/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) CpxTrsMatchBoundsTAProof [FINISHED, 486 ms] (6) BOUNDS(1, n^1) (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (8) TRS for Loop Detection (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (10) BEST (11) proven lower bound (12) LowerBoundPropagationProof [FINISHED, 0 ms] (13) BOUNDS(n^1, INF) (14) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) active(__(X, nil)) -> mark(X) active(__(nil, X)) -> mark(X) active(U11(tt)) -> mark(tt) active(U21(tt, V2)) -> mark(U22(isList(V2))) active(U22(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNeList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt)) -> mark(tt) active(U71(tt, P)) -> mark(U72(isPal(P))) active(U72(tt)) -> mark(tt) active(U81(tt)) -> mark(tt) active(isList(V)) -> mark(U11(isNeList(V))) active(isList(nil)) -> mark(tt) active(isList(__(V1, V2))) -> mark(U21(isList(V1), V2)) active(isNeList(V)) -> mark(U31(isQid(V))) active(isNeList(__(V1, V2))) -> mark(U41(isList(V1), V2)) active(isNeList(__(V1, V2))) -> mark(U51(isNeList(V1), V2)) active(isNePal(V)) -> mark(U61(isQid(V))) active(isNePal(__(I, __(P, I)))) -> mark(U71(isQid(I), P)) active(isPal(V)) -> mark(U81(isNePal(V))) active(isPal(nil)) -> mark(tt) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) active(__(X1, X2)) -> __(active(X1), X2) active(__(X1, X2)) -> __(X1, active(X2)) active(U11(X)) -> U11(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X)) -> U72(active(X)) active(U81(X)) -> U81(active(X)) __(mark(X1), X2) -> mark(__(X1, X2)) __(X1, mark(X2)) -> mark(__(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X)) -> mark(U72(X)) U81(mark(X)) -> mark(U81(X)) proper(__(X1, X2)) -> __(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isList(X)) -> isList(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNeList(X)) -> isNeList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(U61(X)) -> U61(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X)) -> U72(proper(X)) proper(isPal(X)) -> isPal(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(isQid(X)) -> isQid(proper(X)) proper(isNePal(X)) -> isNePal(proper(X)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(o) -> ok(o) proper(u) -> ok(u) __(ok(X1), ok(X2)) -> ok(__(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) isList(ok(X)) -> ok(isList(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNeList(ok(X)) -> ok(isNeList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X)) -> ok(U72(X)) isPal(ok(X)) -> ok(isPal(X)) U81(ok(X)) -> ok(U81(X)) isQid(ok(X)) -> ok(isQid(X)) isNePal(ok(X)) -> ok(isNePal(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) The following defined symbols can occur below the 0th argument of top: proper, active The following defined symbols can occur below the 0th argument of proper: proper, active The following defined symbols can occur below the 0th argument of active: proper, active Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) active(__(X, nil)) -> mark(X) active(__(nil, X)) -> mark(X) active(U11(tt)) -> mark(tt) active(U21(tt, V2)) -> mark(U22(isList(V2))) active(U22(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNeList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt)) -> mark(tt) active(U71(tt, P)) -> mark(U72(isPal(P))) active(U72(tt)) -> mark(tt) active(U81(tt)) -> mark(tt) active(isList(V)) -> mark(U11(isNeList(V))) active(isList(nil)) -> mark(tt) active(isList(__(V1, V2))) -> mark(U21(isList(V1), V2)) active(isNeList(V)) -> mark(U31(isQid(V))) active(isNeList(__(V1, V2))) -> mark(U41(isList(V1), V2)) active(isNeList(__(V1, V2))) -> mark(U51(isNeList(V1), V2)) active(isNePal(V)) -> mark(U61(isQid(V))) active(isNePal(__(I, __(P, I)))) -> mark(U71(isQid(I), P)) active(isPal(V)) -> mark(U81(isNePal(V))) active(isPal(nil)) -> mark(tt) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) active(__(X1, X2)) -> __(active(X1), X2) active(__(X1, X2)) -> __(X1, active(X2)) active(U11(X)) -> U11(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X)) -> U72(active(X)) active(U81(X)) -> U81(active(X)) proper(__(X1, X2)) -> __(proper(X1), proper(X2)) proper(U11(X)) -> U11(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isList(X)) -> isList(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNeList(X)) -> isNeList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(U61(X)) -> U61(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X)) -> U72(proper(X)) proper(isPal(X)) -> isPal(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(isQid(X)) -> isQid(proper(X)) proper(isNePal(X)) -> isNePal(proper(X)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: __(mark(X1), X2) -> mark(__(X1, X2)) __(X1, mark(X2)) -> mark(__(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X)) -> mark(U72(X)) U81(mark(X)) -> mark(U81(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(o) -> ok(o) proper(u) -> ok(u) __(ok(X1), ok(X2)) -> ok(__(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) isList(ok(X)) -> ok(isList(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNeList(ok(X)) -> ok(isNeList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X)) -> ok(U72(X)) isPal(ok(X)) -> ok(isPal(X)) U81(ok(X)) -> ok(U81(X)) isQid(ok(X)) -> ok(isQid(X)) isNePal(ok(X)) -> ok(isNePal(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: __(mark(X1), X2) -> mark(__(X1, X2)) __(X1, mark(X2)) -> mark(__(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X)) -> mark(U72(X)) U81(mark(X)) -> mark(U81(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(o) -> ok(o) proper(u) -> ok(u) __(ok(X1), ok(X2)) -> ok(__(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) isList(ok(X)) -> ok(isList(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNeList(ok(X)) -> ok(isNeList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X)) -> ok(U72(X)) isPal(ok(X)) -> ok(isPal(X)) U81(ok(X)) -> ok(U81(X)) isQid(ok(X)) -> ok(isQid(X)) isNePal(ok(X)) -> ok(isNePal(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (5) CpxTrsMatchBoundsTAProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] transitions: mark0(0) -> 0 nil0() -> 0 ok0(0) -> 0 tt0() -> 0 a0() -> 0 e0() -> 0 i0() -> 0 o0() -> 0 u0() -> 0 active0(0) -> 0 __0(0, 0) -> 1 U110(0) -> 2 U210(0, 0) -> 3 U220(0) -> 4 U310(0) -> 5 U410(0, 0) -> 6 U420(0) -> 7 U510(0, 0) -> 8 U520(0) -> 9 U610(0) -> 10 U710(0, 0) -> 11 U720(0) -> 12 U810(0) -> 13 proper0(0) -> 14 isList0(0) -> 15 isNeList0(0) -> 16 isPal0(0) -> 17 isQid0(0) -> 18 isNePal0(0) -> 19 top0(0) -> 20 __1(0, 0) -> 21 mark1(21) -> 1 U111(0) -> 22 mark1(22) -> 2 U211(0, 0) -> 23 mark1(23) -> 3 U221(0) -> 24 mark1(24) -> 4 U311(0) -> 25 mark1(25) -> 5 U411(0, 0) -> 26 mark1(26) -> 6 U421(0) -> 27 mark1(27) -> 7 U511(0, 0) -> 28 mark1(28) -> 8 U521(0) -> 29 mark1(29) -> 9 U611(0) -> 30 mark1(30) -> 10 U711(0, 0) -> 31 mark1(31) -> 11 U721(0) -> 32 mark1(32) -> 12 U811(0) -> 33 mark1(33) -> 13 nil1() -> 34 ok1(34) -> 14 tt1() -> 35 ok1(35) -> 14 a1() -> 36 ok1(36) -> 14 e1() -> 37 ok1(37) -> 14 i1() -> 38 ok1(38) -> 14 o1() -> 39 ok1(39) -> 14 u1() -> 40 ok1(40) -> 14 __1(0, 0) -> 41 ok1(41) -> 1 U111(0) -> 42 ok1(42) -> 2 U211(0, 0) -> 43 ok1(43) -> 3 U221(0) -> 44 ok1(44) -> 4 isList1(0) -> 45 ok1(45) -> 15 U311(0) -> 46 ok1(46) -> 5 U411(0, 0) -> 47 ok1(47) -> 6 U421(0) -> 48 ok1(48) -> 7 isNeList1(0) -> 49 ok1(49) -> 16 U511(0, 0) -> 50 ok1(50) -> 8 U521(0) -> 51 ok1(51) -> 9 U611(0) -> 52 ok1(52) -> 10 U711(0, 0) -> 53 ok1(53) -> 11 U721(0) -> 54 ok1(54) -> 12 isPal1(0) -> 55 ok1(55) -> 17 U811(0) -> 56 ok1(56) -> 13 isQid1(0) -> 57 ok1(57) -> 18 isNePal1(0) -> 58 ok1(58) -> 19 proper1(0) -> 59 top1(59) -> 20 active1(0) -> 60 top1(60) -> 20 mark1(21) -> 21 mark1(21) -> 41 mark1(22) -> 22 mark1(22) -> 42 mark1(23) -> 23 mark1(23) -> 43 mark1(24) -> 24 mark1(24) -> 44 mark1(25) -> 25 mark1(25) -> 46 mark1(26) -> 26 mark1(26) -> 47 mark1(27) -> 27 mark1(27) -> 48 mark1(28) -> 28 mark1(28) -> 50 mark1(29) -> 29 mark1(29) -> 51 mark1(30) -> 30 mark1(30) -> 52 mark1(31) -> 31 mark1(31) -> 53 mark1(32) -> 32 mark1(32) -> 54 mark1(33) -> 33 mark1(33) -> 56 ok1(34) -> 59 ok1(35) -> 59 ok1(36) -> 59 ok1(37) -> 59 ok1(38) -> 59 ok1(39) -> 59 ok1(40) -> 59 ok1(41) -> 21 ok1(41) -> 41 ok1(42) -> 22 ok1(42) -> 42 ok1(43) -> 23 ok1(43) -> 43 ok1(44) -> 24 ok1(44) -> 44 ok1(45) -> 45 ok1(46) -> 25 