/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^3)) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3). (0) CpxTRS (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] (2) CpxTRS (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 70 ms] (4) CpxTRS (5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (6) CdtProblem (7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CdtProblem (9) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CdtProblem (11) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CdtProblem (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 786 ms] (14) CdtProblem (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 107 ms] (16) CdtProblem (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 65 ms] (18) CdtProblem (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 53 ms] (20) CdtProblem (21) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms] (22) BOUNDS(1, 1) (23) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CpxTRS (25) TypeInferenceProof [BOTH BOUNDS(ID, ID), 9 ms] (26) typed CpxTrs (27) OrderProof [LOWER BOUND(ID), 0 ms] (28) typed CpxTrs (29) RewriteLemmaProof [LOWER BOUND(ID), 509 ms] (30) BEST (31) proven lower bound (32) LowerBoundPropagationProof [FINISHED, 0 ms] (33) BOUNDS(n^1, INF) (34) typed CpxTrs (35) RewriteLemmaProof [LOWER BOUND(ID), 237 ms] (36) typed CpxTrs (37) RewriteLemmaProof [LOWER BOUND(ID), 195 ms] (38) typed CpxTrs (39) RewriteLemmaProof [LOWER BOUND(ID), 101 ms] (40) typed CpxTrs (41) RewriteLemmaProof [LOWER BOUND(ID), 176 ms] (42) typed CpxTrs (43) RewriteLemmaProof [LOWER BOUND(ID), 206 ms] (44) typed CpxTrs (45) RewriteLemmaProof [LOWER BOUND(ID), 204 ms] (46) typed CpxTrs (47) RewriteLemmaProof [LOWER BOUND(ID), 205 ms] (48) typed CpxTrs (49) RewriteLemmaProof [LOWER BOUND(ID), 208 ms] (50) typed CpxTrs (51) RewriteLemmaProof [LOWER BOUND(ID), 127 ms] (52) typed CpxTrs (53) RewriteLemmaProof [LOWER BOUND(ID), 171 ms] (54) typed CpxTrs (55) RewriteLemmaProof [LOWER BOUND(ID), 90 ms] (56) typed CpxTrs (57) RewriteLemmaProof [LOWER BOUND(ID), 219 ms] (58) typed CpxTrs (59) RewriteLemmaProof [LOWER BOUND(ID), 230 ms] (60) typed CpxTrs (61) RewriteLemmaProof [LOWER BOUND(ID), 217 ms] (62) typed CpxTrs (63) RewriteLemmaProof [LOWER BOUND(ID), 167 ms] (64) typed CpxTrs (65) RewriteLemmaProof [LOWER BOUND(ID), 88 ms] (66) typed CpxTrs (67) RewriteLemmaProof [LOWER BOUND(ID), 108 ms] (68) typed CpxTrs (69) RewriteLemmaProof [LOWER BOUND(ID), 71 ms] (70) typed CpxTrs (71) RewriteLemmaProof [LOWER BOUND(ID), 172 ms] (72) typed CpxTrs (73) RewriteLemmaProof [LOWER BOUND(ID), 254 ms] (74) typed CpxTrs (75) RewriteLemmaProof [LOWER BOUND(ID), 181 ms] (76) typed CpxTrs (77) RewriteLemmaProof [LOWER BOUND(ID), 216 ms] (78) typed CpxTrs (79) RewriteLemmaProof [LOWER BOUND(ID), 68 ms] (80) typed CpxTrs (81) RewriteLemmaProof [LOWER BOUND(ID), 49 ms] (82) typed CpxTrs (83) RewriteLemmaProof [LOWER BOUND(ID), 201 ms] (84) typed CpxTrs (85) RewriteLemmaProof [LOWER BOUND(ID), 151 ms] (86) typed CpxTrs (87) RewriteLemmaProof [LOWER BOUND(ID), 163 ms] (88) typed CpxTrs (89) RewriteLemmaProof [LOWER BOUND(ID), 121 ms] (90) typed CpxTrs (91) RewriteLemmaProof [LOWER BOUND(ID), 29 ms] (92) typed CpxTrs (93) RewriteLemmaProof [LOWER BOUND(ID), 100 ms] (94) typed CpxTrs (95) RewriteLemmaProof [LOWER BOUND(ID), 193 ms] (96) typed CpxTrs (97) RewriteLemmaProof [LOWER BOUND(ID), 199 ms] (98) typed CpxTrs (99) RewriteLemmaProof [LOWER BOUND(ID), 136 ms] (100) typed CpxTrs (101) RewriteLemmaProof [LOWER BOUND(ID), 244 ms] (102) typed CpxTrs ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3). The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0) -> ok(0) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^3). The TRS R consists of the following rules: U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(tt) -> ok(tt) proper(0) -> ok(0) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) RcToIrcProof (BOTH BOUNDS(ID, ID)) Converted rc-obligation to irc-obligation. As the TRS is a non-duplicating overlay system, we have rc = irc. ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3). The TRS R consists of the following rules: U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(tt) -> ok(tt) proper(0) -> ok(0) U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (5) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: U101(mark(z0), z1, z2) -> mark(U101(z0, z1, z2)) U101(ok(z0), ok(z1), ok(z2)) -> ok(U101(z0, z1, z2)) U102(mark(z0), z1, z2) -> mark(U102(z0, z1, z2)) U102(ok(z0), ok(z1), ok(z2)) -> ok(U102(z0, z1, z2)) U103(mark(z0), z1, z2) -> mark(U103(z0, z1, z2)) U103(ok(z0), ok(z1), ok(z2)) -> ok(U103(z0, z1, z2)) U104(mark(z0), z1, z2) -> mark(U104(z0, z1, z2)) U104(ok(z0), ok(z1), ok(z2)) -> ok(U104(z0, z1, z2)) plus(mark(z0), z1) -> mark(plus(z0, z1)) plus(z0, mark(z1)) -> mark(plus(z0, z1)) plus(ok(z0), ok(z1)) -> ok(plus(z0, z1)) x(mark(z0), z1) -> mark(x(z0, z1)) x(z0, mark(z1)) -> mark(x(z0, z1)) x(ok(z0), ok(z1)) -> ok(x(z0, z1)) U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2)) U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2)) U12(mark(z0), z1, z2) -> mark(U12(z0, z1, z2)) U12(ok(z0), ok(z1), ok(z2)) -> ok(U12(z0, z1, z2)) U13(mark(z0), z1, z2) -> mark(U13(z0, z1, z2)) U13(ok(z0), ok(z1), ok(z2)) -> ok(U13(z0, z1, z2)) U14(mark(z0), z1, z2) -> mark(U14(z0, z1, z2)) U14(ok(z0), ok(z1), ok(z2)) -> ok(U14(z0, z1, z2)) U15(mark(z0), z1) -> mark(U15(z0, z1)) U15(ok(z0), ok(z1)) -> ok(U15(z0, z1)) U16(mark(z0)) -> mark(U16(z0)) U16(ok(z0)) -> ok(U16(z0)) U21(mark(z0), z1) -> mark(U21(z0, z1)) U21(ok(z0), ok(z1)) -> ok(U21(z0, z1)) U22(mark(z0), z1) -> mark(U22(z0, z1)) U22(ok(z0), ok(z1)) -> ok(U22(z0, z1)) U23(mark(z0)) -> mark(U23(z0)) U23(ok(z0)) -> ok(U23(z0)) U31(mark(z0), z1, z2) -> mark(U31(z0, z1, z2)) U31(ok(z0), ok(z1), ok(z2)) -> ok(U31(z0, z1, z2)) U32(mark(z0), z1, z2) -> mark(U32(z0, z1, z2)) U32(ok(z0), ok(z1), ok(z2)) -> ok(U32(z0, z1, z2)) U33(mark(z0), z1, z2) -> mark(U33(z0, z1, z2)) U33(ok(z0), ok(z1), ok(z2)) -> ok(U33(z0, z1, z2)) U34(mark(z0), z1, z2) -> mark(U34(z0, z1, z2)) U34(ok(z0), ok(z1), ok(z2)) -> ok(U34(z0, z1, z2)) U35(mark(z0), z1) -> mark(U35(z0, z1)) U35(ok(z0), ok(z1)) -> ok(U35(z0, z1)) U36(mark(z0)) -> mark(U36(z0)) U36(ok(z0)) -> ok(U36(z0)) U41(mark(z0), z1) -> mark(U41(z0, z1)) U41(ok(z0), ok(z1)) -> ok(U41(z0, z1)) U42(mark(z0)) -> mark(U42(z0)) U42(ok(z0)) -> ok(U42(z0)) U51(mark(z0)) -> mark(U51(z0)) U51(ok(z0)) -> ok(U51(z0)) U61(mark(z0), z1) -> mark(U61(z0, z1)) U61(ok(z0), ok(z1)) -> ok(U61(z0, z1)) U62(mark(z0)) -> mark(U62(z0)) U62(ok(z0)) -> ok(U62(z0)) U71(mark(z0), z1) -> mark(U71(z0, z1)) U71(ok(z0), ok(z1)) -> ok(U71(z0, z1)) U72(mark(z0), z1) -> mark(U72(z0, z1)) U72(ok(z0), ok(z1)) -> ok(U72(z0, z1)) U81(mark(z0), z1, z2) -> mark(U81(z0, z1, z2)) U81(ok(z0), ok(z1), ok(z2)) -> ok(U81(z0, z1, z2)) U82(mark(z0), z1, z2) -> mark(U82(z0, z1, z2)) U82(ok(z0), ok(z1), ok(z2)) -> ok(U82(z0, z1, z2)) U83(mark(z0), z1, z2) -> mark(U83(z0, z1, z2)) U83(ok(z0), ok(z1), ok(z2)) -> ok(U83(z0, z1, z2)) U84(mark(z0), z1, z2) -> mark(U84(z0, z1, z2)) U84(ok(z0), ok(z1), ok(z2)) -> ok(U84(z0, z1, z2)) s(mark(z0)) -> mark(s(z0)) s(ok(z0)) -> ok(s(z0)) U91(mark(z0), z1) -> mark(U91(z0, z1)) U91(ok(z0), ok(z1)) -> ok(U91(z0, z1)) U92(mark(z0)) -> mark(U92(z0)) U92(ok(z0)) -> ok(U92(z0)) proper(tt) -> ok(tt) proper(0) -> ok(0) isNatKind(ok(z0)) -> ok(isNatKind(z0)) isNat(ok(z0)) -> ok(isNat(z0)) top(mark(z0)) -> top(proper(z0)) top(ok(z0)) -> top(active(z0)) Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) PROPER(tt) -> c72 PROPER(0) -> c73 ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0)), PROPER(z0)) TOP(ok(z0)) -> c77(TOP(active(z0))) S tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) PROPER(tt) -> c72 PROPER(0) -> c73 ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0)), PROPER(z0)) TOP(ok(z0)) -> c77(TOP(active(z0))) K tuples:none Defined Rule Symbols: U101_3, U102_3, U103_3, U104_3, plus_2, x_2, U11_3, U12_3, U13_3, U14_3, U15_2, U16_1, U21_2, U22_2, U23_1, U31_3, U32_3, U33_3, U34_3, U35_2, U36_1, U41_2, U42_1, U51_1, U61_2, U62_1, U71_2, U72_2, U81_3, U82_3, U83_3, U84_3, s_1, U91_2, U92_1, proper_1, isNatKind_1, isNat_1, top_1 Defined Pair Symbols: U101'_3, U102'_3, U103'_3, U104'_3, PLUS_2, X_2, U11'_3, U12'_3, U13'_3, U14'_3, U15'_2, U16'_1, U21'_2, U22'_2, U23'_1, U31'_3, U32'_3, U33'_3, U34'_3, U35'_2, U36'_1, U41'_2, U42'_1, U51'_1, U61'_2, U62'_1, U71'_2, U72'_2, U81'_3, U82'_3, U83'_3, U84'_3, S_1, U91'_2, U92'_1, PROPER_1, ISNATKIND_1, ISNAT_1, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c65_1, c66_1, c67_1, c68_1, c69_1, c70_1, c71_1, c72, c73, c74_1, c75_1, c76_2, c77_1 ---------------------------------------- (7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: PROPER(tt) -> c72 PROPER(0) -> c73 TOP(ok(z0)) -> c77(TOP(active(z0))) ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: U101(mark(z0), z1, z2) -> mark(U101(z0, z1, z2)) U101(ok(z0), ok(z1), ok(z2)) -> ok(U101(z0, z1, z2)) U102(mark(z0), z1, z2) -> mark(U102(z0, z1, z2)) U102(ok(z0), ok(z1), ok(z2)) -> ok(U102(z0, z1, z2)) U103(mark(z0), z1, z2) -> mark(U103(z0, z1, z2)) U103(ok(z0), ok(z1), ok(z2)) -> ok(U103(z0, z1, z2)) U104(mark(z0), z1, z2) -> mark(U104(z0, z1, z2)) U104(ok(z0), ok(z1), ok(z2)) -> ok(U104(z0, z1, z2)) plus(mark(z0), z1) -> mark(plus(z0, z1)) plus(z0, mark(z1)) -> mark(plus(z0, z1)) plus(ok(z0), ok(z1)) -> ok(plus(z0, z1)) x(mark(z0), z1) -> mark(x(z0, z1)) x(z0, mark(z1)) -> mark(x(z0, z1)) x(ok(z0), ok(z1)) -> ok(x(z0, z1)) U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2)) U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2)) U12(mark(z0), z1, z2) -> mark(U12(z0, z1, z2)) U12(ok(z0), ok(z1), ok(z2)) -> ok(U12(z0, z1, z2)) U13(mark(z0), z1, z2) -> mark(U13(z0, z1, z2)) U13(ok(z0), ok(z1), ok(z2)) -> ok(U13(z0, z1, z2)) U14(mark(z0), z1, z2) -> mark(U14(z0, z1, z2)) U14(ok(z0), ok(z1), ok(z2)) -> ok(U14(z0, z1, z2)) U15(mark(z0), z1) -> mark(U15(z0, z1)) U15(ok(z0), ok(z1)) -> ok(U15(z0, z1)) U16(mark(z0)) -> mark(U16(z0)) U16(ok(z0)) -> ok(U16(z0)) U21(mark(z0), z1) -> mark(U21(z0, z1)) U21(ok(z0), ok(z1)) -> ok(U21(z0, z1)) U22(mark(z0), z1) -> mark(U22(z0, z1)) U22(ok(z0), ok(z1)) -> ok(U22(z0, z1)) U23(mark(z0)) -> mark(U23(z0)) U23(ok(z0)) -> ok(U23(z0)) U31(mark(z0), z1, z2) -> mark(U31(z0, z1, z2)) U31(ok(z0), ok(z1), ok(z2)) -> ok(U31(z0, z1, z2)) U32(mark(z0), z1, z2) -> mark(U32(z0, z1, z2)) U32(ok(z0), ok(z1), ok(z2)) -> ok(U32(z0, z1, z2)) U33(mark(z0), z1, z2) -> mark(U33(z0, z1, z2)) U33(ok(z0), ok(z1), ok(z2)) -> ok(U33(z0, z1, z2)) U34(mark(z0), z1, z2) -> mark(U34(z0, z1, z2)) U34(ok(z0), ok(z1), ok(z2)) -> ok(U34(z0, z1, z2)) U35(mark(z0), z1) -> mark(U35(z0, z1)) U35(ok(z0), ok(z1)) -> ok(U35(z0, z1)) U36(mark(z0)) -> mark(U36(z0)) U36(ok(z0)) -> ok(U36(z0)) U41(mark(z0), z1) -> mark(U41(z0, z1)) U41(ok(z0), ok(z1)) -> ok(U41(z0, z1)) U42(mark(z0)) -> mark(U42(z0)) U42(ok(z0)) -> ok(U42(z0)) U51(mark(z0)) -> mark(U51(z0)) U51(ok(z0)) -> ok(U51(z0)) U61(mark(z0), z1) -> mark(U61(z0, z1)) U61(ok(z0), ok(z1)) -> ok(U61(z0, z1)) U62(mark(z0)) -> mark(U62(z0)) U62(ok(z0)) -> ok(U62(z0)) U71(mark(z0), z1) -> mark(U71(z0, z1)) U71(ok(z0), ok(z1)) -> ok(U71(z0, z1)) U72(mark(z0), z1) -> mark(U72(z0, z1)) U72(ok(z0), ok(z1)) -> ok(U72(z0, z1)) U81(mark(z0), z1, z2) -> mark(U81(z0, z1, z2)) U81(ok(z0), ok(z1), ok(z2)) -> ok(U81(z0, z1, z2)) U82(mark(z0), z1, z2) -> mark(U82(z0, z1, z2)) U82(ok(z0), ok(z1), ok(z2)) -> ok(U82(z0, z1, z2)) U83(mark(z0), z1, z2) -> mark(U83(z0, z1, z2)) U83(ok(z0), ok(z1), ok(z2)) -> ok(U83(z0, z1, z2)) U84(mark(z0), z1, z2) -> mark(U84(z0, z1, z2)) U84(ok(z0), ok(z1), ok(z2)) -> ok(U84(z0, z1, z2)) s(mark(z0)) -> mark(s(z0)) s(ok(z0)) -> ok(s(z0)) U91(mark(z0), z1) -> mark(U91(z0, z1)) U91(ok(z0), ok(z1)) -> ok(U91(z0, z1)) U92(mark(z0)) -> mark(U92(z0)) U92(ok(z0)) -> ok(U92(z0)) proper(tt) -> ok(tt) proper(0) -> ok(0) isNatKind(ok(z0)) -> ok(isNatKind(z0)) isNat(ok(z0)) -> ok(isNat(z0)) top(mark(z0)) -> top(proper(z0)) top(ok(z0)) -> top(active(z0)) Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0)), PROPER(z0)) S tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0)), PROPER(z0)) K tuples:none Defined Rule Symbols: U101_3, U102_3, U103_3, U104_3, plus_2, x_2, U11_3, U12_3, U13_3, U14_3, U15_2, U16_1, U21_2, U22_2, U23_1, U31_3, U32_3, U33_3, U34_3, U35_2, U36_1, U41_2, U42_1, U51_1, U61_2, U62_1, U71_2, U72_2, U81_3, U82_3, U83_3, U84_3, s_1, U91_2, U92_1, proper_1, isNatKind_1, isNat_1, top_1 