/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 16 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) CpxTrsMatchBoundsTAProof [FINISHED, 546 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), 429 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), 136 ms] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 81 ms] (22) typed CpxTrs (23) RewriteLemmaProof [LOWER BOUND(ID), 126 ms] (24) typed CpxTrs (25) RewriteLemmaProof [LOWER BOUND(ID), 111 ms] (26) typed CpxTrs (27) RewriteLemmaProof [LOWER BOUND(ID), 46 ms] (28) typed CpxTrs (29) RewriteLemmaProof [LOWER BOUND(ID), 84 ms] (30) typed CpxTrs (31) RewriteLemmaProof [LOWER BOUND(ID), 157 ms] (32) typed CpxTrs (33) RewriteLemmaProof [LOWER BOUND(ID), 177 ms] (34) typed CpxTrs (35) RewriteLemmaProof [LOWER BOUND(ID), 189 ms] (36) typed CpxTrs (37) RewriteLemmaProof [LOWER BOUND(ID), 171 ms] (38) typed CpxTrs (39) RewriteLemmaProof [LOWER BOUND(ID), 129 ms] (40) typed CpxTrs (41) RewriteLemmaProof [LOWER BOUND(ID), 129 ms] (42) typed CpxTrs (43) RewriteLemmaProof [LOWER BOUND(ID), 152 ms] (44) typed CpxTrs (45) RewriteLemmaProof [LOWER BOUND(ID), 62 ms] (46) typed CpxTrs (47) RewriteLemmaProof [LOWER BOUND(ID), 171 ms] (48) 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(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2)) active(U32(tt, V2)) -> mark(U33(isNat(V2))) active(U33(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(plus(N, 0)) -> mark(U41(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(x(N, 0)) -> mark(U61(and(isNat(N), isNatKind(N)))) active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(U61(X)) -> U61(proper(X)) proper(0) -> ok(0) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(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(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2)) active(U32(tt, V2)) -> mark(U33(isNat(V2))) active(U33(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(plus(N, 0)) -> mark(U41(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(x(N, 0)) -> mark(U61(and(isNat(N), isNatKind(N)))) active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(U61(X)) -> U61(proper(X)) proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(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: U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(tt) -> ok(tt) proper(0) -> ok(0) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(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: U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(tt) -> ok(tt) proper(0) -> ok(0) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(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 tt0() -> 0 ok0(0) -> 0 00() -> 0 active0(0) -> 0 U110(0, 0, 0) -> 1 U120(0, 0) -> 2 U130(0) -> 3 U210(0, 0) -> 4 U220(0) -> 5 U310(0, 0, 0) -> 6 U320(0, 0) -> 7 U330(0) -> 8 U410(0, 0) -> 9 U510(0, 0, 0) -> 10 s0(0) -> 11 plus0(0, 0) -> 12 U610(0) -> 13 U710(0, 0, 0) -> 14 x0(0, 0) -> 15 and0(0, 0) -> 16 proper0(0) -> 17 isNat0(0) -> 18 isNatKind0(0) -> 19 top0(0) -> 20 U111(0, 0, 0) -> 21 mark1(21) -> 1 U121(0, 0) -> 22 mark1(22) -> 2 U131(0) -> 23 mark1(23) -> 3 U211(0, 0) -> 24 mark1(24) -> 4 U221(0) -> 25 mark1(25) -> 5 U311(0, 0, 0) -> 26 mark1(26) -> 6 U321(0, 0) -> 27 mark1(27) -> 7 U331(0) -> 28 mark1(28) -> 8 U411(0, 0) -> 29 mark1(29) -> 9 U511(0, 0, 0) -> 30 mark1(30) -> 10 s1(0) -> 31 mark1(31) -> 11 plus1(0, 0) -> 32 mark1(32) -> 12 U611(0) -> 33 mark1(33) -> 13 U711(0, 0, 0) -> 34 mark1(34) -> 14 x1(0, 0) -> 35 mark1(35) -> 15 and1(0, 0) -> 36 mark1(36) -> 16 tt1() -> 37 ok1(37) -> 17 01() -> 38 ok1(38) -> 17 U111(0, 0, 0) -> 39 ok1(39) -> 1 U121(0, 0) -> 40 ok1(40) -> 2 isNat1(0) -> 41 ok1(41) -> 18 U131(0) -> 42 ok1(42) -> 3 U211(0, 0) -> 43 ok1(43) -> 4 U221(0) -> 44 ok1(44) -> 5 U311(0, 0, 0) -> 45 ok1(45) -> 6 U321(0, 0) -> 46 ok1(46) -> 7 U331(0) -> 47 ok1(47) -> 8 U411(0, 0) -> 48 ok1(48) -> 9 U511(0, 0, 0) -> 49 ok1(49) -> 10 s1(0) -> 50 ok1(50) -> 11 plus1(0, 0) -> 51 ok1(51) -> 12 U611(0) -> 52 ok1(52) -> 13 U711(0, 0, 0) -> 53 ok1(53) -> 14 x1(0, 0) -> 54 ok1(54) -> 15 and1(0, 0) -> 55 ok1(55) -> 16 isNatKind1(0) -> 56 ok1(56) -> 19 proper1(0) -> 57 top1(57) -> 20 active1(0) -> 58 top1(58) -> 20 mark1(21) -> 21 mark1(21) -> 39 mark1(22) -> 22 mark1(22) -> 40 mark1(23) -> 23 mark1(23) -> 42 mark1(24) -> 24 mark1(24) -> 43 mark1(25) -> 25 mark1(25) -> 44 mark1(26) -> 26 mark1(26) -> 45 mark1(27) -> 27 mark1(27) -> 46 mark1(28) -> 28 mark1(28) -> 47 mark1(29) -> 29 mark1(29) -> 48 mark1(30) -> 30 mark1(30) -> 49 mark1(31) -> 31 mark1(31) -> 50 mark1(32) -> 32 mark1(32) -> 51 mark1(33) -> 33 mark1(33) -> 52 mark1(34) -> 34 mark1(34) -> 53 mark1(35) -> 35 mark1(35) -> 54 mark1(36) -> 36 mark1(36) -> 55 ok1(37) -> 57 ok1(38) -> 57 ok1(39) -> 21 ok1(39) -> 39 ok1(40) -> 22 ok1(40) -> 40 ok1(41) -> 41 ok1(42) -> 23 ok1(42) -> 42 ok1(43) -> 24 ok1(43) -> 43 ok1(44) -> 25 ok1(44) -> 44 ok1(45) -> 26 ok1(45) -> 45 ok1(46) -> 27 ok1(46) -> 46 ok1(47) -> 28 ok1(47) -> 47 ok1(48) -> 29 ok1(48) -> 48 ok1(49) -> 30 ok1(49) -> 49 ok1(50) -> 31 ok1(50) -> 50 ok1(51) -> 32 ok1(51) -> 51 ok1(52) -> 33 ok1(52) -> 52 ok1(53) -> 34 ok1(53) -> 53 ok1(54) -> 35 ok1(54) -> 54 ok1(55) -> 36 ok1(55) -> 55 ok1(56) -> 56 active2(37) -> 59 top2(59) -> 20 active2(38) -> 59 ---------------------------------------- (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(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2)) active(U32(tt, V2)) -> mark(U33(isNat(V2))) active(U33(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0') active(U71(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N)))) active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(U61(X)) -> U61(proper(X)) proper(0') -> ok(0') proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(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(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2)) active(U32(tt, V2)) -> mark(U33(isNat(V2))) active(U33(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0') active(U71(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N)))) active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(U61(X)) -> U61(proper(X)) proper(0') -> ok(0') proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok ---------------------------------------- (11) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: active, U12, isNat, U13, U22, U32, U33, s, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top They will be analysed ascendingly in the following order: U12 < active isNat < active U13 < active U22 < active U32 < active U33 < active s < active plus < active x < active U11 < active and < active isNatKind < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active active < top U12 < proper isNat < proper U13 < proper U22 < proper U32 < proper U33 < proper s < proper plus < proper x < proper U11 < proper and < proper isNatKind < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper proper < top ---------------------------------------- (12) Obligation: TRS: Rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2)) active(U32(tt, V2)) -> mark(U33(isNat(V2))) active(U33(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0') active(U71(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N)))) active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(U61(X)) -> U61(proper(X)) proper(0') -> ok(0') proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: U12, active, isNat, U13, U22, U32, U33, s, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top They will be analysed ascendingly in the following order: U12 < active isNat < active U13 < active U22 < active U32 < active U33 < active s < active plus < active x < active U11 < active and < active isNatKind < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active active < top U12 < proper isNat < proper U13 < proper U22 < proper U32 < proper U33 < proper s < proper plus < proper x < proper U11 < proper and < proper isNatKind < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper proper < top ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) Induction Base: U12(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b)) Induction Step: U12(gen_tt:mark:0':ok3_0(+(1, +(n5_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':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(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2)) active(U32(tt, V2)) -> mark(U33(isNat(V2))) active(U33(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0') active(U71(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N)))) active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(U61(X)) -> U61(proper(X)) proper(0') -> ok(0') proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: U12, active, isNat, U13, U22, U32, U33, s, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top They will be analysed ascendingly in the following order: U12 < active isNat < active U13 < active U22 < active U32 < active U33 < active s < active plus < active x < active U11 < active and < active isNatKind < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active active < top U12 < proper isNat < proper U13 < proper U22 < proper U32 < proper U33 < proper s < proper plus < proper x < proper U11 < proper and < proper isNatKind < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper proper < top ---------------------------------------- (16) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (17) BOUNDS(n^1, INF) ---------------------------------------- (18) Obligation: TRS: Rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2)) active(U32(tt, V2)) -> mark(U33(isNat(V2))) active(U33(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0') active(U71(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N)))) active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(U61(X)) -> U61(proper(X)) proper(0') -> ok(0') proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: isNat, active, U13, U22, U32, U33, s, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top They will be analysed ascendingly in the following order: isNat < active U13 < active U22 < active U32 < active U33 < active s < active plus < active x < active U11 < active and < active isNatKind < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active active < top isNat < proper U13 < proper U22 < proper U32 < proper U33 < proper s < proper plus < proper x < proper U11 < proper and < proper isNatKind < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper proper < top ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0) Induction Base: U13(gen_tt:mark:0':ok3_0(+(1, 0))) Induction Step: U13(gen_tt:mark:0':ok3_0(+(1, +(n1863_0, 