ok1(46) -> 46 ok1(47) -> 26 ok1(47) -> 47 ok1(48) -> 27 ok1(48) -> 48 ok1(49) -> 49 ok1(50) -> 28 ok1(50) -> 50 ok1(51) -> 29 ok1(51) -> 51 ok1(52) -> 30 ok1(52) -> 52 ok1(53) -> 31 ok1(53) -> 53 ok1(54) -> 32 ok1(54) -> 54 ok1(55) -> 55 ok1(56) -> 33 ok1(56) -> 56 ok1(57) -> 57 ok1(58) -> 58 active2(34) -> 61 top2(61) -> 20 active2(35) -> 61 active2(36) -> 61 active2(37) -> 61 active2(38) -> 61 active2(39) -> 61 active2(40) -> 61 ---------------------------------------- (6) BOUNDS(1, n^1) ---------------------------------------- (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (8) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) active(__(X, nil)) -> mark(X) active(__(nil, X)) -> mark(X) active(U11(tt)) -> mark(tt) active(U21(tt, V2)) -> mark(U22(isList(V2))) active(U22(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNeList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt)) -> mark(tt) active(U71(tt, P)) -> mark(U72(isPal(P))) active(U72(tt)) -> mark(tt) active(U81(tt)) -> mark(tt) active(isList(V)) -> mark(U11(isNeList(V))) active(isList(nil)) -> mark(tt) active(isList(__(V1, V2))) -> mark(U21(isList(V1), V2)) active(isNeList(V)) -> mark(U31(isQid(V))) active(isNeList(__(V1, V2))) -> mark(U41(isList(V1), V2)) active(isNeList(__(V1, V2))) -> mark(U51(isNeList(V1), V2)) active(isNePal(V)) -> mark(U61(isQid(V))) active(isNePal(__(I, __(P, I)))) -> mark(U71(isQid(I), P)) active(isPal(V)) -> mark(U81(isNePal(V))) active(isPal(nil)) -> mark(tt) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) active(__(X1, X2)) -> __(active(X1), X2) active(__(X1, X2)) -> __(X1, active(X2)) active(U11(X)) -> U11(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X)) -> U72(active(X)) active(U81(X)) -> U81(active(X)) __(mark(X1), X2) -> mark(__(X1, X2)) __(X1, mark(X2)) -> mark(__(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X)) -> mark(U72(X)) U81(mark(X)) -> mark(U81(X)) proper(__(X1, X2)) -> __(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isList(X)) -> isList(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNeList(X)) -> isNeList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(U61(X)) -> U61(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X)) -> U72(proper(X)) proper(isPal(X)) -> isPal(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(isQid(X)) -> isQid(proper(X)) proper(isNePal(X)) -> isNePal(proper(X)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(o) -> ok(o) proper(u) -> ok(u) __(ok(X1), ok(X2)) -> ok(__(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) isList(ok(X)) -> ok(isList(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNeList(ok(X)) -> ok(isNeList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X)) -> ok(U72(X)) isPal(ok(X)) -> ok(isPal(X)) U81(ok(X)) -> ok(U81(X)) isQid(ok(X)) -> ok(isQid(X)) isNePal(ok(X)) -> ok(isNePal(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (9) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence U31(ok(X)) ->^+ ok(U31(X)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / ok(X)]. The result substitution is [ ]. ---------------------------------------- (10) Complex Obligation (BEST) ---------------------------------------- (11) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) active(__(X, nil)) -> mark(X) active(__(nil, X)) -> mark(X) active(U11(tt)) -> mark(tt) active(U21(tt, V2)) -> mark(U22(isList(V2))) active(U22(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNeList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt)) -> mark(tt) active(U71(tt, P)) -> mark(U72(isPal(P))) active(U72(tt)) -> mark(tt) active(U81(tt)) -> mark(tt) active(isList(V)) -> mark(U11(isNeList(V))) active(isList(nil)) -> mark(tt) active(isList(__(V1, V2))) -> mark(U21(isList(V1), V2)) active(isNeList(V)) -> mark(U31(isQid(V))) active(isNeList(__(V1, V2))) -> mark(U41(isList(V1), V2)) active(isNeList(__(V1, V2))) -> mark(U51(isNeList(V1), V2)) active(isNePal(V)) -> mark(U61(isQid(V))) active(isNePal(__(I, __(P, I)))) -> mark(U71(isQid(I), P)) active(isPal(V)) -> mark(U81(isNePal(V))) active(isPal(nil)) -> mark(tt) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) active(__(X1, X2)) -> __(active(X1