Defined Pair Symbols: U101'_3, U102'_3, U103'_3, U104'_3, PLUS_2, X_2, U11'_3, U12'_3, U13'_3, U14'_3, U15'_2, U16'_1, U21'_2, U22'_2, U23'_1, U31'_3, U32'_3, U33'_3, U34'_3, U35'_2, U36'_1, U41'_2, U42'_1, U51'_1, U61'_2, U62'_1, U71'_2, U72'_2, U81'_3, U82'_3, U83'_3, U84'_3, S_1, U91'_2, U92'_1, ISNATKIND_1, ISNAT_1, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c65_1, c66_1, c67_1, c68_1, c69_1, c70_1, c71_1, c74_1, c75_1, c76_2 ---------------------------------------- (9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules: U101(mark(z0), z1, z2) -> mark(U101(z0, z1, z2)) U101(ok(z0), ok(z1), ok(z2)) -> ok(U101(z0, z1, z2)) U102(mark(z0), z1, z2) -> mark(U102(z0, z1, z2)) U102(ok(z0), ok(z1), ok(z2)) -> ok(U102(z0, z1, z2)) U103(mark(z0), z1, z2) -> mark(U103(z0, z1, z2)) U103(ok(z0), ok(z1), ok(z2)) -> ok(U103(z0, z1, z2)) U104(mark(z0), z1, z2) -> mark(U104(z0, z1, z2)) U104(ok(z0), ok(z1), ok(z2)) -> ok(U104(z0, z1, z2)) plus(mark(z0), z1) -> mark(plus(z0, z1)) plus(z0, mark(z1)) -> mark(plus(z0, z1)) plus(ok(z0), ok(z1)) -> ok(plus(z0, z1)) x(mark(z0), z1) -> mark(x(z0, z1)) x(z0, mark(z1)) -> mark(x(z0, z1)) x(ok(z0), ok(z1)) -> ok(x(z0, z1)) U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2)) U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2)) U12(mark(z0), z1, z2) -> mark(U12(z0, z1, z2)) U12(ok(z0), ok(z1), ok(z2)) -> ok(U12(z0, z1, z2)) U13(mark(z0), z1, z2) -> mark(U13(z0, z1, z2)) U13(ok(z0), ok(z1), ok(z2)) -> ok(U13(z0, z1, z2)) U14(mark(z0), z1, z2) -> mark(U14(z0, z1, z2)) U14(ok(z0), ok(z1), ok(z2)) -> ok(U14(z0, z1, z2)) U15(mark(z0), z1) -> mark(U15(z0, z1)) U15(ok(z0), ok(z1)) -> ok(U15(z0, z1)) U16(mark(z0)) -> mark(U16(z0)) U16(ok(z0)) -> ok(U16(z0)) U21(mark(z0), z1) -> mark(U21(z0, z1)) U21(ok(z0), ok(z1)) -> ok(U21(z0, z1)) U22(mark(z0), z1) -> mark(U22(z0, z1)) U22(ok(z0), ok(z1)) -> ok(U22(z0, z1)) U23(mark(z0)) -> mark(U23(z0)) U23(ok(z0)) -> ok(U23(z0)) U31(mark(z0), z1, z2) -> mark(U31(z0, z1, z2)) U31(ok(z0), ok(z1), ok(z2)) -> ok(U31(z0, z1, z2)) U32(mark(z0), z1, z2) -> mark(U32(z0, z1, z2)) U32(ok(z0), ok(z1), ok(z2)) -> ok(U32(z0, z1, z2)) U33(mark(z0), z1, z2) -> mark(U33(z0, z1, z2)) U33(ok(z0), ok(z1), ok(z2)) -> ok(U33(z0, z1, z2)) U34(mark(z0), z1, z2) -> mark(U34(z0, z1, z2)) U34(ok(z0), ok(z1), ok(z2)) -> ok(U34(z0, z1, z2)) U35(mark(z0), z1) -> mark(U35(z0, z1)) U35(ok(z0), ok(z1)) -> ok(U35(z0, z1)) U36(mark(z0)) -> mark(U36(z0)) U36(ok(z0)) -> ok(U36(z0)) U41(mark(z0), z1) -> mark(U41(z0, z1)) U41(ok(z0), ok(z1)) -> ok(U41(z0, z1)) U42(mark(z0)) -> mark(U42(z0)) U42(ok(z0)) -> ok(U42(z0)) U51(mark(z0)) -> mark(U51(z0)) U51(ok(z0)) -> ok(U51(z0)) U61(mark(z0), z1) -> mark(U61(z0, z1)) U61(ok(z0), ok(z1)) -> ok(U61(z0, z1)) U62(mark(z0)) -> mark(U62(z0)) U62(ok(z0)) -> ok(U62(z0)) U71(mark(z0), z1) -> mark(U71(z0, z1)) U71(ok(z0), ok(z1)) -> ok(U71(z0, z1)) U72(mark(z0), z1) -> mark(U72(z0, z1)) U72(ok(z0), ok(z1)) -> ok(U72(z0, z1)) U81(mark(z0), z1, z2) -> mark(U81(z0, z1, z2)) U81(ok(z0), ok(z1), ok(z2)) -> ok(U81(z0, z1, z2)) U82(mark(z0), z1, z2) -> mark(U82(z0, z1, z2)) U82(ok(z0), ok(z1), ok(z2)) -> ok(U82(z0, z1, z2)) U83(mark(z0), z1, z2) -> mark(U83(z0, z1, z2)) U83(ok(z0), ok(z1), ok(z2)) -> ok(U83(z0, z1, z2)) U84(mark(z0), z1, z2) -> mark(U84(z0, z1, z2)) U84(ok(z0), ok(z1), ok(z2)) -> ok(U84(z0, z1, z2)) s(mark(z0)) -> mark(s(z0)) s(ok(z0)) -> ok(s(z0)) U91(mark(z0), z1) -> mark(U91(z0, z1)) U91(ok(z0), ok(z1)) -> ok(U91(z0, z1)) U92(mark(z0)) -> mark(U92(z0)) U92(ok(z0)) -> ok(U92(z0)) proper(tt) -> ok(tt) proper(0) -> ok(0) isNatKind(ok(z0)) -> ok(isNatKind(z0)) isNat(ok(z0)) -> ok(isNat(z0)) top(mark(z0)) -> top(proper(z0)) top(ok(z0)) -> top(active(z0)) Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) S tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) K tuples:none Defined Rule Symbols: U101_3, U102_3, U103_3, U104_3, plus_2, x_2, U11_3, U12_3, U13_3, U14_3, U15_2, U16_1, U21_2, U22_2, U23_1, U31_3, U32_3, U33_3, U34_3, U35_2, U36_1, U41_2, U42_1, U51_1, U61_2, U62_1, U71_2, U72_2, U81_3, U82_3, U83_3, U84_3, s_1, U91_2, U92_1, proper_1, isNatKind_1, isNat_1, top_1 Defined Pair Symbols: U101'_3, U102'_3, U103'_3, U104'_3, PLUS_2, X_2, U11'_3, U12'_3, U13'_3, U14'_3, U15'_2, U16'_1, U21'_2, U22'_2, U23'_1, U31'_3, U32'_3, U33'_3, U34'_3, U35'_2, U36'_1, U41'_2, U42'_1, U51'_1, U61'_2, U62'_1, U71'_2, U72'_2, U81'_3, U82'_3, U83'_3, U84'_3, S_1, U91'_2, U92'_1, ISNATKIND_1, ISNAT_1, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c65_1, c66_1, c67_1, c68_1, c69_1, c70_1, c71_1, c74_1, c75_1, c76_1 ---------------------------------------- (11) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: U101(mark(z0), z1, z2) -> mark(U101(z0, z1, z2)) U101(ok(z0), ok(z1), ok(z2)) -> ok(U101(z0, z1, z2)) U102(mark(z0), z1, z2) -> mark(U102(z0, z1, z2)) U102(ok(z0), ok(z1), ok(z2)) -> ok(U102(z0, z1, z2)) U103(mark(z0), z1, z2) -> mark(U103(z0, z1, z2)) U103(ok(z0), ok(z1), ok(z2)) -> ok(U103(z0, z1, z2)) U104(mark(z0), z1, z2) -> mark(U104(z0, z1, z2)) U104(ok(z0), ok(z1), ok(z2)) -> ok(U104(z0, z1, z2)) plus(mark(z0), z1) -> mark(plus(z0, z1)) plus(z0, mark(z1)) -> mark(plus(z0, z1)) plus(ok(z0), ok(z1)) -> ok(plus(z0, z1)) x(mark(z0), z1) -> mark(x(z0, z1)) x(z0, mark(z1)) -> mark(x(z0, z1)) x(ok(z0), ok(z1)) -> ok(x(z0, z1)) U11(mark(z0), z1, z2) -> mark(U11(z0, z1, z2)) U11(ok(z0), ok(z1), ok(z2)) -> ok(U11(z0, z1, z2)) U12(mark(z0), z1, z2) -> mark(U12(z0, z1, z2)) U12(ok(z0), ok(z1), ok(z2)) -> ok(U12(z0, z1, z2)) U13(mark(z0), z1, z2) -> mark(U13(z0, z1, z2)) U13(ok(z0), ok(z1), ok(z2)) -> ok(U13(z0, z1, z2)) U14(mark(z0), z1, z2) -> mark(U14(z0, z1, z2)) U14(ok(z0), ok(z1), ok(z2)) -> ok(U14(z0, z1, z2)) U15(mark(z0), z1) -> mark(U15(z0, z1)) U15(ok(z0), ok(z1)) -> ok(U15(z0, z1)) U16(mark(z0)) -> mark(U16(z0)) U16(ok(z0)) -> ok(U16(z0)) U21(mark(z0), z1) -> mark(U21(z0, z1)) U21(ok(z0), ok(z1)) -> ok(U21(z0, z1)) U22(mark(z0), z1) -> mark(U22(z0, z1)) U22(ok(z0), ok(z1)) -> ok(U22(z0, z1)) U23(mark(z0)) -> mark(U23(z0)) U23(ok(z0)) -> ok(U23(z0)) U31(mark(z0), z1, z2) -> mark(U31(z0, z1, z2)) U31(ok(z0), ok(z1), ok(z2)) -> ok(U31(z0, z1, z2)) U32(mark(z0), z1, z2) -> mark(U32(z0, z1, z2)) U32(ok(z0), ok(z1), ok(z2)) -> ok(U32(z0, z1, z2)) U33(mark(z0), z1, z2) -> mark(U33(z0, z1, z2)) U33(ok(z0), ok(z1), ok(z2)) -> ok(U33(z0, z1, z2)) U34(mark(z0), z1, z2) -> mark(U34(z0, z1, z2)) U34(ok(z0), ok(z1), ok(z2)) -> ok(U34(z0, z1, z2)) U35(mark(z0), z1) -> mark(U35(z0, z1)) U35(ok(z0), ok(z1)) -> ok(U35(z0, z1)) U36(mark(z0)) -> mark(U36(z0)) U36(ok(z0)) -> ok(U36(z0)) U41(mark(z0), z1) -> mark(U41(z0, z1)) U41(ok(z0), ok(z1)) -> ok(U41(z0, z1)) U42(mark(z0)) -> mark(U42(z0)) U42(ok(z0)) -> ok(U42(z0)) U51(mark(z0)) -> mark(U51(z0)) U51(ok(z0)) -> ok(U51(z0)) U61(mark(z0), z1) -> mark(U61(z0, z1)) U61(ok(z0), ok(z1)) -> ok(U61(z0, z1)) U62(mark(z0)) -> mark(U62(z0)) U62(ok(z0)) -> ok(U62(z0)) U71(mark(z0), z1) -> mark(U71(z0, z1)) U71(ok(z0), ok(z1)) -> ok(U71(z0, z1)) U72(mark(z0), z1) -> mark(U72(z0, z1)) U72(ok(z0), ok(z1)) -> ok(U72(z0, z1)) U81(mark(z0), z1, z2) -> mark(U81(z0, z1, z2)) U81(ok(z0), ok(z1), ok(z2)) -> ok(U81(z0, z1, z2)) U82(mark(z0), z1, z2) -> mark(U82(z0, z1, z2)) U82(ok(z0), ok(z1), ok(z2)) -> ok(U82(z0, z1, z2)) U83(mark(z0), z1, z2) -> mark(U83(z0, z1, z2)) U83(ok(z0), ok(z1), ok(z2)) -> ok(U83(z0, z1, z2)) U84(mark(z0), z1, z2) -> mark(U84(z0, z1, z2)) U84(ok(z0), ok(z1), ok(z2)) -> ok(U84(z0, z1, z2)) s(mark(z0)) -> mark(s(z0)) s(ok(z0)) -> ok(s(z0)) U91(mark(z0), z1) -> mark(U91(z0, z1)) U91(ok(z0), ok(z1)) -> ok(U91(z0, z1)) U92(mark(z0)) -> mark(U92(z0)) U92(ok(z0)) -> ok(U92(z0)) isNatKind(ok(z0)) -> ok(isNatKind(z0)) isNat(ok(z0)) -> ok(isNat(z0)) top(mark(z0)) -> top(proper(z0)) top(ok(z0)) -> top(active(z0)) ---------------------------------------- (12) Obligation: Complexity Dependency Tuples Problem Rules: proper(tt) -> ok(tt) proper(0) -> ok(0) Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) S tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) K tuples:none Defined Rule Symbols: proper_1 Defined Pair Symbols: U101'_3, U102'_3, U103'_3, U104'_3, PLUS_2, X_2, U11'_3, U12'_3, U13'_3, U14'_3, U15'_2, U16'_1, U21'_2, U22'_2, U23'_1, U31'_3, U32'_3, U33'_3, U34'_3, U35'_2, U36'_1, U41'_2, U42'_1, U51'_1, U61'_2, U62'_1, U71'_2, U72'_2, U81'_3, U82'_3, U83'_3, U84'_3, S_1, U91'_2, U92'_1, ISNATKIND_1, ISNAT_1, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c65_1, c66_1, c67_1, c68_1, c69_1, c70_1, c71_1, c74_1, c75_1, c76_1 ---------------------------------------- (13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U16'(ok(z0)) -> c25(U16'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) S(ok(z0)) -> c67(S(z0)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) We considered the (Usable) Rules:none And the Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(ISNAT(x_1)) = x_1 + x_1^2 + x_1^3 POL(ISNATKIND(x_1)) = x_1 + x_1^2 + x_1^3 POL(PLUS(x_1, x_2)) = 0 POL(S(x_1)) = x_1 + x_1^2 + x_1^3 POL(TOP(x_1)) = 0 POL(U101'(x_1, x_2, x_3)) = 0 POL(U102'(x_1, x_2, x_3)) = 0 POL(U103'(x_1, x_2, x_3)) = 0 POL(U104'(x_1, x_2, x_3)) = 0 POL(U11'(x_1, x_2, x_3)) = 0 POL(U12'(x_1, x_2, x_3)) = 0 POL(U13'(x_1, x_2, x_3)) = 0 POL(U14'(x_1, x_2, x_3)) = x_1^2 POL(U15'(x_1, x_2)) = 0 POL(U16'(x_1)) = x_1 + x_1^2 + x_1^3 POL(U21'(x_1, x_2)) = 0 POL(U22'(x_1, x_2)) = 0 POL(U23'(x_1)) = x_1 + x_1^2 + x_1^3 POL(U31'(x_1, x_2, x_3)) = 0 POL(U32'(x_1, x_2, x_3)) = 0 POL(U33'(x_1, x_2, x_3)) = 0 POL(U34'(x_1, x_2, x_3)) = 0 POL(U35'(x_1, x_2)) = 0 POL(U36'(x_1)) = x_1 + x_1^2 + x_1^3 POL(U41'(x_1, x_2)) = 0 POL(U42'(x_1)) = x_1 + x_1^2 + x_1^3 POL(U51'(x_1)) = x_1 + x_1^2 + x_1^3 POL(U61'(x_1, x_2)) = 0 POL(U62'(x_1)) = x_1 + x_1^2 + x_1^3 POL(U71'(x_1, x_2)) = 0 POL(U72'(x_1, x_2)) = 0 POL(U81'(x_1, x_2, x_3)) = 0 POL(U82'(x_1, x_2, x_3)) = 0 POL(U83'(x_1, x_2, x_3)) = 0 POL(U84'(x_1, x_2, x_3)) = 0 POL(U91'(x_1, x_2)) = x_1 + x_2 + x_2^2 + x_1*x_2 + x_1^2 + x_1^3 + x_1^2*x_2 + x_1*x_2^2 + x_2^3 POL(U92'(x_1)) = x_1 + x_1^2 + x_1^3 POL(X(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c27(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c33(x_1)) = x_1 POL(c34(x_1)) = x_1 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c37(x_1)) = x_1 POL(c38(x_1)) = x_1 POL(c39(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c40(x_1)) = x_1 POL(c41(x_1)) = x_1 POL(c42(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c44(x_1)) = x_1 POL(c45(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(c47(x_1)) = x_1 POL(c48(x_1)) = x_1 POL(c49(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c50(x_1)) = x_1 POL(c51(x_1)) = x_1 POL(c52(x_1)) = x_1 POL(c53(x_1)) = x_1 POL(c54(x_1)) = x_1 POL(c55(x_1)) = x_1 POL(c56(x_1)) = x_1 POL(c57(x_1)) = x_1 POL(c58(x_1)) = x_1 POL(c59(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c60(x_1)) = x_1 POL(c61(x_1)) = x_1 POL(c62(x_1)) = x_1 POL(c63(x_1)) = x_1 POL(c64(x_1)) = x_1 POL(c65(x_1)) = x_1 POL(c66(x_1)) = x_1 POL(c67(x_1)) = x_1 POL(c68(x_1)) = x_1 POL(c69(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c70(x_1)) = x_1 POL(c71(x_1)) = x_1 POL(c74(x_1)) = x_1 POL(c75(x_1)) = x_1 POL(c76(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(mark(x_1)) = x_1 POL(ok(x_1)) = [1] + x_1 POL(proper(x_1)) = x_1^2 + x_1^3 POL(tt) = 0 ---------------------------------------- (14) Obligation: Complexity Dependency Tuples Problem Rules: proper(tt) -> ok(tt) proper(0) -> ok(0) Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) S tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) K tuples: U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U16'(ok(z0)) -> c25(U16'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) S(ok(z0)) -> c67(S(z0)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) Defined Rule Symbols: proper_1 Defined Pair Symbols: U101'_3, U102'_3, U103'_3, U104'_3, PLUS_2, X_2, U11'_3, U12'_3, U13'_3, U14'_3, U15'_2, U16'_1, U21'_2, U22'_2, U23'_1, U31'_3, U32'_3, U33'_3, U34'_3, U35'_2, U36'_1, U41'_2, U42'_1, U51'_1, U61'_2, U62'_1, U71'_2, U72'_2, U81'_3, U82'_3, U83'_3, U84'_3, S_1, U91'_2, U92'_1, ISNATKIND_1, ISNAT_1, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c65_1, c66_1, c67_1, c68_1, c69_1, c70_1, c71_1, c74_1, c75_1, c76_1 ---------------------------------------- (15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) We considered the (Usable) Rules:none And the Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [3] POL(ISNAT(x_1)) = [3]x_1 POL(ISNATKIND(x_1)) = [3]x_1 POL(PLUS(x_1, x_2)) = 0 POL(S(x_1)) = [3]x_1 POL(TOP(x_1)) = 0 POL(U101'(x_1, x_2, x_3)) = 0 POL(U102'(x_1, x_2, x_3)) = 0 POL(U103'(x_1, x_2, x_3)) = 0 POL(U104'(x_1, x_2, x_3)) = 0 POL(U11'(x_1, x_2, x_3)) = 0 POL(U12'(x_1, x_2, x_3)) = 0 POL(U13'(x_1, x_2, x_3)) = 0 POL(U14'(x_1, x_2, x_3)) = [3]x_1 + [3]x_2 + [3]x_3 POL(U15'(x_1, x_2)) = 0 POL(U16'(x_1)) = [3]x_1 POL(U21'(x_1, x_2)) = 0 POL(U22'(x_1, x_2)) = 0 POL(U23'(x_1)) = [3]x_1 POL(U31'(x_1, x_2, x_3)) = 0 POL(U32'(x_1, x_2, x_3)) = 0 POL(U33'(x_1, x_2, x_3)) = 0 POL(U34'(x_1, x_2, x_3)) = 0 POL(U35'(x_1, x_2)) = 0 POL(U36'(x_1)) = [3]x_1 POL(U41'(x_1, x_2)) = 0 POL(U42'(x_1)) = [3]x_1 POL(U51'(x_1)) = [3]x_1 POL(U61'(x_1, x_2)) = 0 POL(U62'(x_1)) = [3]x_1 POL(U71'(x_1, x_2)) = 0 POL(U72'(x_1, x_2)) = x_2 POL(U81'(x_1, x_2, x_3)) = 0 POL(U82'(x_1, x_2, x_3)) = 0 POL(U83'(x_1, x_2, x_3)) = 0 POL(U84'(x_1, x_2, x_3)) = 0 POL(U91'(x_1, x_2)) = [3]x_1 + [3]x_2 POL(U92'(x_1)) = [3]x_1 POL(X(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c27(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c33(x_1)) = x_1 POL(c34(x_1)) = x_1 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c37(x_1)) = x_1 POL(c38(x_1)) = x_1 POL(c39(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c40(x_1)) = x_1 POL(c41(x_1)) = x_1 POL(c42(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c44(x_1)) = x_1 POL(c45(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(c47(x_1)) = x_1 POL(c48(x_1)) = x_1 POL(c49(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c50(x_1)) = x_1 POL(c51(x_1)) = x_1 POL(c52(x_1)) = x_1 POL(c53(x_1)) = x_1 POL(c54(x_1)) = x_1 POL(c55(x_1)) = x_1 POL(c56(x_1)) = x_1 POL(c57(x_1)) = x_1 POL(c58(x_1)) = x_1 POL(c59(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c60(x_1)) = x_1 POL(c61(x_1)) = x_1 POL(c62(x_1)) = x_1 POL(c63(x_1)) = x_1 POL(c64(x_1)) = x_1 POL(c65(x_1)) = x_1 POL(c66(x_1)) = x_1 POL(c67(x_1)) = x_1 POL(c68(x_1)) = x_1 POL(c69(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c70(x_1)) = x_1 POL(c71(x_1)) = x_1 POL(c74(x_1)) = x_1 POL(c75(x_1)) = x_1 POL(c76(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(mark(x_1)) = x_1 POL(ok(x_1)) = [1] + x_1 POL(proper(x_1)) = [1] + [3]x_1 POL(tt) = 0 ---------------------------------------- (16) Obligation: Complexity Dependency Tuples Problem Rules: proper(tt) -> ok(tt) proper(0) -> ok(0) Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) S tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) K tuples: U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U16'(ok(z0)) -> c25(U16'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) S(ok(z0)) -> c67(S(z0)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) Defined Rule Symbols: proper_1 Defined Pair Symbols: U101'_3, U102'_3, U103'_3, U104'_3, PLUS_2, X_2, U11'_3, U12'_3, U13'_3, U14'_3, U15'_2, U16'_1, U21'_2, U22'_2, U23'_1, U31'_3, U32'_3, U33'_3, U34'_3, U35'_2, U36'_1, U41'_2, U42'_1, U51'_1, U61'_2, U62'_1, U71'_2, U72'_2, U81'_3, U82'_3, U83'_3, U84'_3, S_1, U91'_2, U92'_1, ISNATKIND_1, ISNAT_1, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c65_1, c66_1, c67_1, c68_1, c69_1, c70_1, c71_1, c74_1, c75_1, c76_1 ---------------------------------------- (17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) We considered the (Usable) Rules: proper(tt) -> ok(tt) proper(0) -> ok(0) And the Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(ISNAT(x_1)) = x_1 POL(ISNATKIND(x_1)) = x_1 POL(PLUS(x_1, x_2)) = x_1 + x_2 POL(S(x_1)) = x_1 POL(TOP(x_1)) = x_1 POL(U101'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U102'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U103'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U104'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U11'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U12'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U13'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U14'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U15'(x_1, x_2)) = x_1 + x_2 POL(U16'(x_1)) = x_1 POL(U21'(x_1, x_2)) = x_1 + x_2 POL(U22'(x_1, x_2)) = x_1 + x_2 POL(U23'(x_1)) = x_1 POL(U31'(x_1, x_2, x_3)) = x_1 + x_3 POL(U32'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U33'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U34'(x_1, x_2, x_3)) = x_1 + x_3 POL(U35'(x_1, x_2)) = x_1 + x_2 POL(U36'(x_1)) = x_1 POL(U41'(x_1, x_2)) = x_1 + x_2 POL(U42'(x_1)) = x_1 POL(U51'(x_1)) = x_1 POL(U61'(x_1, x_2)) = x_1 + x_2 POL(U62'(x_1)) = x_1 POL(U71'(x_1, x_2)) = x_1 + x_2 POL(U72'(x_1, x_2)) = x_1 + x_2 POL(U81'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U82'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U83'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U84'(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U91'(x_1, x_2)) = x_1 + x_2 POL(U92'(x_1)) = x_1 POL(X(x_1, x_2)) = x_1 + x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c27(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c33(x_1)) = x_1 POL(c34(x_1)) = x_1 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c37(x_1)) = x_1 POL(c38(x_1)) = x_1 POL(c39(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c40(x_1)) = x_1 POL(c41(x_1)) = x_1 POL(c42(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c44(x_1)) = x_1 POL(c45(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(c47(x_1)) = x_1 POL(c48(x_1)) = x_1 POL(c49(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c50(x_1)) = x_1 POL(c51(x_1)) = x_1 POL(c52(x_1)) = x_1 POL(c53(x_1)) = x_1 POL(c54(x_1)) = x_1 POL(c55(x_1)) = x_1 POL(c56(x_1)) = x_1 POL(c57(x_1)) = x_1 POL(c58(x_1)) = x_1 POL(c59(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c60(x_1)) = x_1 POL(c61(x_1)) = x_1 POL(c62(x_1)) = x_1 POL(c63(x_1)) = x_1 POL(c64(x_1)) = x_1 POL(c65(x_1)) = x_1 POL(c66(x_1)) = x_1 POL(c67(x_1)) = x_1 POL(c68(x_1)) = x_1 POL(c69(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c70(x_1)) = x_1 POL(c71(x_1)) = x_1 POL(c74(x_1)) = x_1 POL(c75(x_1)) = x_1 