1)))) ->_R^Omega(1) mark(U13(gen_tt:mark:0':ok3_0(+(1, n1863_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(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2)) active(U32(tt, V2)) -> mark(U33(isNat(V2))) active(U33(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0') active(U71(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N)))) active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(U61(X)) -> U61(proper(X)) proper(0') -> ok(0') proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0) Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: U22, active, U32, U33, s, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top They will be analysed ascendingly in the following order: U22 < active U32 < active U33 < active s < active plus < active x < active U11 < active and < active isNatKind < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active active < top U22 < proper U32 < proper U33 < proper s < proper plus < proper x < proper U11 < proper and < proper isNatKind < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper proper < top ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0) Induction Base: U22(gen_tt:mark:0':ok3_0(+(1, 0))) Induction Step: U22(gen_tt:mark:0':ok3_0(+(1, +(n2583_0, 1)))) ->_R^Omega(1) mark(U22(gen_tt:mark:0':ok3_0(+(1, n2583_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(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2)) active(U32(tt, V2)) -> mark(U33(isNat(V2))) active(U33(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0') active(U71(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N)))) active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(U61(X)) -> U61(proper(X)) proper(0') -> ok(0') proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0) U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0) Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: U32, active, U33, s, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top They will be analysed ascendingly in the following order: U32 < active U33 < active s < active plus < active x < active U11 < active and < active isNatKind < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active active < top U32 < proper U33 < proper s < proper plus < proper x < proper U11 < proper and < proper isNatKind < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper proper < top ---------------------------------------- (23) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0) Induction Base: U32(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b)) Induction Step: U32(gen_tt:mark:0':ok3_0(+(1, +(n3404_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':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). ---------------------------------------- (24) Obligation: TRS: Rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2)) active(U32(tt, V2)) -> mark(U33(isNat(V2))) active(U33(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0') active(U71(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N)))) active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(U61(X)) -> U61(proper(X)) proper(0') -> ok(0') proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0) U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0) U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0) Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: U33, active, s, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top They will be analysed ascendingly in the following order: U33 < active s < active plus < active x < active U11 < active and < active isNatKind < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active active < top U33 < proper s < proper plus < proper x < proper U11 < proper and < proper isNatKind < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper proper < top ---------------------------------------- (25) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0) Induction Base: U33(gen_tt:mark:0':ok3_0(+(1, 0))) Induction Step: U33(gen_tt:mark:0':ok3_0(+(1, +(n5975_0, 1)))) ->_R^Omega(1) mark(U33(gen_tt:mark:0':ok3_0(+(1, n5975_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(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2)) active(U32(tt, V2)) -> mark(U33(isNat(V2))) active(U33(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0') active(U71(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N)))) active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(U61(X)) -> U61(proper(X)) proper(0') -> ok(0') proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0) U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0) U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0) U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0) Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: s, active, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top They will be analysed ascendingly in the following order: s < active plus < active x < active U11 < active and < active isNatKind < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active active < top s < proper plus < proper x < proper U11 < proper and < proper isNatKind < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper proper < top ---------------------------------------- (27) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0) Induction Base: s(gen_tt:mark:0':ok3_0(+(1, 0))) Induction Step: s(gen_tt:mark:0':ok3_0(+(1, +(n7047_0, 1)))) ->_R^Omega(1) mark(s(gen_tt:mark:0':ok3_0(+(1, n7047_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(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2)) active(U32(tt, V2)) -> mark(U33(isNat(V2))) active(U33(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0') active(U71(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N)))) active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(U61(X)) -> U61(proper(X)) proper(0') -> ok(0') proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0) U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0) U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0) U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0) s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0) Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: plus, active, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top They will be analysed ascendingly in the following order: plus < active x < active U11 < active and < active isNatKind < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active active < top plus < proper x < proper U11 < proper and < proper isNatKind < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper proper < top ---------------------------------------- (29) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n8220_0) Induction Base: plus(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b)) Induction Step: plus(gen_tt:mark:0':ok3_0(+(1, +(n8220_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':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(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2)) active(U32(tt, V2)) -> mark(U33(isNat(V2))) active(U33(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0') active(U71(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N)))) active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(U61(X)) -> U61(proper(X)) proper(0') -> ok(0') proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0) U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0) U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0) U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0) s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0) plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n8220_0) Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: x, active, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top They will be analysed ascendingly in the following order: x < active U11 < active and < active isNatKind < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active active < top x < proper U11 < proper and < proper isNatKind < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper proper < top ---------------------------------------- (31) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: x(gen_tt:mark:0':ok3_0(+(1, n11712_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n11712_0) Induction Base: x(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b)) Induction Step: x(gen_tt:mark:0':ok3_0(+(1, +(n11712_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(x(gen_tt:mark:0':ok3_0(+(1, n11712_0)), gen_tt:mark:0':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). ---------------------------------------- (32) Obligation: TRS: Rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2)) active(U32(tt, V2)) -> mark(U33(isNat(V2))) active(U33(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0') active(U71(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N)))) active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(U61(X)) -> U61(proper(X)) proper(0') -> ok(0') proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0) U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0) U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0) U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0) s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0) plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n8220_0) x(gen_tt:mark:0':ok3_0(+(1, n11712_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n11712_0) Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: U11, active, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top They will be analysed ascendingly in the following order: U11 < active and < active isNatKind < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active active < top U11 < proper and < proper isNatKind < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper proper < top ---------------------------------------- (33) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U11(gen_tt:mark:0':ok3_0(+(1, n15510_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n15510_0) Induction Base: U11(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) Induction Step: U11(gen_tt:mark:0':ok3_0(+(1, +(n15510_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1) mark(U11(gen_tt:mark:0':ok3_0(+(1, n15510_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_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(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2)) active(U32(tt, V2)) -> mark(U33(isNat(V2))) active(U33(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0') active(U71(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N)))) active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(U61(X)) -> U61(proper(X)) proper(0') -> ok(0') proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0) U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0) U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0) U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0) s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0) plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n8220_0) x(gen_tt:mark:0':ok3_0(+(1, n11712_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n11712_0) U11(gen_tt:mark:0':ok3_0(+(1, n15510_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n15510_0) Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: and, active, isNatKind, U21, U31, U41, U51, U61, U71, proper, top They will be analysed ascendingly in the following order: and < active isNatKind < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active active < top and < proper isNatKind < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper proper < top ---------------------------------------- (35) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: and(gen_tt:mark:0':ok3_0(+(1, n21746_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n21746_0) Induction Base: and(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b)) Induction Step: and(gen_tt:mark:0':ok3_0(+(1, +(n21746_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(and(gen_tt:mark:0':ok3_0(+(1, n21746_0)), gen_tt:mark:0':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(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2)) active(U32(tt, V2)) -> mark(U33(isNat(V2))) active(U33(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0') active(U71(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N)))) active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(U61(X)) -> U61(proper(X)) proper(0') -> ok(0') proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0) U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0) U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0) U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0) s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0) plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n8220_0) x(gen_tt:mark:0':ok3_0(+(1, n11712_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n11712_0) U11(gen_tt:mark:0':ok3_0(+(1, n15510_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n15510_0) and(gen_tt:mark:0':ok3_0(+(1, n21746_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n21746_0) Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: isNatKind, active, U21, U31, U41, U51, U61, U71, proper, top They will be analysed ascendingly in the following order: isNatKind < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active active < top isNatKind < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper proper < top ---------------------------------------- (37) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U21(gen_tt:mark:0':ok3_0(+(1, n26103_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n26103_0) Induction Base: U21(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b)) Induction Step: U21(gen_tt:mark:0':ok3_0(+(1, +(n26103_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(U21(gen_tt:mark:0':ok3_0(+(1, n26103_0)), gen_tt:mark:0':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). ---------------------------------------- (38) Obligation: TRS: Rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2)) active(U32(tt, V2)) -> mark(U33(isNat(V2))) active(U33(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0') active(U71(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N)))) active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(U61(X)) -> U61(proper(X)) proper(0') -> ok(0') proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0) U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0) U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0) U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0) s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0) plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n8220_0) x(gen_tt:mark:0':ok3_0(+(1, n11712_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n11712_0) U11(gen_tt:mark:0':ok3_0(+(1, n15510_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n15510_0) and(gen_tt:mark:0':ok3_0(+(1, n21746_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n21746_0) U21(gen_tt:mark:0':ok3_0(+(1, n26103_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n26103_0) Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: U31, active, U41, U51, U61, U71, proper, top They will be analysed ascendingly in the following order: U31 < active U41 < active U51 < active U61 < active U71 < active active < top U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper proper < top ---------------------------------------- (39) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U31(gen_tt:mark:0':ok3_0(+(1, n30718_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n30718_0) Induction Base: U31(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) Induction Step: U31(gen_tt:mark:0':ok3_0(+(1, +(n30718_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1) mark(U31(gen_tt:mark:0':ok3_0(+(1, n30718_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_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(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2)) active(U32(tt, V2)) -> mark(U33(isNat(V2))) active(U33(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0') active(U71(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N)))) active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(U61(X)) -> U61(proper(X)) proper(0') -> ok(0') proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0) U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0) U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0) U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0) s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0) plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n8220_0) x(gen_tt:mark:0':ok3_0(+(1, n11712_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n11712_0) U11(gen_tt:mark:0':ok3_0(+(1, n15510_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n15510_0) and(gen_tt:mark:0':ok3_0(+(1, n21746_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n21746_0) U21(gen_tt:mark:0':ok3_0(+(1, n26103_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n26103_0) U31(gen_tt:mark:0':ok3_0(+(1, n30718_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n30718_0) Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: U41, active, U51, U61, U71, proper, top They will be analysed ascendingly in the following order: U41 < active U51 < active U61 < active U71 < active active < top U41 < proper U51 < proper U61 < proper U71 < proper proper < top ---------------------------------------- (41) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U41(gen_tt:mark:0':ok3_0(+(1, n38487_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n38487_0) Induction Base: U41(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b)) Induction Step: U41(gen_tt:mark:0':ok3_0(+(1, +(n38487_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(U41(gen_tt:mark:0':ok3_0(+(1, n38487_0)), gen_tt:mark:0':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). ---------------------------------------- (42) Obligation: TRS: Rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2)) active(U32(tt, V2)) -> mark(U33(isNat(V2))) active(U33(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0') active(U71(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N)))) active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(U61(X)) -> U61(proper(X)) proper(0') -> ok(0') proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0) U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0) U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0) U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0) s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0) plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n8220_0) x(gen_tt:mark:0':ok3_0(+(1, n11712_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n11712_0) U11(gen_tt:mark:0':ok3_0(+(1, n15510_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n15510_0) and(gen_tt:mark:0':ok3_0(+(1, n21746_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n21746_0) U21(gen_tt:mark:0':ok3_0(+(1, n26103_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n26103_0) U31(gen_tt:mark:0':ok3_0(+(1, n30718_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n30718_0) U41(gen_tt:mark:0':ok3_0(+(1, n38487_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n38487_0) Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: U51, active, U61, U71, proper, top They will be analysed ascendingly in the following order: U51 < active U61 < active U71 < active active < top U51 < proper U61 < proper U71 < proper proper < top ---------------------------------------- (43) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U51(gen_tt:mark:0':ok3_0(+(1, n43812_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n43812_0) Induction Base: U51(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) Induction Step: U51(gen_tt:mark:0':ok3_0(+(1, +(n43812_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1) mark(U51(gen_tt:mark:0':ok3_0(+(1, n43812_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (44) Obligation: TRS: Rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2)) active(U32(tt, V2)) -> mark(U33(isNat(V2))) active(U33(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0') active(U71(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N)))) active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(U61(X)) -> U61(proper(X)) proper(0') -> ok(0') proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0) U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0) U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0) U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0) s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0) plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n8220_0) x(gen_tt:mark:0':ok3_0(+(1, n11712_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n11712_0) U11(gen_tt:mark:0':ok3_0(+(1, n15510_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n15510_0) and(gen_tt:mark:0':ok3_0(+(1, n21746_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n21746_0) U21(gen_tt:mark:0':ok3_0(+(1, n26103_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n26103_0) U31(gen_tt:mark:0':ok3_0(+(1, n30718_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n30718_0) U41(gen_tt:mark:0':ok3_0(+(1, n38487_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n38487_0) U51(gen_tt:mark:0':ok3_0(+(1, n43812_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n43812_0) Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: U61, active, U71, proper, top They will be analysed ascendingly in the following order: U61 < active U71 < active active < top U61 < proper U71 < proper proper < top ---------------------------------------- (45) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U61(gen_tt:mark:0':ok3_0(+(1, n52652_0))) -> *4_0, rt in Omega(n52652_0) Induction Base: U61(gen_tt:mark:0':ok3_0(+(1, 0))) Induction Step: U61(gen_tt:mark:0':ok3_0(+(1, +(n52652_0, 1)))) ->_R^Omega(1) mark(U61(gen_tt:mark:0':ok3_0(+(1, n52652_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). ---------------------------------------- (46) Obligation: TRS: Rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2)) active(U32(tt, V2)) -> mark(U33(isNat(V2))) active(U33(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0') active(U71(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N)))) active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(U61(X)) -> U61(proper(X)) proper(0') -> ok(0') proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0) U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0) U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0) U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0) s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0) plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n8220_0) x(gen_tt:mark:0':ok3_0(+(1, n11712_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n11712_0) U11(gen_tt:mark:0':ok3_0(+(1, n15510_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n15510_0) and(gen_tt:mark:0':ok3_0(+(1, n21746_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n21746_0) U21(gen_tt:mark:0':ok3_0(+(1, n26103_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n26103_0) U31(gen_tt:mark:0':ok3_0(+(1, n30718_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n30718_0) U41(gen_tt:mark:0':ok3_0(+(1, n38487_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n38487_0) U51(gen_tt:mark:0':ok3_0(+(1, n43812_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n43812_0) U61(gen_tt:mark:0':ok3_0(+(1, n52652_0))) -> *4_0, rt in Omega(n52652_0) Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: U71, active, proper, top They will be analysed ascendingly in the following order: U71 < active active < top U71 < proper proper < top ---------------------------------------- (47) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U71(gen_tt:mark:0':ok3_0(+(1, n55273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n55273_0) Induction Base: U71(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) Induction Step: U71(gen_tt:mark:0':ok3_0(+(1, +(n55273_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1) mark(U71(gen_tt:mark:0':ok3_0(+(1, n55273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (48) Obligation: TRS: Rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2)) active(U32(tt, V2)) -> mark(U33(isNat(V2))) active(U33(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0') active(U71(tt, M, N)) -> mark(plus(x(N, M), N)) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N)))) active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2)) -> U32(active(X1), X2) active(U33(X)) -> U33(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(U61(X)) -> U61(active(X)) active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2) -> mark(U32(X1, X2)) U33(mark(X)) -> mark(U33(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) U61(mark(X)) -> mark(U61(X)) U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) proper(U33(X)) -> U33(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(U61(X)) -> U61(proper(X)) proper(0') -> ok(0') proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) U33(ok(X)) -> ok(U33(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) U61(ok(X)) -> ok(U61(X)) U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0) U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0) U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0) U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0) s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0) plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n8220_0) x(gen_tt:mark:0':ok3_0(+(1, n11712_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n11712_0) U11(gen_tt:mark:0':ok3_0(+(1, n15510_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n15510_0) and(gen_tt:mark:0':ok3_0(+(1, n21746_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n21746_0) U21(gen_tt:mark:0':ok3_0(+(1, n26103_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n26103_0) U31(gen_tt:mark:0':ok3_0(+(1, n30718_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n30718_0) U41(gen_tt:mark:0':ok3_0(+(1, n38487_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n38487_0) U51(gen_tt:mark:0':ok3_0(+(1, n43812_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n43812_0) U61(gen_tt:mark:0':ok3_0(+(1, n52652_0))) -> *4_0, rt in Omega(n52652_0) U71(gen_tt:mark:0':ok3_0(+(1, n55273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n55273_0) Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: active, proper, top They will be analysed ascendingly in the following order: active < top proper < top