), X2) active(__(X1, X2)) -> __(X1, active(X2)) active(U11(X)) -> U11(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X)) -> U72(active(X)) active(U81(X)) -> U81(active(X)) __(mark(X1), X2) -> mark(__(X1, X2)) __(X1, mark(X2)) -> mark(__(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X)) -> mark(U72(X)) U81(mark(X)) -> mark(U81(X)) proper(__(X1, X2)) -> __(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isList(X)) -> isList(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNeList(X)) -> isNeList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(U61(X)) -> U61(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X)) -> U72(proper(X)) proper(isPal(X)) -> isPal(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(isQid(X)) -> isQid(proper(X)) proper(isNePal(X)) -> isNePal(proper(X)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(o) -> ok(o) proper(u) -> ok(u) __(ok(X1), ok(X2)) -> ok(__(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) isList(ok(X)) -> ok(isList(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNeList(ok(X)) -> ok(isNeList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X)) -> ok(U72(X)) isPal(ok(X)) -> ok(isPal(X)) U81(ok(X)) -> ok(U81(X)) isQid(ok(X)) -> ok(isQid(X)) isNePal(ok(X)) -> ok(isNePal(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (12) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (13) BOUNDS(n^1, INF) ---------------------------------------- (14) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) active(__(X, nil)) -> mark(X) active(__(nil, X)) -> mark(X) active(U11(tt)) -> mark(tt) active(U21(tt, V2)) -> mark(U22(isList(V2))) active(U22(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNeList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt)) -> mark(tt) active(U71(tt, P)) -> mark(U72(isPal(P))) active(U72(tt)) -> mark(tt) active(U81(tt)) -> mark(tt) active(isList(V)) -> mark(U11(isNeList(V))) active(isList(nil)) -> mark(tt) active(isList(__(V1, V2))) -> mark(U21(isList(V1), V2)) active(isNeList(V)) -> mark(U31(isQid(V))) active(isNeList(__(V1, V2))) -> mark(U41(isList(V1), V2)) active(isNeList(__(V1, V2))) -> mark(U51(isNeList(V1), V2)) active(isNePal(V)) -> mark(U61(isQid(V))) active(isNePal(__(I, __(P, I)))) -> mark(U71(isQid(I), P)) active(isPal(V)) -> mark(U81(isNePal(V))) active(isPal(nil)) -> mark(tt) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) active(__(X1, X2)) -> __(active(X1), X2) active(__(X1, X2)) -> __(X1, active(X2)) active(U11(X)) -> U11(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X1, X2)) -> U51(active(X1), X2) active(U52(X)) -> U52(active(X)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X)) -> U72(active(X)) active(U81(X)) -> U81(active(X)) __(mark(X1), X2) -> mark(__(X1, X2)) __(X1, mark(X2)) -> mark(__(X1, X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1), X2) -> mark(U51(X1, X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X)) -> mark(U72(X)) U81(mark(X)) -> mark(U81(X)) proper(__(X1, X2)) -> __(proper(X1), proper(X2)) proper(nil) -> ok(nil) proper(U11(X)) -> U11(proper(X)) proper(tt) -> ok(tt) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isList(X)) -> isList(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNeList(X)) -> isNeList(proper(X)) proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(U61(X)) -> U61(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X)) -> U72(proper(X)) proper(isPal(X)) -> isPal(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(isQid(X)) -> isQid(proper(X)) proper(isNePal(X)) -> isNePal(proper(X)) proper(a) -> ok(a) proper(e) -> ok(e) proper(i) -> ok(i) proper(o) -> ok(o) proper(u) -> ok(u) __(ok(X1), ok(X2)) -> ok(__(X1, X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) isList(ok(X)) -> ok(isList(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) isNeList(ok(X)) -> ok(isNeList(X)) U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) U52(ok(X)) -> ok(U52(X)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X)) -> ok(U72(X)) isPal(ok(X)) -> ok(isPal(X)) U81(ok(X)) -> ok(U81(X)) isQid(ok(X)) -> ok(isQid(X)) isNePal(ok(X)) -> ok(isNePal(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL