/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).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, 460 ms] (6) BOUNDS(1, n^1) (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTRS (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) typed CpxTrs (11) OrderProof [LOWER BOUND(ID), 0 ms] (12) typed CpxTrs (13) RewriteLemmaProof [LOWER BOUND(ID), 485 ms] (14) BEST (15) proven lower bound (16) LowerBoundPropagationProof [FINISHED, 0 ms] (17) BOUNDS(n^1, INF) (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 101 ms] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 104 ms] (22) typed CpxTrs (23) RewriteLemmaProof [LOWER BOUND(ID), 99 ms] (24) typed CpxTrs (25) RewriteLemmaProof [LOWER BOUND(ID), 119 ms] (26) typed CpxTrs (27) RewriteLemmaProof [LOWER BOUND(ID), 69 ms] (28) typed CpxTrs (29) RewriteLemmaProof [LOWER BOUND(ID), 110 ms] (30) typed CpxTrs (31) RewriteLemmaProof [LOWER BOUND(ID), 54 ms] (32) typed CpxTrs (33) RewriteLemmaProof [LOWER BOUND(ID), 139 ms] (34) typed CpxTrs (35) RewriteLemmaProof [LOWER BOUND(ID), 169 ms] (36) typed CpxTrs (37) RewriteLemmaProof [LOWER BOUND(ID), 87 ms] (38) typed CpxTrs (39) RewriteLemmaProof [LOWER BOUND(ID), 113 ms] (40) typed CpxTrs (41) RewriteLemmaProof [LOWER BOUND(ID), 75 ms] (42) typed CpxTrs ---------------------------------------- (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) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (8) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 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) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: TRS: 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)) Types: active :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok __ :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok mark :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok nil :: mark:nil:tt:a:e:i:o:u:ok U11 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok tt :: mark:nil:tt:a:e:i:o:u:ok U21 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U22 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U31 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U41 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U42 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNeList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U51 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U52 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U61 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U71 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U72 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isPal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U81 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isQid :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNePal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok a :: mark:nil:tt:a:e:i:o:u:ok e :: mark:nil:tt:a:e:i:o:u:ok i :: mark:nil:tt:a:e:i:o:u:ok o :: mark:nil:tt:a:e:i:o:u:ok u :: mark:nil:tt:a:e:i:o:u:ok proper :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok ok :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok top :: mark:nil:tt:a:e:i:o:u:ok -> top hole_mark:nil:tt:a:e:i:o:u:ok1_0 :: mark:nil:tt:a:e:i:o:u:ok hole_top2_0 :: top gen_mark:nil:tt:a:e:i:o:u:ok3_0 :: Nat -> mark:nil:tt:a:e:i:o:u:ok ---------------------------------------- (11) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: active, __, U22, isList, U42, isNeList, U52, U72, isPal, U11, U21, U31, isQid, U41, U51, U61, U71, U81, isNePal, proper, top They will be analysed ascendingly in the following order: __ < active U22 < active isList < active U42 < active isNeList < active U52 < active U72 < active isPal < active U11 < active U21 < active U31 < active isQid < active U41 < active U51 < active U61 < active U71 < active U81 < active isNePal < active active < top __ < proper U22 < proper isList < proper U42 < proper isNeList < proper U52 < proper U72 < proper isPal < proper U11 < proper U21 < proper U31 < proper isQid < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper isNePal < proper proper < top ---------------------------------------- (12) Obligation: TRS: 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)) Types: active :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok __ :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok mark :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok nil :: mark:nil:tt:a:e:i:o:u:ok U11 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok tt :: mark:nil:tt:a:e:i:o:u:ok U21 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U22 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U31 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U41 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U42 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNeList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U51 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U52 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U61 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U71 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U72 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isPal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U81 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isQid :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNePal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok a :: mark:nil:tt:a:e:i:o:u:ok e :: mark:nil:tt:a:e:i:o:u:ok i :: mark:nil:tt:a:e:i:o:u:ok o :: mark:nil:tt:a:e:i:o:u:ok u :: mark:nil:tt:a:e:i:o:u:ok proper :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok ok :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok top :: mark:nil:tt:a:e:i:o:u:ok -> top hole_mark:nil:tt:a:e:i:o:u:ok1_0 :: mark:nil:tt:a:e:i:o:u:ok hole_top2_0 :: top gen_mark:nil:tt:a:e:i:o:u:ok3_0 :: Nat -> mark:nil:tt:a:e:i:o:u:ok Generator Equations: gen_mark:nil:tt:a:e:i:o:u:ok3_0(0) <=> nil gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:a:e:i:o:u:ok3_0(x)) The following defined symbols remain to be analysed: __, active, U22, isList, U42, isNeList, U52, U72, isPal, U11, U21, U31, isQid, U41, U51, U61, U71, U81, isNePal, proper, top They will be analysed ascendingly in the following order: __ < active U22 < active isList < active U42 < active isNeList < active U52 < active U72 < active isPal < active U11 < active U21 < active U31 < active isQid < active U41 < active U51 < active U61 < active U71 < active U81 < active isNePal < active active < top __ < proper U22 < proper isList < proper U42 < proper isNeList < proper U52 < proper U72 < proper isPal < proper U11 < proper U21 < proper U31 < proper isQid < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper isNePal < proper proper < top ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: __(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n5_0) Induction Base: __(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, 0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) Induction Step: __(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, +(n5_0, 1))), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) ->_R^Omega(1) mark(__(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (14) Complex Obligation (BEST) ---------------------------------------- (15) Obligation: Proved the lower bound n^1 for the following obligation: TRS: 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)) Types: active :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok __ :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok mark :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok nil :: mark:nil:tt:a:e:i:o:u:ok U11 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok tt :: mark:nil:tt:a:e:i:o:u:ok U21 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U22 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U31 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U41 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U42 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNeList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U51 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U52 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U61 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U71 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U72 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isPal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U81 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isQid :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNePal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok a :: mark:nil:tt:a:e:i:o:u:ok e :: mark:nil:tt:a:e:i:o:u:ok i :: mark:nil:tt:a:e:i:o:u:ok o :: mark:nil:tt:a:e:i:o:u:ok u :: mark:nil:tt:a:e:i:o:u:ok proper :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok ok :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok top :: mark:nil:tt:a:e:i:o:u:ok -> top hole_mark:nil:tt:a:e:i:o:u:ok1_0 :: mark:nil:tt:a:e:i:o:u:ok hole_top2_0 :: top gen_mark:nil:tt:a:e:i:o:u:ok3_0 :: Nat -> mark:nil:tt:a:e:i:o:u:ok Generator Equations: gen_mark:nil:tt:a:e:i:o:u:ok3_0(0) <=> nil gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:a:e:i:o:u:ok3_0(x)) The following defined symbols remain to be analysed: __, active, U22, isList, U42, isNeList, U52, U72, isPal, U11, U21, U31, isQid, U41, U51, U61, U71, U81, isNePal, proper, top They will be analysed ascendingly in the following order: __ < active U22 < active isList < active U42 < active isNeList < active U52 < active U72 < active isPal < active U11 < active U21 < active U31 < active isQid < active U41 < active U51 < active U61 < active U71 < active U81 < active isNePal < active active < top __ < proper U22 < proper isList < proper U42 < proper isNeList < proper U52 < proper U72 < proper isPal < proper U11 < proper U21 < proper U31 < proper isQid < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper isNePal < proper proper < top ---------------------------------------- (16) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (17) BOUNDS(n^1, INF) ---------------------------------------- (18) Obligation: TRS: 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)) Types: active :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok __ :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok mark :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok nil :: mark:nil:tt:a:e:i:o:u:ok U11 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok tt :: mark:nil:tt:a:e:i:o:u:ok U21 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U22 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U31 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U41 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U42 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNeList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U51 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U52 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U61 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U71 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U72 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isPal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U81 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isQid :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNePal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok a :: mark:nil:tt:a:e:i:o:u:ok e :: mark:nil:tt:a:e:i:o:u:ok i :: mark:nil:tt:a:e:i:o:u:ok o :: mark:nil:tt:a:e:i:o:u:ok u :: mark:nil:tt:a:e:i:o:u:ok proper :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok ok :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok top :: mark:nil:tt:a:e:i:o:u:ok -> top hole_mark:nil:tt:a:e:i:o:u:ok1_0 :: mark:nil:tt:a:e:i:o:u:ok hole_top2_0 :: top gen_mark:nil:tt:a:e:i:o:u:ok3_0 :: Nat -> mark:nil:tt:a:e:i:o:u:ok Lemmas: __(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n5_0) Generator Equations: gen_mark:nil:tt:a:e:i:o:u:ok3_0(0) <=> nil gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:a:e:i:o:u:ok3_0(x)) The following defined symbols remain to be analysed: U22, active, isList, U42, isNeList, U52, U72, isPal, U11, U21, U31, isQid, U41, U51, U61, U71, U81, isNePal, proper, top They will be analysed ascendingly in the following order: U22 < active isList < active U42 < active isNeList < active U52 < active U72 < active isPal < active U11 < active U21 < active U31 < active isQid < active U41 < active U51 < active U61 < active U71 < active U81 < active isNePal < active active < top U22 < proper isList < proper U42 < proper isNeList < proper U52 < proper U72 < proper isPal < proper U11 < proper U21 < proper U31 < proper isQid < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper isNePal < proper proper < top ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U22(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2081_0))) -> *4_0, rt in Omega(n2081_0) Induction Base: U22(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, 0))) Induction Step: U22(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, +(n2081_0, 1)))) ->_R^Omega(1) mark(U22(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2081_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (20) Obligation: TRS: 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)) Types: active :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok __ :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok mark :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok nil :: mark:nil:tt:a:e:i:o:u:ok U11 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok tt :: mark:nil:tt:a:e:i:o:u:ok U21 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U22 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U31 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U41 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U42 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNeList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U51 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U52 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U61 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U71 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U72 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isPal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U81 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isQid :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNePal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok a :: mark:nil:tt:a:e:i:o:u:ok e :: mark:nil:tt:a:e:i:o:u:ok i :: mark:nil:tt:a:e:i:o:u:ok o :: mark:nil:tt:a:e:i:o:u:ok u :: mark:nil:tt:a:e:i:o:u:ok proper :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok ok :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok top :: mark:nil:tt:a:e:i:o:u:ok -> top hole_mark:nil:tt:a:e:i:o:u:ok1_0 :: mark:nil:tt:a:e:i:o:u:ok hole_top2_0 :: top gen_mark:nil:tt:a:e:i:o:u:ok3_0 :: Nat -> mark:nil:tt:a:e:i:o:u:ok Lemmas: __(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U22(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2081_0))) -> *4_0, rt in Omega(n2081_0) Generator Equations: gen_mark:nil:tt:a:e:i:o:u:ok3_0(0) <=> nil gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:a:e:i:o:u:ok3_0(x)) The following defined symbols remain to be analysed: isList, active, U42, isNeList, U52, U72, isPal, U11, U21, U31, isQid, U41, U51, U61, U71, U81, isNePal, proper, top They will be analysed ascendingly in the following order: isList < active U42 < active isNeList < active U52 < active U72 < active isPal < active U11 < active U21 < active U31 < active isQid < active U41 < active U51 < active U61 < active U71 < active U81 < active isNePal < active active < top isList < proper U42 < proper isNeList < proper U52 < proper U72 < proper isPal < proper U11 < proper U21 < proper U31 < proper isQid < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper isNePal < proper proper < top ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U42(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2819_0))) -> *4_0, rt in Omega(n2819_0) Induction Base: U42(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, 0))) Induction Step: U42(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, +(n2819_0, 1)))) ->_R^Omega(1) mark(U42(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2819_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (22) Obligation: TRS: 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)) Types: active :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok __ :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok mark :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok nil :: mark:nil:tt:a:e:i:o:u:ok U11 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok tt :: mark:nil:tt:a:e:i:o:u:ok U21 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U22 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U31 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U41 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U42 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNeList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U51 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U52 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U61 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U71 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U72 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isPal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U81 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isQid :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNePal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok a :: mark:nil:tt:a:e:i:o:u:ok e :: mark:nil:tt:a:e:i:o:u:ok i :: mark:nil:tt:a:e:i:o:u:ok o :: mark:nil:tt:a:e:i:o:u:ok u :: mark:nil:tt:a:e:i:o:u:ok proper :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok ok :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok top :: mark:nil:tt:a:e:i:o:u:ok -> top hole_mark:nil:tt:a:e:i:o:u:ok1_0 :: mark:nil:tt:a:e:i:o:u:ok hole_top2_0 :: top gen_mark:nil:tt:a:e:i:o:u:ok3_0 :: Nat -> mark:nil:tt:a:e:i:o:u:ok Lemmas: __(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U22(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2081_0))) -> *4_0, rt in Omega(n2081_0) U42(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2819_0))) -> *4_0, rt in Omega(n2819_0) Generator Equations: gen_mark:nil:tt:a:e:i:o:u:ok3_0(0) <=> nil gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:a:e:i:o:u:ok3_0(x)) The following defined symbols remain to be analysed: isNeList, active, U52, U72, isPal, U11, U21, U31, isQid, U41, U51, U61, U71, U81, isNePal, proper, top They will be analysed ascendingly in the following order: isNeList < active U52 < active U72 < active isPal < active U11 < active U21 < active U31 < active isQid < active U41 < active U51 < active U61 < active U71 < active U81 < active isNePal < active active < top isNeList < proper U52 < proper U72 < proper isPal < proper U11 < proper U21 < proper U31 < proper isQid < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper isNePal < proper proper < top ---------------------------------------- (23) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U52(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n3661_0))) -> *4_0, rt in Omega(n3661_0) Induction Base: U52(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, 0))) Induction Step: U52(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, +(n3661_0, 1)))) ->_R^Omega(1) mark(U52(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n3661_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (24) Obligation: TRS: 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)) Types: active :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok __ :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok mark :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok nil :: mark:nil:tt:a:e:i:o:u:ok U11 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok tt :: mark:nil:tt:a:e:i:o:u:ok U21 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U22 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U31 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U41 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U42 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNeList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U51 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U52 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U61 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U71 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U72 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isPal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U81 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isQid :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNePal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok a :: mark:nil:tt:a:e:i:o:u:ok e :: mark:nil:tt:a:e:i:o:u:ok i :: mark:nil:tt:a:e:i:o:u:ok o :: mark:nil:tt:a:e:i:o:u:ok u :: mark:nil:tt:a:e:i:o:u:ok proper :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok ok :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok top :: mark:nil:tt:a:e:i:o:u:ok -> top hole_mark:nil:tt:a:e:i:o:u:ok1_0 :: mark:nil:tt:a:e:i:o:u:ok hole_top2_0 :: top gen_mark:nil:tt:a:e:i:o:u:ok3_0 :: Nat -> mark:nil:tt:a:e:i:o:u:ok Lemmas: __(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U22(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2081_0))) -> *4_0, rt in Omega(n2081_0) U42(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2819_0))) -> *4_0, rt in Omega(n2819_0) U52(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n3661_0))) -> *4_0, rt in Omega(n3661_0) Generator Equations: gen_mark:nil:tt:a:e:i:o:u:ok3_0(0) <=> nil gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:a:e:i:o:u:ok3_0(x)) The following defined symbols remain to be analysed: U72, active, isPal, U11, U21, U31, isQid, U41, U51, U61, U71, U81, isNePal, proper, top They will be analysed ascendingly in the following order: U72 < active isPal < active U11 < active U21 < active U31 < active isQid < active U41 < active U51 < active U61 < active U71 < active U81 < active isNePal < active active < top U72 < proper isPal < proper U11 < proper U21 < proper U31 < proper isQid < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper isNePal < proper proper < top ---------------------------------------- (25) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U72(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n4589_0))) -> *4_0, rt in Omega(n4589_0) Induction Base: U72(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, 0))) Induction Step: U72(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, +(n4589_0, 1)))) ->_R^Omega(1) mark(U72(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n4589_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (26) Obligation: TRS: 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)) Types: active :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok __ :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok mark :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok nil :: mark:nil:tt:a:e:i:o:u:ok U11 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok tt :: mark:nil:tt:a:e:i:o:u:ok U21 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U22 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U31 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U41 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U42 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNeList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U51 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U52 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U61 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U71 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U72 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isPal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U81 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isQid :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNePal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok a :: mark:nil:tt:a:e:i:o:u:ok e :: mark:nil:tt:a:e:i:o:u:ok i :: mark:nil:tt:a:e:i:o:u:ok o :: mark:nil:tt:a:e:i:o:u:ok u :: mark:nil:tt:a:e:i:o:u:ok proper :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok ok :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok top :: mark:nil:tt:a:e:i:o:u:ok -> top hole_mark:nil:tt:a:e:i:o:u:ok1_0 :: mark:nil:tt:a:e:i:o:u:ok hole_top2_0 :: top gen_mark:nil:tt:a:e:i:o:u:ok3_0 :: Nat -> mark:nil:tt:a:e:i:o:u:ok Lemmas: __(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U22(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2081_0))) -> *4_0, rt in Omega(n2081_0) U42(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2819_0))) -> *4_0, rt in Omega(n2819_0) U52(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n3661_0))) -> *4_0, rt in Omega(n3661_0) U72(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n4589_0))) -> *4_0, rt in Omega(n4589_0) Generator Equations: gen_mark:nil:tt:a:e:i:o:u:ok3_0(0) <=> nil gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:a:e:i:o:u:ok3_0(x)) The following defined symbols remain to be analysed: isPal, active, U11, U21, U31, isQid, U41, U51, U61, U71, U81, isNePal, proper, top They will be analysed ascendingly in the following order: isPal < active U11 < active U21 < active U31 < active isQid < active U41 < active U51 < active U61 < active U71 < active U81 < active isNePal < active active < top isPal < proper U11 < proper U21 < proper U31 < proper isQid < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper isNePal < proper proper < top ---------------------------------------- (27) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U11(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5639_0))) -> *4_0, rt in Omega(n5639_0) Induction Base: U11(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, 0))) Induction Step: U11(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, +(n5639_0, 1)))) ->_R^Omega(1) mark(U11(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5639_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (28) Obligation: TRS: 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)) Types: active :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok __ :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok mark :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok nil :: mark:nil:tt:a:e:i:o:u:ok U11 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok tt :: mark:nil:tt:a:e:i:o:u:ok U21 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U22 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U31 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U41 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U42 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNeList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U51 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U52 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U61 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U71 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U72 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isPal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U81 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isQid :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNePal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok a :: mark:nil:tt:a:e:i:o:u:ok e :: mark:nil:tt:a:e:i:o:u:ok i :: mark:nil:tt:a:e:i:o:u:ok o :: mark:nil:tt:a:e:i:o:u:ok u :: mark:nil:tt:a:e:i:o:u:ok proper :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok ok :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok top :: mark:nil:tt:a:e:i:o:u:ok -> top hole_mark:nil:tt:a:e:i:o:u:ok1_0 :: mark:nil:tt:a:e:i:o:u:ok hole_top2_0 :: top gen_mark:nil:tt:a:e:i:o:u:ok3_0 :: Nat -> mark:nil:tt:a:e:i:o:u:ok Lemmas: __(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U22(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2081_0))) -> *4_0, rt in Omega(n2081_0) U42(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2819_0))) -> *4_0, rt in Omega(n2819_0) U52(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n3661_0))) -> *4_0, rt in Omega(n3661_0) U72(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n4589_0))) -> *4_0, rt in Omega(n4589_0) U11(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5639_0))) -> *4_0, rt in Omega(n5639_0) Generator Equations: gen_mark:nil:tt:a:e:i:o:u:ok3_0(0) <=> nil gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:a:e:i:o:u:ok3_0(x)) The following defined symbols remain to be analysed: U21, active, U31, isQid, U41, U51, U61, U71, U81, isNePal, proper, top They will be analysed ascendingly in the following order: U21 < active U31 < active isQid < active U41 < active U51 < active U61 < active U71 < active U81 < active isNePal < active active < top U21 < proper U31 < proper isQid < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper isNePal < proper proper < top ---------------------------------------- (29) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U21(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n6769_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n6769_0) Induction Base: U21(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, 0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) Induction Step: U21(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, +(n6769_0, 1))), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) ->_R^Omega(1) mark(U21(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n6769_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (30) Obligation: TRS: 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)) Types: active :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok __ :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok mark :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok nil :: mark:nil:tt:a:e:i:o:u:ok U11 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok tt :: mark:nil:tt:a:e:i:o:u:ok U21 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U22 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U31 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U41 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U42 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNeList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U51 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U52 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U61 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U71 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U72 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isPal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U81 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isQid :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNePal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok a :: mark:nil:tt:a:e:i:o:u:ok e :: mark:nil:tt:a:e:i:o:u:ok i :: mark:nil:tt:a:e:i:o:u:ok o :: mark:nil:tt:a:e:i:o:u:ok u :: mark:nil:tt:a:e:i:o:u:ok proper :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok ok :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok top :: mark:nil:tt:a:e:i:o:u:ok -> top hole_mark:nil:tt:a:e:i:o:u:ok1_0 :: mark:nil:tt:a:e:i:o:u:ok hole_top2_0 :: top gen_mark:nil:tt:a:e:i:o:u:ok3_0 :: Nat -> mark:nil:tt:a:e:i:o:u:ok Lemmas: __(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U22(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2081_0))) -> *4_0, rt in Omega(n2081_0) U42(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2819_0))) -> *4_0, rt in Omega(n2819_0) U52(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n3661_0))) -> *4_0, rt in Omega(n3661_0) U72(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n4589_0))) -> *4_0, rt in Omega(n4589_0) U11(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5639_0))) -> *4_0, rt in Omega(n5639_0) U21(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n6769_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n6769_0) Generator Equations: gen_mark:nil:tt:a:e:i:o:u:ok3_0(0) <=> nil gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:a:e:i:o:u:ok3_0(x)) The following defined symbols remain to be analysed: U31, active, isQid, U41, U51, U61, U71, U81, isNePal, proper, top They will be analysed ascendingly in the following order: U31 < active isQid < active U41 < active U51 < active U61 < active U71 < active U81 < active isNePal < active active < top U31 < proper isQid < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper isNePal < proper proper < top ---------------------------------------- (31) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U31(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n9992_0))) -> *4_0, rt in Omega(n9992_0) Induction Base: U31(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, 0))) Induction Step: U31(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, +(n9992_0, 1)))) ->_R^Omega(1) mark(U31(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n9992_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (32) Obligation: TRS: 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)) Types: active :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok __ :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok mark :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok nil :: mark:nil:tt:a:e:i:o:u:ok U11 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok tt :: mark:nil:tt:a:e:i:o:u:ok U21 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U22 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U31 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U41 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U42 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNeList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U51 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U52 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U61 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U71 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U72 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isPal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U81 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isQid :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNePal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok a :: mark:nil:tt:a:e:i:o:u:ok e :: mark:nil:tt:a:e:i:o:u:ok i :: mark:nil:tt:a:e:i:o:u:ok o :: mark:nil:tt:a:e:i:o:u:ok u :: mark:nil:tt:a:e:i:o:u:ok proper :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok ok :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok top :: mark:nil:tt:a:e:i:o:u:ok -> top hole_mark:nil:tt:a:e:i:o:u:ok1_0 :: mark:nil:tt:a:e:i:o:u:ok hole_top2_0 :: top gen_mark:nil:tt:a:e:i:o:u:ok3_0 :: Nat -> mark:nil:tt:a:e:i:o:u:ok Lemmas: __(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U22(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2081_0))) -> *4_0, rt in Omega(n2081_0) U42(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2819_0))) -> *4_0, rt in Omega(n2819_0) U52(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n3661_0))) -> *4_0, rt in Omega(n3661_0) U72(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n4589_0))) -> *4_0, rt in Omega(n4589_0) U11(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5639_0))) -> *4_0, rt in Omega(n5639_0) U21(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n6769_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n6769_0) U31(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n9992_0))) -> *4_0, rt in Omega(n9992_0) Generator Equations: gen_mark:nil:tt:a:e:i:o:u:ok3_0(0) <=> nil gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:a:e:i:o:u:ok3_0(x)) The following defined symbols remain to be analysed: isQid, active, U41, U51, U61, U71, U81, isNePal, proper, top They will be analysed ascendingly in the following order: isQid < active U41 < active U51 < active U61 < active U71 < active U81 < active isNePal < active active < top isQid < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper isNePal < proper proper < top ---------------------------------------- (33) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U41(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n11405_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n11405_0) Induction Base: U41(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, 0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) Induction Step: U41(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, +(n11405_0, 1))), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) ->_R^Omega(1) mark(U41(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n11405_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (34) Obligation: TRS: 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)) Types: active :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok __ :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok mark :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok nil :: mark:nil:tt:a:e:i:o:u:ok U11 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok tt :: mark:nil:tt:a:e:i:o:u:ok U21 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U22 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U31 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U41 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U42 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNeList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U51 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U52 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U61 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U71 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U72 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isPal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U81 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isQid :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNePal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok a :: mark:nil:tt:a:e:i:o:u:ok e :: mark:nil:tt:a:e:i:o:u:ok i :: mark:nil:tt:a:e:i:o:u:ok o :: mark:nil:tt:a:e:i:o:u:ok u :: mark:nil:tt:a:e:i:o:u:ok proper :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok ok :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok top :: mark:nil:tt:a:e:i:o:u:ok -> top hole_mark:nil:tt:a:e:i:o:u:ok1_0 :: mark:nil:tt:a:e:i:o:u:ok hole_top2_0 :: top gen_mark:nil:tt:a:e:i:o:u:ok3_0 :: Nat -> mark:nil:tt:a:e:i:o:u:ok Lemmas: __(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U22(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2081_0))) -> *4_0, rt in Omega(n2081_0) U42(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2819_0))) -> *4_0, rt in Omega(n2819_0) U52(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n3661_0))) -> *4_0, rt in Omega(n3661_0) U72(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n4589_0))) -> *4_0, rt in Omega(n4589_0) U11(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5639_0))) -> *4_0, rt in Omega(n5639_0) U21(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n6769_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n6769_0) U31(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n9992_0))) -> *4_0, rt in Omega(n9992_0) U41(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n11405_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n11405_0) Generator Equations: gen_mark:nil:tt:a:e:i:o:u:ok3_0(0) <=> nil gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:a:e:i:o:u:ok3_0(x)) The following defined symbols remain to be analysed: U51, active, U61, U71, U81, isNePal, proper, top They will be analysed ascendingly in the following order: U51 < active U61 < active U71 < active U81 < active isNePal < active active < top U51 < proper U61 < proper U71 < proper U81 < proper isNePal < proper proper < top ---------------------------------------- (35) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U51(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n15142_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n15142_0) Induction Base: U51(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, 0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) Induction Step: U51(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, +(n15142_0, 1))), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) ->_R^Omega(1) mark(U51(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n15142_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (36) Obligation: TRS: 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)) Types: active :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok __ :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok mark :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok nil :: mark:nil:tt:a:e:i:o:u:ok U11 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok tt :: mark:nil:tt:a:e:i:o:u:ok U21 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U22 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U31 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U41 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U42 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNeList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U51 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U52 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U61 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U71 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U72 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isPal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U81 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isQid :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNePal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok a :: mark:nil:tt:a:e:i:o:u:ok e :: mark:nil:tt:a:e:i:o:u:ok i :: mark:nil:tt:a:e:i:o:u:ok o :: mark:nil:tt:a:e:i:o:u:ok u :: mark:nil:tt:a:e:i:o:u:ok proper :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok ok :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok top :: mark:nil:tt:a:e:i:o:u:ok -> top hole_mark:nil:tt:a:e:i:o:u:ok1_0 :: mark:nil:tt:a:e:i:o:u:ok hole_top2_0 :: top gen_mark:nil:tt:a:e:i:o:u:ok3_0 :: Nat -> mark:nil:tt:a:e:i:o:u:ok Lemmas: __(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U22(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2081_0))) -> *4_0, rt in Omega(n2081_0) U42(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2819_0))) -> *4_0, rt in Omega(n2819_0) U52(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n3661_0))) -> *4_0, rt in Omega(n3661_0) U72(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n4589_0))) -> *4_0, rt in Omega(n4589_0) U11(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5639_0))) -> *4_0, rt in Omega(n5639_0) U21(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n6769_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n6769_0) U31(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n9992_0))) -> *4_0, rt in Omega(n9992_0) U41(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n11405_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n11405_0) U51(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n15142_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n15142_0) Generator Equations: gen_mark:nil:tt:a:e:i:o:u:ok3_0(0) <=> nil gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:a:e:i:o:u:ok3_0(x)) The following defined symbols remain to be analysed: U61, active, U71, U81, isNePal, proper, top They will be analysed ascendingly in the following order: U61 < active U71 < active U81 < active isNePal < active active < top U61 < proper U71 < proper U81 < proper isNePal < proper proper < top ---------------------------------------- (37) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U61(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n19185_0))) -> *4_0, rt in Omega(n19185_0) Induction Base: U61(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, 0))) Induction Step: U61(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, +(n19185_0, 1)))) ->_R^Omega(1) mark(U61(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n19185_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (38) Obligation: TRS: 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)) Types: active :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok __ :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok mark :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok nil :: mark:nil:tt:a:e:i:o:u:ok U11 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok tt :: mark:nil:tt:a:e:i:o:u:ok U21 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U22 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U31 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U41 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U42 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNeList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U51 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U52 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U61 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U71 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U72 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isPal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U81 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isQid :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNePal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok a :: mark:nil:tt:a:e:i:o:u:ok e :: mark:nil:tt:a:e:i:o:u:ok i :: mark:nil:tt:a:e:i:o:u:ok o :: mark:nil:tt:a:e:i:o:u:ok u :: mark:nil:tt:a:e:i:o:u:ok proper :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok ok :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok top :: mark:nil:tt:a:e:i:o:u:ok -> top hole_mark:nil:tt:a:e:i:o:u:ok1_0 :: mark:nil:tt:a:e:i:o:u:ok hole_top2_0 :: top gen_mark:nil:tt:a:e:i:o:u:ok3_0 :: Nat -> mark:nil:tt:a:e:i:o:u:ok Lemmas: __(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U22(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2081_0))) -> *4_0, rt in Omega(n2081_0) U42(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2819_0))) -> *4_0, rt in Omega(n2819_0) U52(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n3661_0))) -> *4_0, rt in Omega(n3661_0) U72(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n4589_0))) -> *4_0, rt in Omega(n4589_0) U11(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5639_0))) -> *4_0, rt in Omega(n5639_0) U21(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n6769_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n6769_0) U31(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n9992_0))) -> *4_0, rt in Omega(n9992_0) U41(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n11405_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n11405_0) U51(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n15142_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n15142_0) U61(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n19185_0))) -> *4_0, rt in Omega(n19185_0) Generator Equations: gen_mark:nil:tt:a:e:i:o:u:ok3_0(0) <=> nil gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:a:e:i:o:u:ok3_0(x)) The following defined symbols remain to be analysed: U71, active, U81, isNePal, proper, top They will be analysed ascendingly in the following order: U71 < active U81 < active isNePal < active active < top U71 < proper U81 < proper isNePal < proper proper < top ---------------------------------------- (39) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U71(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n20967_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n20967_0) Induction Base: U71(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, 0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) Induction Step: U71(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, +(n20967_0, 1))), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) ->_R^Omega(1) mark(U71(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n20967_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (40) Obligation: TRS: 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)) Types: active :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok __ :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok mark :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok nil :: mark:nil:tt:a:e:i:o:u:ok U11 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok tt :: mark:nil:tt:a:e:i:o:u:ok U21 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U22 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U31 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U41 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U42 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNeList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U51 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U52 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U61 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U71 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U72 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isPal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U81 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isQid :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNePal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok a :: mark:nil:tt:a:e:i:o:u:ok e :: mark:nil:tt:a:e:i:o:u:ok i :: mark:nil:tt:a:e:i:o:u:ok o :: mark:nil:tt:a:e:i:o:u:ok u :: mark:nil:tt:a:e:i:o:u:ok proper :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok ok :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok top :: mark:nil:tt:a:e:i:o:u:ok -> top hole_mark:nil:tt:a:e:i:o:u:ok1_0 :: mark:nil:tt:a:e:i:o:u:ok hole_top2_0 :: top gen_mark:nil:tt:a:e:i:o:u:ok3_0 :: Nat -> mark:nil:tt:a:e:i:o:u:ok Lemmas: __(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U22(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2081_0))) -> *4_0, rt in Omega(n2081_0) U42(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2819_0))) -> *4_0, rt in Omega(n2819_0) U52(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n3661_0))) -> *4_0, rt in Omega(n3661_0) U72(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n4589_0))) -> *4_0, rt in Omega(n4589_0) U11(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5639_0))) -> *4_0, rt in Omega(n5639_0) U21(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n6769_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n6769_0) U31(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n9992_0))) -> *4_0, rt in Omega(n9992_0) U41(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n11405_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n11405_0) U51(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n15142_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n15142_0) U61(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n19185_0))) -> *4_0, rt in Omega(n19185_0) U71(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n20967_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n20967_0) Generator Equations: gen_mark:nil:tt:a:e:i:o:u:ok3_0(0) <=> nil gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:a:e:i:o:u:ok3_0(x)) The following defined symbols remain to be analysed: U81, active, isNePal, proper, top They will be analysed ascendingly in the following order: U81 < active isNePal < active active < top U81 < proper isNePal < proper proper < top ---------------------------------------- (41) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U81(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n25524_0))) -> *4_0, rt in Omega(n25524_0) Induction Base: U81(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, 0))) Induction Step: U81(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, +(n25524_0, 1)))) ->_R^Omega(1) mark(U81(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n25524_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (42) Obligation: TRS: 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)) Types: active :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok __ :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok mark :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok nil :: mark:nil:tt:a:e:i:o:u:ok U11 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok tt :: mark:nil:tt:a:e:i:o:u:ok U21 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U22 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U31 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U41 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U42 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNeList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U51 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U52 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U61 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U71 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U72 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isPal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok U81 :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isQid :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok isNePal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok a :: mark:nil:tt:a:e:i:o:u:ok e :: mark:nil:tt:a:e:i:o:u:ok i :: mark:nil:tt:a:e:i:o:u:ok o :: mark:nil:tt:a:e:i:o:u:ok u :: mark:nil:tt:a:e:i:o:u:ok proper :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok ok :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok top :: mark:nil:tt:a:e:i:o:u:ok -> top hole_mark:nil:tt:a:e:i:o:u:ok1_0 :: mark:nil:tt:a:e:i:o:u:ok hole_top2_0 :: top gen_mark:nil:tt:a:e:i:o:u:ok3_0 :: Nat -> mark:nil:tt:a:e:i:o:u:ok Lemmas: __(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n5_0) U22(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2081_0))) -> *4_0, rt in Omega(n2081_0) U42(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n2819_0))) -> *4_0, rt in Omega(n2819_0) U52(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n3661_0))) -> *4_0, rt in Omega(n3661_0) U72(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n4589_0))) -> *4_0, rt in Omega(n4589_0) U11(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5639_0))) -> *4_0, rt in Omega(n5639_0) U21(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n6769_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n6769_0) U31(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n9992_0))) -> *4_0, rt in Omega(n9992_0) U41(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n11405_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n11405_0) U51(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n15142_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n15142_0) U61(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n19185_0))) -> *4_0, rt in Omega(n19185_0) U71(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n20967_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n20967_0) U81(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n25524_0))) -> *4_0, rt in Omega(n25524_0) Generator Equations: gen_mark:nil:tt:a:e:i:o:u:ok3_0(0) <=> nil gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:a:e:i:o:u:ok3_0(x)) The following defined symbols remain to be analysed: isNePal, active, proper, top They will be analysed ascendingly in the following order: isNePal < active active < top isNePal < proper proper < top