons(z0,z1)) -> selects(z0,Nil(),z1) select(Nil()) -> Nil() selects(z0,z1,Cons(z2,z3)) -> Cons(Cons(z0,revapp(z1,Cons(z2,z3))),selects(z2,Cons(z0,z1),z3)) selects(z0,z1,Nil()) -> Cons(Cons(z0,revapp(z1,Nil())),Nil()) - Signature: {REVAPP/2,SELECT/1,SELECTS/3,revapp/2,select/1,selects/3} / {Cons/2,Nil/0,c/1,c1/1,c2/1,c3/1,c4/0,c5/1,c6/0} - Obligation: innermost runtime complexity wrt. defined symbols {REVAPP,SELECT,SELECTS,revapp,select ,selects} and constructors {Cons,Nil,c,c1,c2,c3,c4,c5,c6} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: REVAPP(y,z){y -> Cons(x,y)} = REVAPP(Cons(x,y),z) ->^+ c5(REVAPP(y,Cons(x,z))) = C[REVAPP(y,Cons(x,z)) = REVAPP(y,z){z -> Cons(x,z)}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: REVAPP(Cons(z0,z1),z2) -> c5(REVAPP(z1,Cons(z0,z2))) REVAPP(Nil(),z0) -> c6() SELECT(Cons(z0,z1)) -> c3(SELECTS(z0,Nil(),z1)) SELECT(Nil()) -> c4() SELECTS(z0,z1,Cons(z2,z3)) -> c(REVAPP(z1,Cons(z2,z3))) SELECTS(z0,z1,Cons(z2,z3)) -> c1(SELECTS(z2,Cons(z0,z1),z3)) SELECTS(z0,z1,Nil()) -> c2(REVAPP(z1,Nil())) - Weak TRS: revapp(Cons(z0,z1),z2) -> revapp(z1,Cons(z0,z2)) revapp(Nil(),z0) -> z0 select(Cons(z0,z1)) -> selects(z0,Nil(),z1) select(Nil()) -> Nil() selects(z0,z1,Cons(z2,z3)) -> Cons(Cons(z0,revapp(z1,Cons(z2,z3))),selects(z2,Cons(z0,z1),z3)) selects(z0,z1,Nil()) -> Cons(Cons(z0,revapp(z1,Nil())),Nil()) - Signature: {REVAPP/2,SELECT/1,SELECTS/3,revapp/2,select/1,selects/3} / {Cons/2,Nil/0,c/1,c1/1,c2/1,c3/1,c4/0,c5/1,c6/0} - Obligation: innermost runtime complexity wrt. defined symbols {REVAPP,SELECT,SELECTS,revapp,select ,selects} and constructors {Cons,Nil,c,c1,c2,c3,c4,c5,c6} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs REVAPP#(Cons(z0,z1),z2) -> c_1(REVAPP#(z1,Cons(z0,z2))) REVAPP#(Nil(),z0) -> c_2() SELECT#(Cons(z0,z1)) -> c_3(SELECTS#(z0,Nil(),z1)) SELECT#(Nil()) -> c_4() SELECTS#(z0,z1,Cons(z2,z3)) -> c_5(REVAPP#(z1,Cons(z2,z3))) SELECTS#(z0,z1,Cons(z2,z3)) -> c_6(SELECTS#(z2,Cons(z0,z1),z3)) SELECTS#(z0,z1,Nil()) -> c_7(REVAPP#(z1,Nil())) Weak DPs revapp#(Cons(z0,z1),z2) -> c_8(revapp#(z1,Cons(z0,z2))) revapp#(Nil(),z0) -> c_9() select#(Cons(z0,z1)) -> c_10(selects#(z0,Nil(),z1)) select#(Nil()) -> c_11() selects#(z0,z1,Cons(z2,z3)) -> c_12(revapp#(z1,Cons(z2,z3)),selects#(z2,Cons(z0,z1),z3)) selects#(z0,z1,Nil()) -> c_13(revapp#(z1,Nil())) and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: REVAPP#(Cons(z0,z1),z2) -> c_1(REVAPP#(z1,Cons(z0,z2))) REVAPP#(Nil(),z0) -> c_2() SELECT#(Cons(z0,z1)) -> c_3(SELECTS#(z0,Nil(),z1)) SELECT#(Nil()) -> c_4() SELECTS#(z0,z1,Cons(z2,z3)) -> c_5(REVAPP#(z1,Cons(z2,z3))) SELECTS#(z0,z1,Cons(z2,z3)) -> c_6(SELECTS#(z2,Cons(z0,z1),z3)) SELECTS#(z0,z1,Nil()) -> c_7(REVAPP#(z1,Nil())) - Weak DPs: revapp#(Cons(z0,z1),z2) -> c_8(revapp#(z1,Cons(z0,z2))) revapp#(Nil(),z0) -> c_9() select#(Cons(z0,z1)) -> c_10(selects#(z0,Nil(),z1)) select#(Nil()) -> c_11() selects#(z0,z1,Cons(z2,z3)) -> c_12(revapp#(z1,Cons(z2,z3)),selects#(z2,Cons(z0,z1),z3)) selects#(z0,z1,Nil()) -> c_13(revapp#(z1,Nil())) - Weak TRS: REVAPP(Cons(z0,z1),z2) -> c5(REVAPP(z1,Cons(z0,z2))) REVAPP(Nil(),z0) -> c6() SELECT(Cons(z0,z1)) -> c3(SELECTS(z0,Nil(),z1)) SELECT(Nil()) -> c4() SELECTS(z0,z1,Cons(z2,z3)) -> c(REVAPP(z1,Cons(z2,z3))) SELECTS(z0,z1,Cons(z2,z3)) -> c1(SELECTS(z2,Cons(z0,z1),z3)) SELECTS(z0,z1,Nil()) -> c2(REVAPP(z1,Nil())) revapp(Cons(z0,z1),z2) -> revapp(z1,Cons(z0,z2)) revapp(Nil(),z0) -> z0 select(Cons(z0,z1)) -> selects(z0,Nil(),z1) select(Nil()) -> Nil() selects(z0,z1,Cons(z2,z3)) -> Cons(Cons(z0,revapp(z1,Cons(z2,z3))),selects(z2,Cons(z0,z1),z3)) selects(z0,z1,Nil()) -> Cons(Cons(z0,revapp(z1,Nil())),Nil()) - Signature: {REVAPP/2,SELECT/1,SELECTS/3,revapp/2,select/1,selects/3,REVAPP#/2,SELECT#/1,SELECTS#/3,revapp#/2,select#/1 ,selects#/3} / {Cons/2,Nil/0,c/1,c1/1,c2/1,c3/1,c4/0,c5/1,c6/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/1,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {REVAPP#,SELECT#,SELECTS#,revapp#,select# ,selects#} and constructors {Cons,Nil,c,c1,c2,c3,c4,c5,c6} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,4} by application of Pre({2,4}) = {1,5,7}. Here rules are labelled as follows: 1: REVAPP#(Cons(z0,z1),z2) -> c_1(REVAPP#(z1,Cons(z0,z2))) 2: REVAPP#(Nil(),z0) -> c_2() 3: SELECT#(Cons(z0,z1)) -> c_3(SELECTS#(z0,Nil(),z1)) 4: SELECT#(Nil()) -> c_4() 5: SELECTS#(z0,z1,Cons(z2,z3)) -> c_5(REVAPP#(z1,Cons(z2,z3))) 6: SELECTS#(z0,z1,Cons(z2,z3)) -> c_6(SELECTS#(z2,Cons(z0,z1),z3)) 7: SELECTS#(z0,z1,Nil()) -> c_7(REVAPP#(z1,Nil())) 8: revapp#(Cons(z0,z1),z2) -> c_8(revapp#(z1,Cons(z0,z2))) 9: revapp#(Nil(),z0) -> c_9() 10: select#(Cons(z0,z1)) -> c_10(selects#(z0,Nil(),z1)) 11: select#(Nil()) -> c_11() 12: selects#(z0,z1,Cons(z2,z3)) -> c_12(revapp#(z1,Cons(z2,z3)),selects#(z2,Cons(z0,z1),z3)) 13: selects#(z0,z1,Nil()) -> c_13(revapp#(z1,Nil())) ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: REVAPP#(Cons(z0,z1),z2) -> c_1(REVAPP#(z1,Cons(z0,z2))) SELECT#(Cons(z0,z1)) -> c_3(SELECTS#(z0,Nil(),z1)) SELECTS#(z0,z1,Cons(z2,z3)) -> c_5(REVAPP#(z1,Cons(z2,z3))) SELECTS#(z0,z1,Cons(z2,z3)) -> c_6(SELECTS#(z2,Cons(z0,z1),z3)) SELECTS#(z0,z1,Nil()) -> c_7(REVAPP#(z1,Nil())) - Weak DPs: REVAPP#(Nil(),z0) -> c_2() SELECT#(Nil()) -> c_4() revapp#(Cons(z0,z1),z2) -> c_8(revapp#(z1,Cons(z0,z2))) revapp#(Nil(),z0) -> c_9() select#(Cons(z0,z1)) -> c_10(selects#(z0,Nil(),z1)) select#(Nil()) -> c_11() selects#(z0,z1,Cons(z2,z3)) -> c_12(revapp#(z1,Cons(z2,z3)),selects#(z2,Cons(z0,z1),z3)) selects#(z0,z1,Nil()) -> c_13(revapp#(z1,Nil())) - Weak TRS: REVAPP(Cons(z0,z1),z2) -> c5(REVAPP(z1,Cons(z0,z2))) REVAPP(Nil(),z0) -> c6() SELECT(Cons(z0,z1)) -> c3(SELECTS(z0,Nil(),z1)) SELECT(Nil()) -> c4() SELECTS(z0,z1,Cons(z2,z3)) -> c(REVAPP(z1,Cons(z2,z3))) SELECTS(z0,z1,Cons(z2,z3)) -> c1(SELECTS(z2,Cons(z0,z1),z3)) SELECTS(z0,z1,Nil()) -> c2(REVAPP(z1,Nil())) revapp(Cons(z0,z1),z2) -> revapp(z1,Cons(z0,z2)) revapp(Nil(),z0) -> z0 select(Cons(z0,z1)) -> selects(z0,Nil(),z1) select(Nil()) -> Nil() selects(z0,z1,Cons(z2,z3)) -> Cons(Cons(z0,revapp(z1,Cons(z2,z3))),selects(z2,Cons(z0,z1),z3)) selects(z0,z1,Nil()) -> Cons(Cons(z0,revapp(z1,Nil())),Nil()) - Signature: {REVAPP/2,SELECT/1,SELECTS/3,revapp/2,select/1,selects/3,REVAPP#/2,SELECT#/1,SELECTS#/3,revapp#/2,select#/1 ,selects#/3} / {Cons/2,Nil/0,c/1,c1/1,c2/1,c3/1,c4/0,c5/1,c6/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/1,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {REVAPP#,SELECT#,SELECTS#,revapp#,select# ,selects#} and constructors {Cons,Nil,c,c1,c2,c3,c4,c5,c6} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:REVAPP#(Cons(z0,z1),z2) -> c_1(REVAPP#(z1,Cons(z0,z2))) -->_1 REVAPP#(Nil(),z0) -> c_2():6 -->_1 REVAPP#(Cons(z0,z1),z2) -> c_1(REVAPP#(z1,Cons(z0,z2))):1 2:S:SELECT#(Cons(z0,z1)) -> c_3(SELECTS#(z0,Nil(),z1)) -->_1 SELECTS#(z0,z1,Nil()) -> c_7(REVAPP#(z1,Nil())):5 -->_1 SELECTS#(z0,z1,Cons(z2,z3)) -> c_6(SELECTS#(z2,Cons(z0,z1),z3)):4 -->_1 SELECTS#(z0,z1,Cons(z2,z3)) -> c_5(REVAPP#(z1,Cons(z2,z3))):3 3:S:SELECTS#(z0,z1,Cons(z2,z3)) -> c_5(REVAPP#(z1,Cons(z2,z3))) -->_1 REVAPP#(Nil(),z0) -> c_2():6 -->_1 REVAPP#(Cons(z0,z1),z2) -> c_1(REVAPP#(z1,Cons(z0,z2))):1 4:S:SELECTS#(z0,z1,Cons(z2,z3)) -> c_6(SELECTS#(z2,Cons(z0,z1),z3)) -->_1 SELECTS#(z0,z1,Nil()) -> c_7(REVAPP#(z1,Nil())):5 -->_1 SELECTS#(z0,z1,Cons(z2,z3)) -> c_6(SELECTS#(z2,Cons(z0,z1),z3)):4 -->_1 SELECTS#(z0,z1,Cons(z2,z3)) -> c_5(REVAPP#(z1,Cons(z2,z3))):3 5:S:SELECTS#(z0,z1,Nil()) -> c_7(REVAPP#(z1,Nil())) -->_1 REVAPP#(Nil(),z0) -> c_2():6 -->_1 REVAPP#(Cons(z0,z1),z2) -> c_1(REVAPP#(z1,Cons(z0,z2))):1 6:W:REVAPP#(Nil(),z0) -> c_2() 7:W:SELECT#(Nil()) -> c_4() 8:W:revapp#(Cons(z0,z1),z2) -> c_8(revapp#(z1,Cons(z0,z2))) -->_1 revapp#(Nil(),z0) -> c_9():9 -->_1 revapp#(Cons(z0,z1),z2) -> c_8(revapp#(z1,Cons(z0,z2))):8 9:W:revapp#(Nil(),z0) -> c_9() 10:W:select#(Cons(z0,z1)) -> c_10(selects#(z0,Nil(),z1)) -->_1 selects#(z0,z1,Nil()) -> c_13(revapp#(z1,Nil())):13 -->_1 selects#(z0,z1,Cons(z2,z3)) -> c_12(revapp#(z1,Cons(z2,z3)),selects#(z2,Cons(z0,z1),z3)):12 11:W:select#(Nil()) -> c_11() 12:W:selects#(z0,z1,Cons(z2,z3)) -> c_12(revapp#(z1,Cons(z2,z3)),selects#(z2,Cons(z0,z1),z3)) -->_2 selects#(z0,z1,Nil()) -> c_13(revapp#(z1,Nil())):13 -->_2 selects#(z0,z1,Cons(z2,z3)) -> c_12(revapp#(z1,Cons(z2,z3)),selects#(z2,Cons(z0,z1),z3)):12 -->_1 revapp#(Nil(),z0) -> c_9():9 -->_1 revapp#(Cons(z0,z1),z2) -> c_8(revapp#(z1,Cons(z0,z2))):8 13:W:selects#(z0,z1,Nil()) -> c_13(revapp#(z1,Nil())) -->_1 revapp#(Nil(),z0) -> c_9():9 -->_1 revapp#(Cons(z0,z1),z2) -> c_8(revapp#(z1,Cons(z0,z2))):8 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 11: select#(Nil()) -> c_11() 10: select#(Cons(z0,z1)) -> c_10(selects#(z0,Nil(),z1)) 12: selects#(z0,z1,Cons(z2,z3)) -> c_12(revapp#(z1,Cons(z2,z3)),selects#(z2,Cons(z0,z1),z3)) 13: selects#(z0,z1,Nil()) -> c_13(revapp#(z1,Nil())) 8: revapp#(Cons(z0,z1),z2) -> c_8(revapp#(z1,Cons(z0,z2))) 9: revapp#(Nil(),z0) -> c_9() 7: SELECT#(Nil()) -> c_4() 6: REVAPP#(Nil(),z0) -> c_2() ** Step 1.b:4: RemoveHeads. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: REVAPP#(Cons(z0,z1),z2) -> c_1(REVAPP#(z1,Cons(z0,z2))) SELECT#(Cons(z0,z1)) -> c_3(SELECTS#(z0,Nil(),z1)) SELECTS#(z0,z1,Cons(z2,z3)) -> c_5(REVAPP#(z1,Cons(z2,z3))) SELECTS#(z0,z1,Cons(z2,z3)) -> c_6(SELECTS#(z2,Cons(z0,z1),z3)) SELECTS#(z0,z1,Nil()) -> c_7(REVAPP#(z1,Nil())) - Weak TRS: REVAPP(Cons(z0,z1),z2) -> c5(REVAPP(z1,Cons(z0,z2))) REVAPP(Nil(),z0) -> c6() SELECT(Cons(z0,z1)) -> c3(SELECTS(z0,Nil(),z1)) SELECT(Nil()) -> c4() SELECTS(z0,z1,Cons(z2,z3)) -> c(REVAPP(z1,Cons(z2,z3))) SELECTS(z0,z1,Cons(z2,z3)) -> c1(SELECTS(z2,Cons(z0,z1),z3)) SELECTS(z0,z1,Nil()) -> c2(REVAPP(z1,Nil())) revapp(Cons(z0,z1),z2) -> revapp(z1,Cons(z0,z2)) revapp(Nil(),z0) -> z0 select(Cons(z0,z1)) -> selects(z0,Nil(),z1) select(Nil()) -> Nil() selects(z0,z1,Cons(z2,z3)) -> Cons(Cons(z0,revapp(z1,Cons(z2,z3))),selects(z2,Cons(z0,z1),z3)) selects(z0,z1,Nil()) -> Cons(Cons(z0,revapp(z1,Nil())),Nil()) - Signature: {REVAPP/2,SELECT/1,SELECTS/3,revapp/2,select/1,selects/3,REVAPP#/2,SELECT#/1,SELECTS#/3,revapp#/2,select#/1 ,selects#/3} / {Cons/2,Nil/0,c/1,c1/1,c2/1,c3/1,c4/0,c5/1,c6/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/1,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {REVAPP#,SELECT#,SELECTS#,revapp#,select# ,selects#} and constructors {Cons,Nil,c,c1,c2,c3,c4,c5,c6} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:REVAPP#(Cons(z0,z1),z2) -> c_1(REVAPP#(z1,Cons(z0,z2))) -->_1 REVAPP#(Cons(z0,z1),z2) -> c_1(REVAPP#(z1,Cons(z0,z2))):1 2:S:SELECT#(Cons(z0,z1)) -> c_3(SELECTS#(z0,Nil(),z1)) -->_1 SELECTS#(z0,z1,Nil()) -> c_7(REVAPP#(z1,Nil())):5 -->_1 SELECTS#(z0,z1,Cons(z2,z3)) -> c_6(SELECTS#(z2,Cons(z0,z1),z3)):4 -->_1 SELECTS#(z0,z1,Cons(z2,z3)) -> c_5(REVAPP#(z1,Cons(z2,z3))):3 3:S:SELECTS#(z0,z1,Cons(z2,z3)) -> c_5(REVAPP#(z1,Cons(z2,z3))) -->_1 REVAPP#(Cons(z0,z1),z2) -> c_1(REVAPP#(z1,Cons(z0,z2))):1 4:S:SELECTS#(z0,z1,Cons(z2,z3)) -> c_6(SELECTS#(z2,Cons(z0,z1),z3)) -->_1 SELECTS#(z0,z1,Nil()) -> c_7(REVAPP#(z1,Nil())):5 -->_1 SELECTS#(z0,z1,Cons(z2,z3)) -> c_6(SELECTS#(z2,Cons(z0,z1),z3)):4 -->_1 SELECTS#(z0,z1,Cons(z2,z3)) -> c_5(REVAPP#(z1,Cons(z2,z3))):3 5:S:SELECTS#(z0,z1,Nil()) -> c_7(REVAPP#(z1,Nil())) -->_1 REVAPP#(Cons(z0,z1),z2) -> c_1(REVAPP#(z1,Cons(z0,z2))):1 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(2,SELECT#(Cons(z0,z1)) -> c_3(SELECTS#(z0,Nil(),z1)))] ** Step 1.b:5: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: REVAPP#(Cons(z0,z1),z2) -> c_1(REVAPP#(z1,Cons(z0,z2))) SELECTS#(z0,z1,Cons(z2,z3)) -> c_5(REVAPP#(z1,Cons(z2,z3))) SELECTS#(z0,z1,Cons(z2,z3)) -> c_6(SELECTS#(z2,Cons(z0,z1),z3)) SELECTS#(z0,z1,Nil()) -> c_7(REVAPP#(z1,Nil())) - Weak TRS: REVAPP(Cons(z0,z1),z2) -> c5(REVAPP(z1,Cons(z0,z2))) REVAPP(Nil(),z0) -> c6() SELECT(Cons(z0,z1)) -> c3(SELECTS(z0,Nil(),z1)) SELECT(Nil()) -> c4() SELECTS(z0,z1,Cons(z2,z3)) -> c(REVAPP(z1,Cons(z2,z3))) SELECTS(z0,z1,Cons(z2,z3)) -> c1(SELECTS(z2,Cons(z0,z1),z3)) SELECTS(z0,z1,Nil()) -> c2(REVAPP(z1,Nil())) revapp(Cons(z0,z1),z2) -> revapp(z1,Cons(z0,z2)) revapp(Nil(),z0) -> z0 select(Cons(z0,z1)) -> selects(z0,Nil(),z1) select(Nil()) -> Nil() selects(z0,z1,Cons(z2,z3)) -> Cons(Cons(z0,revapp(z1,Cons(z2,z3))),selects(z2,Cons(z0,z1),z3)) selects(z0,z1,Nil()) -> Cons(Cons(z0,revapp(z1,Nil())),Nil()) - Signature: {REVAPP/2,SELECT/1,SELECTS/3,revapp/2,select/1,selects/3,REVAPP#/2,SELECT#/1,SELECTS#/3,revapp#/2,select#/1 ,selects#/3} / {Cons/2,Nil/0,c/1,c1/1,c2/1,c3/1,c4/0,c5/1,c6/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/1,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {REVAPP#,SELECT#,SELECTS#,revapp#,select# ,selects#} and constructors {Cons,Nil,c,c1,c2,c3,c4,c5,c6} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: REVAPP#(Cons(z0,z1),z2) -> c_1(REVAPP#(z1,Cons(z0,z2))) SELECTS#(z0,z1,Cons(z2,z3)) -> c_5(REVAPP#(z1,Cons(z2,z3))) SELECTS#(z0,z1,Cons(z2,z3)) -> c_6(SELECTS#(z2,Cons(z0,z1),z3)) SELECTS#(z0,z1,Nil()) -> c_7(REVAPP#(z1,Nil())) ** Step 1.b:6: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: REVAPP#(Cons(z0,z1),z2) -> c_1(REVAPP#(z1,Cons(z0,z2))) SELECTS#(z0,z1,Cons(z2,z3)) -> c_5(REVAPP#(z1,Cons(z2,z3))) SELECTS#(z0,z1,Cons(z2,z3)) -> c_6(SELECTS#(z2,Cons(z0,z1),z3)) SELECTS#(z0,z1,Nil()) -> c_7(REVAPP#(z1,Nil())) - Signature: {REVAPP/2,SELECT/1,SELECTS/3,revapp/2,select/1,selects/3,REVAPP#/2,SELECT#/1,SELECTS#/3,revapp#/2,select#/1 ,selects#/3} / {Cons/2,Nil/0,c/1,c1/1,c2/1,c3/1,c4/0,c5/1,c6/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/1,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {REVAPP#,SELECT#,SELECTS#,revapp#,select# ,selects#} and constructors {Cons,Nil,c,c1,c2,c3,c4,c5,c6} + 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_1) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1} Following symbols are considered usable: {REVAPP#,SELECT#,SELECTS#,revapp#,select#,selects#} TcT has computed the following interpretation: p(Cons) = [1] x1 + [1] x2 + [1] p(Nil) = [2] p(REVAPP) = [1] x2 + [8] p(SELECT) = [4] x1 + [1] p(SELECTS) = [1] x2 + [1] x3 + [1] p(c) = [1] x1 + [8] p(c1) = [0] p(c2) = [2] p(c3) = [0] p(c4) = [1] p(c5) = [1] p(c6) = [8] p(revapp) = [2] x1 + [0] p(select) = [1] p(selects) = [1] x1 + [4] x3 + [1] p(REVAPP#) = [0] p(SELECT#) = [2] x1 + [0] p(SELECTS#) = [4] x1 + [1] x2 + [8] x3 + [0] p(revapp#) = [1] x1 + [4] x2 + [1] p(select#) = [1] x1 + [0] p(selects#) = [4] x1 + [1] x3 + [1] p(c_1) = [8] x1 + [0] p(c_2) = [1] p(c_3) = [4] p(c_4) = [0] p(c_5) = [1] x1 + [4] p(c_6) = [1] x1 + [5] p(c_7) = [4] x1 + [6] p(c_8) = [0] p(c_9) = [1] p(c_10) = [1] p(c_11) = [1] p(c_12) = [1] x1 + [1] p(c_13) = [0] Following rules are strictly oriented: SELECTS#(z0,z1,Cons(z2,z3)) = [4] z0 + [1] z1 + [8] z2 + [8] z3 + [8] > [4] = c_5(REVAPP#(z1,Cons(z2,z3))) SELECTS#(z0,z1,Cons(z2,z3)) = [4] z0 + [1] z1 + [8] z2 + [8] z3 + [8] > [1] z0 + [1] z1 + [4] z2 + [8] z3 + [6] = c_6(SELECTS#(z2,Cons(z0,z1),z3)) SELECTS#(z0,z1,Nil()) = [4] z0 + [1] z1 + [16] > [6] = c_7(REVAPP#(z1,Nil())) Following rules are (at-least) weakly oriented: REVAPP#(Cons(z0,z1),z2) = [0] >= [0] = c_1(REVAPP#(z1,Cons(z0,z2))) ** Step 1.b:7: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: REVAPP#(Cons(z0,z1),z2) -> c_1(REVAPP#(z1,Cons(z0,z2))) - Weak DPs: SELECTS#(z0,z1,Cons(z2,z3)) -> c_5(REVAPP#(z1,Cons(z2,z3))) SELECTS#(z0,z1,Cons(z2,z3)) -> c_6(SELECTS#(z2,Cons(z0,z1),z3)) SELECTS#(z0,z1,Nil()) -> c_7(REVAPP#(z1,Nil())) - Signature: {REVAPP/2,SELECT/1,SELECTS/3,revapp/2,select/1,selects/3,REVAPP#/2,SELECT#/1,SELECTS#/3,revapp#/2,select#/1 ,selects#/3} / {Cons/2,Nil/0,c/1,c1/1,c2/1,c3/1,c4/0,c5/1,c6/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/1,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {REVAPP#,SELECT#,SELECTS#,revapp#,select# ,selects#} and constructors {Cons,Nil,c,c1,c2,c3,c4,c5,c6} + 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_1) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1} Following symbols are considered usable: {REVAPP#,SELECT#,SELECTS#,revapp#,select#,selects#} TcT has computed the following interpretation: p(Cons) = [1] x2 + [3] p(Nil) = [2] p(REVAPP) = [1] x1 + [0] p(SELECT) = [2] x1 + [1] p(SELECTS) = [2] x3 + [0] p(c) = [1] p(c1) = [0] p(c2) = [1] x1 + [0] p(c3) = [4] p(c4) = [0] p(c5) = [8] p(c6) = [2] p(revapp) = [1] x1 + [4] x2 + [4] p(select) = [0] p(selects) = [1] x2 + [1] p(REVAPP#) = [3] x1 + [5] p(SELECT#) = [1] x1 + [1] p(SELECTS#) = [8] x2 + [8] x3 + [1] p(revapp#) = [1] x1 + [1] x2 + [0] p(select#) = [1] x1 + [2] p(selects#) = [1] x1 + [1] x3 + [1] p(c_1) = [1] x1 + [6] p(c_2) = [0] p(c_3) = [8] p(c_4) = [8] p(c_5) = [1] x1 + [1] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [12] p(c_8) = [1] x1 + [1] p(c_9) = [8] p(c_10) = [8] p(c_11) = [1] p(c_12) = [1] x1 + [2] x2 + [0] p(c_13) = [0] Following rules are strictly oriented: REVAPP#(Cons(z0,z1),z2) = [3] z1 + [14] > [3] z1 + [11] = c_1(REVAPP#(z1,Cons(z0,z2))) Following rules are (at-least) weakly oriented: SELECTS#(z0,z1,Cons(z2,z3)) = [8] z1 + [8] z3 + [25] >= [3] z1 + [6] = c_5(REVAPP#(z1,Cons(z2,z3))) SELECTS#(z0,z1,Cons(z2,z3)) = [8] z1 + [8] z3 + [25] >= [8] z1 + [8] z3 + [25] = c_6(SELECTS#(z2,Cons(z0,z1),z3)) SELECTS#(z0,z1,Nil()) = [8] z1 + [17] >= [3] z1 + [17] = c_7(REVAPP#(z1,Nil())) ** Step 1.b:8: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: REVAPP#(Cons(z0,z1),z2) -> c_1(REVAPP#(z1,Cons(z0,z2))) SELECTS#(z0,z1,Cons(z2,z3)) -> c_5(REVAPP#(z1,Cons(z2,z3))) SELECTS#(z0,z1,Cons(z2,z3)) -> c_6(SELECTS#(z2,Cons(z0,z1),z3)) SELECTS#(z0,z1,Nil()) -> c_7(REVAPP#(z1,Nil())) - Signature: {REVAPP/2,SELECT/1,SELECTS/3,revapp/2,select/1,selects/3,REVAPP#/2,SELECT#/1,SELECTS#/3,revapp#/2,select#/1 ,selects#/3} / {Cons/2,Nil/0,c/1,c1/1,c2/1,c3/1,c4/0,c5/1,c6/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/1,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {REVAPP#,SELECT#,SELECTS#,revapp#,select# ,selects#} and constructors {Cons,Nil,c,c1,c2,c3,c4,c5,c6} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))