POL(c76(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(mark(x_1)) = [1] + x_1 POL(ok(x_1)) = [1] + x_1 POL(proper(x_1)) = [1] + x_1 POL(tt) = [1] ---------------------------------------- (18) Obligation: Complexity Dependency Tuples Problem Rules: proper(tt) -> ok(tt) proper(0) -> ok(0) Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) S tuples: TOP(mark(z0)) -> c76(TOP(proper(z0))) K tuples: U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U16'(ok(z0)) -> c25(U16'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) S(ok(z0)) -> c67(S(z0)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) Defined Rule Symbols: proper_1 Defined Pair Symbols: U101'_3, U102'_3, U103'_3, U104'_3, PLUS_2, X_2, U11'_3, U12'_3, U13'_3, U14'_3, U15'_2, U16'_1, U21'_2, U22'_2, U23'_1, U31'_3, U32'_3, U33'_3, U34'_3, U35'_2, U36'_1, U41'_2, U42'_1, U51'_1, U61'_2, U62'_1, U71'_2, U72'_2, U81'_3, U82'_3, U83'_3, U84'_3, S_1, U91'_2, U92'_1, ISNATKIND_1, ISNAT_1, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c65_1, c66_1, c67_1, c68_1, c69_1, c70_1, c71_1, c74_1, c75_1, c76_1 ---------------------------------------- (19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. TOP(mark(z0)) -> c76(TOP(proper(z0))) We considered the (Usable) Rules: proper(tt) -> ok(tt) proper(0) -> ok(0) And the Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(ISNAT(x_1)) = 0 POL(ISNATKIND(x_1)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(S(x_1)) = 0 POL(TOP(x_1)) = x_1 POL(U101'(x_1, x_2, x_3)) = 0 POL(U102'(x_1, x_2, x_3)) = 0 POL(U103'(x_1, x_2, x_3)) = 0 POL(U104'(x_1, x_2, x_3)) = 0 POL(U11'(x_1, x_2, x_3)) = 0 POL(U12'(x_1, x_2, x_3)) = 0 POL(U13'(x_1, x_2, x_3)) = 0 POL(U14'(x_1, x_2, x_3)) = 0 POL(U15'(x_1, x_2)) = 0 POL(U16'(x_1)) = 0 POL(U21'(x_1, x_2)) = 0 POL(U22'(x_1, x_2)) = 0 POL(U23'(x_1)) = 0 POL(U31'(x_1, x_2, x_3)) = 0 POL(U32'(x_1, x_2, x_3)) = 0 POL(U33'(x_1, x_2, x_3)) = 0 POL(U34'(x_1, x_2, x_3)) = 0 POL(U35'(x_1, x_2)) = 0 POL(U36'(x_1)) = 0 POL(U41'(x_1, x_2)) = 0 POL(U42'(x_1)) = 0 POL(U51'(x_1)) = 0 POL(U61'(x_1, x_2)) = 0 POL(U62'(x_1)) = 0 POL(U71'(x_1, x_2)) = 0 POL(U72'(x_1, x_2)) = 0 POL(U81'(x_1, x_2, x_3)) = 0 POL(U82'(x_1, x_2, x_3)) = 0 POL(U83'(x_1, x_2, x_3)) = 0 POL(U84'(x_1, x_2, x_3)) = 0 POL(U91'(x_1, x_2)) = 0 POL(U92'(x_1)) = 0 POL(X(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c18(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c22(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c25(x_1)) = x_1 POL(c26(x_1)) = x_1 POL(c27(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c30(x_1)) = x_1 POL(c31(x_1)) = x_1 POL(c32(x_1)) = x_1 POL(c33(x_1)) = x_1 POL(c34(x_1)) = x_1 POL(c35(x_1)) = x_1 POL(c36(x_1)) = x_1 POL(c37(x_1)) = x_1 POL(c38(x_1)) = x_1 POL(c39(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c40(x_1)) = x_1 POL(c41(x_1)) = x_1 POL(c42(x_1)) = x_1 POL(c43(x_1)) = x_1 POL(c44(x_1)) = x_1 POL(c45(x_1)) = x_1 POL(c46(x_1)) = x_1 POL(c47(x_1)) = x_1 POL(c48(x_1)) = x_1 POL(c49(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c50(x_1)) = x_1 POL(c51(x_1)) = x_1 POL(c52(x_1)) = x_1 POL(c53(x_1)) = x_1 POL(c54(x_1)) = x_1 POL(c55(x_1)) = x_1 POL(c56(x_1)) = x_1 POL(c57(x_1)) = x_1 POL(c58(x_1)) = x_1 POL(c59(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c60(x_1)) = x_1 POL(c61(x_1)) = x_1 POL(c62(x_1)) = x_1 POL(c63(x_1)) = x_1 POL(c64(x_1)) = x_1 POL(c65(x_1)) = x_1 POL(c66(x_1)) = x_1 POL(c67(x_1)) = x_1 POL(c68(x_1)) = x_1 POL(c69(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c70(x_1)) = x_1 POL(c71(x_1)) = x_1 POL(c74(x_1)) = x_1 POL(c75(x_1)) = x_1 POL(c76(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(mark(x_1)) = [1] + x_1 POL(ok(x_1)) = 0 POL(proper(x_1)) = 0 POL(tt) = 0 ---------------------------------------- (20) Obligation: Complexity Dependency Tuples Problem Rules: proper(tt) -> ok(tt) proper(0) -> ok(0) Tuples: U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U16'(ok(z0)) -> c25(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) S(ok(z0)) -> c67(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) S tuples:none K tuples: U14'(ok(z0), ok(z1), ok(z2)) -> c21(U14'(z0, z1, z2)) U16'(ok(z0)) -> c25(U16'(z0)) U23'(ok(z0)) -> c31(U23'(z0)) U36'(ok(z0)) -> c43(U36'(z0)) U42'(ok(z0)) -> c47(U42'(z0)) U51'(ok(z0)) -> c49(U51'(z0)) U62'(ok(z0)) -> c53(U62'(z0)) S(ok(z0)) -> c67(S(z0)) U91'(ok(z0), ok(z1)) -> c69(U91'(z0, z1)) U92'(ok(z0)) -> c71(U92'(z0)) ISNATKIND(ok(z0)) -> c74(ISNATKIND(z0)) ISNAT(ok(z0)) -> c75(ISNAT(z0)) U72'(ok(z0), ok(z1)) -> c57(U72'(z0, z1)) U101'(mark(z0), z1, z2) -> c(U101'(z0, z1, z2)) U101'(ok(z0), ok(z1), ok(z2)) -> c1(U101'(z0, z1, z2)) U102'(mark(z0), z1, z2) -> c2(U102'(z0, z1, z2)) U102'(ok(z0), ok(z1), ok(z2)) -> c3(U102'(z0, z1, z2)) U103'(mark(z0), z1, z2) -> c4(U103'(z0, z1, z2)) U103'(ok(z0), ok(z1), ok(z2)) -> c5(U103'(z0, z1, z2)) U104'(mark(z0), z1, z2) -> c6(U104'(z0, z1, z2)) U104'(ok(z0), ok(z1), ok(z2)) -> c7(U104'(z0, z1, z2)) PLUS(mark(z0), z1) -> c8(PLUS(z0, z1)) PLUS(z0, mark(z1)) -> c9(PLUS(z0, z1)) PLUS(ok(z0), ok(z1)) -> c10(PLUS(z0, z1)) X(mark(z0), z1) -> c11(X(z0, z1)) X(z0, mark(z1)) -> c12(X(z0, z1)) X(ok(z0), ok(z1)) -> c13(X(z0, z1)) U11'(mark(z0), z1, z2) -> c14(U11'(z0, z1, z2)) U11'(ok(z0), ok(z1), ok(z2)) -> c15(U11'(z0, z1, z2)) U12'(mark(z0), z1, z2) -> c16(U12'(z0, z1, z2)) U12'(ok(z0), ok(z1), ok(z2)) -> c17(U12'(z0, z1, z2)) U13'(mark(z0), z1, z2) -> c18(U13'(z0, z1, z2)) U13'(ok(z0), ok(z1), ok(z2)) -> c19(U13'(z0, z1, z2)) U14'(mark(z0), z1, z2) -> c20(U14'(z0, z1, z2)) U15'(mark(z0), z1) -> c22(U15'(z0, z1)) U15'(ok(z0), ok(z1)) -> c23(U15'(z0, z1)) U16'(mark(z0)) -> c24(U16'(z0)) U21'(mark(z0), z1) -> c26(U21'(z0, z1)) U21'(ok(z0), ok(z1)) -> c27(U21'(z0, z1)) U22'(mark(z0), z1) -> c28(U22'(z0, z1)) U22'(ok(z0), ok(z1)) -> c29(U22'(z0, z1)) U23'(mark(z0)) -> c30(U23'(z0)) U31'(mark(z0), z1, z2) -> c32(U31'(z0, z1, z2)) U31'(ok(z0), ok(z1), ok(z2)) -> c33(U31'(z0, z1, z2)) U32'(mark(z0), z1, z2) -> c34(U32'(z0, z1, z2)) U32'(ok(z0), ok(z1), ok(z2)) -> c35(U32'(z0, z1, z2)) U33'(mark(z0), z1, z2) -> c36(U33'(z0, z1, z2)) U33'(ok(z0), ok(z1), ok(z2)) -> c37(U33'(z0, z1, z2)) U34'(mark(z0), z1, z2) -> c38(U34'(z0, z1, z2)) U34'(ok(z0), ok(z1), ok(z2)) -> c39(U34'(z0, z1, z2)) U35'(mark(z0), z1) -> c40(U35'(z0, z1)) U35'(ok(z0), ok(z1)) -> c41(U35'(z0, z1)) U36'(mark(z0)) -> c42(U36'(z0)) U41'(mark(z0), z1) -> c44(U41'(z0, z1)) U41'(ok(z0), ok(z1)) -> c45(U41'(z0, z1)) U42'(mark(z0)) -> c46(U42'(z0)) U51'(mark(z0)) -> c48(U51'(z0)) U61'(mark(z0), z1) -> c50(U61'(z0, z1)) U61'(ok(z0), ok(z1)) -> c51(U61'(z0, z1)) U62'(mark(z0)) -> c52(U62'(z0)) U71'(mark(z0), z1) -> c54(U71'(z0, z1)) U71'(ok(z0), ok(z1)) -> c55(U71'(z0, z1)) U72'(mark(z0), z1) -> c56(U72'(z0, z1)) U81'(mark(z0), z1, z2) -> c58(U81'(z0, z1, z2)) U81'(ok(z0), ok(z1), ok(z2)) -> c59(U81'(z0, z1, z2)) U82'(mark(z0), z1, z2) -> c60(U82'(z0, z1, z2)) U82'(ok(z0), ok(z1), ok(z2)) -> c61(U82'(z0, z1, z2)) U83'(mark(z0), z1, z2) -> c62(U83'(z0, z1, z2)) U83'(ok(z0), ok(z1), ok(z2)) -> c63(U83'(z0, z1, z2)) U84'(mark(z0), z1, z2) -> c64(U84'(z0, z1, z2)) U84'(ok(z0), ok(z1), ok(z2)) -> c65(U84'(z0, z1, z2)) S(mark(z0)) -> c66(S(z0)) U91'(mark(z0), z1) -> c68(U91'(z0, z1)) U92'(mark(z0)) -> c70(U92'(z0)) TOP(mark(z0)) -> c76(TOP(proper(z0))) Defined Rule Symbols: proper_1 Defined Pair Symbols: U101'_3, U102'_3, U103'_3, U104'_3, PLUS_2, X_2, U11'_3, U12'_3, U13'_3, U14'_3, U15'_2, U16'_1, U21'_2, U22'_2, U23'_1, U31'_3, U32'_3, U33'_3, U34'_3, U35'_2, U36'_1, U41'_2, U42'_1, U51'_1, U61'_2, U62'_1, U71'_2, U72'_2, U81'_3, U82'_3, U83'_3, U84'_3, S_1, U91'_2, U92'_1, ISNATKIND_1, ISNAT_1, TOP_1 Compound Symbols: c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_1, c7_1, c8_1, c9_1, c10_1, c11_1, c12_1, c13_1, c14_1, c15_1, c16_1, c17_1, c18_1, c19_1, c20_1, c21_1, c22_1, c23_1, c24_1, c25_1, c26_1, c27_1, c28_1, c29_1, c30_1, c31_1, c32_1, c33_1, c34_1, c35_1, c36_1, c37_1, c38_1, c39_1, c40_1, c41_1, c42_1, c43_1, c44_1, c45_1, c46_1, c47_1, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_1, c59_1, c60_1, c61_1, c62_1, c63_1, c64_1, c65_1, c66_1, c67_1, c68_1, c69_1, c70_1, c71_1, c74_1, c75_1, c76_1 ---------------------------------------- (21) SIsEmptyProof (BOTH BOUNDS(ID, ID)) The set S is empty ---------------------------------------- (22) BOUNDS(1, 1) ---------------------------------------- (23) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (24) 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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (25) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (26) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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 ---------------------------------------- (27) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: active, U102, isNatKind, U103, isNat, U104, plus, x, U12, U13, U14, U15, U16, U22, U23, U32, U33, U34, U35, U36, U42, U62, U72, U82, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U102 < active isNatKind < active U103 < active isNat < active U104 < active plus < active x < active U12 < active U13 < active U14 < active U15 < active U16 < active U22 < active U23 < active U32 < active U33 < active U34 < active U35 < active U36 < active U42 < active U62 < active U72 < active U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U102 < proper isNatKind < proper U103 < proper isNat < proper U104 < proper plus < proper x < proper U12 < proper U13 < proper U14 < proper U15 < proper U16 < proper U22 < proper U23 < proper U32 < proper U33 < proper U34 < proper U35 < proper U36 < proper U42 < proper U62 < proper U72 < proper U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (28) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102, active, isNatKind, U103, isNat, U104, plus, x, U12, U13, U14, U15, U16, U22, U23, U32, U33, U34, U35, U36, U42, U62, U72, U82, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U102 < active isNatKind < active U103 < active isNat < active U104 < active plus < active x < active U12 < active U13 < active U14 < active U15 < active U16 < active U22 < active U23 < active U32 < active U33 < active U34 < active U35 < active U36 < active U42 < active U62 < active U72 < active U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U102 < proper isNatKind < proper U103 < proper isNat < proper U104 < proper plus < proper x < proper U12 < proper U13 < proper U14 < proper U15 < proper U16 < proper U22 < proper U23 < proper U32 < proper U33 < proper U34 < proper U35 < proper U36 < proper U42 < proper U62 < proper U72 < proper U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (29) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) Induction Base: U102(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) Induction Step: U102(gen_tt:mark:0':ok3_0(+(1, +(n5_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1) mark(U102(gen_tt:mark:0':ok3_0(+(1, n5_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). ---------------------------------------- (30) Complex Obligation (BEST) ---------------------------------------- (31) Obligation: Proved the lower bound n^1 for the following obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102, active, isNatKind, U103, isNat, U104, plus, x, U12, U13, U14, U15, U16, U22, U23, U32, U33, U34, U35, U36, U42, U62, U72, U82, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U102 < active isNatKind < active U103 < active isNat < active U104 < active plus < active x < active U12 < active U13 < active U14 < active U15 < active U16 < active U22 < active U23 < active U32 < active U33 < active U34 < active U35 < active U36 < active U42 < active U62 < active U72 < active U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U102 < proper isNatKind < proper U103 < proper isNat < proper U104 < proper plus < proper x < proper U12 < proper U13 < proper U14 < proper U15 < proper U16 < proper U22 < proper U23 < proper U32 < proper U33 < proper U34 < proper U35 < proper U36 < proper U42 < proper U62 < proper U72 < proper U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (32) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (33) BOUNDS(n^1, INF) ---------------------------------------- (34) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *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: isNatKind, active, U103, isNat, U104, plus, x, U12, U13, U14, U15, U16, U22, U23, U32, U33, U34, U35, U36, U42, U62, U72, U82, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: isNatKind < active U103 < active isNat < active U104 < active plus < active x < active U12 < active U13 < active U14 < active U15 < active U16 < active U22 < active U23 < active U32 < active U33 < active U34 < active U35 < active U36 < active U42 < active U62 < active U72 < active U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top isNatKind < proper U103 < proper isNat < proper U104 < proper plus < proper x < proper U12 < proper U13 < proper U14 < proper U15 < proper U16 < proper U22 < proper U23 < proper U32 < proper U33 < proper U34 < proper U35 < proper U36 < proper U42 < proper U62 < proper U72 < proper U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (35) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) Induction Base: U103(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) Induction Step: U103(gen_tt:mark:0':ok3_0(+(1, +(n5519_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1) mark(U103(gen_tt:mark:0':ok3_0(+(1, n5519_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). ---------------------------------------- (36) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_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, U104, plus, x, U12, U13, U14, U15, U16, U22, U23, U32, U33, U34, U35, U36, U42, U62, U72, U82, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: isNat < active U104 < active plus < active x < active U12 < active U13 < active U14 < active U15 < active U16 < active U22 < active U23 < active U32 < active U33 < active U34 < active U35 < active U36 < active U42 < active U62 < active U72 < active U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top isNat < proper U104 < proper plus < proper x < proper U12 < proper U13 < proper U14 < proper U15 < proper U16 < proper U22 < proper U23 < proper U32 < proper U33 < proper U34 < proper U35 < proper U36 < proper U42 < proper U62 < proper U72 < proper U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (37) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) Induction Base: U104(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) Induction Step: U104(gen_tt:mark:0':ok3_0(+(1, +(n11649_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1) mark(U104(gen_tt:mark:0':ok3_0(+(1, n11649_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). ---------------------------------------- (38) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_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, U12, U13, U14, U15, U16, U22, U23, U32, U33, U34, U35, U36, U42, U62, U72, U82, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: plus < active x < active U12 < active U13 < active U14 < active U15 < active U16 < active U22 < active U23 < active U32 < active U33 < active U34 < active U35 < active U36 < active U42 < active U62 < active U72 < active U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top plus < proper x < proper U12 < proper U13 < proper U14 < proper U15 < proper U16 < proper U22 < proper U23 < proper U32 < proper U33 < proper U34 < proper U35 < proper U36 < proper U42 < proper U62 < proper U72 < proper U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (39) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_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, +(n18370_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(plus(gen_tt:mark:0':ok3_0(+(1, n18370_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). ---------------------------------------- (40) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_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, U12, U13, U14, U15, U16, U22, U23, U32, U33, U34, U35, U36, U42, U62, U72, U82, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: x < active U12 < active U13 < active U14 < active U15 < active U16 < active U22 < active U23 < active U32 < active U33 < active U34 < active U35 < active U36 < active U42 < active U62 < active U72 < active U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top x < proper U12 < proper U13 < proper U14 < proper U15 < proper U16 < proper U22 < proper U23 < proper U32 < proper U33 < proper U34 < proper U35 < proper U36 < proper U42 < proper U62 < proper U72 < proper U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (41) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_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, +(n22960_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(x(gen_tt:mark:0':ok3_0(+(1, n22960_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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_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: U12, active, U13, U14, U15, U16, U22, U23, U32, U33, U34, U35, U36, U42, U62, U72, U82, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U12 < active U13 < active U14 < active U15 < active U16 < active U22 < active U23 < active U32 < active U33 < active U34 < active U35 < active U36 < active U42 < active U62 < active U72 < active U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U12 < proper U13 < proper U14 < proper U15 < proper U16 < proper U22 < proper U23 < proper U32 < proper U33 < proper U34 < proper U35 < proper U36 < proper U42 < proper U62 < proper U72 < proper U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (43) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) Induction Base: U12(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) Induction Step: U12(gen_tt:mark:0':ok3_0(+(1, +(n27856_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1) mark(U12(gen_tt:mark:0':ok3_0(+(1, n27856_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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_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: U13, active, U14, U15, U16, U22, U23, U32, U33, U34, U35, U36, U42, U62, U72, U82, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U13 < active U14 < active U15 < active U16 < active U22 < active U23 < active U32 < active U33 < active U34 < active U35 < active U36 < active U42 < active U62 < active U72 < active U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U13 < proper U14 < proper U15 < proper U16 < proper U22 < proper U23 < proper U32 < proper U33 < proper U34 < proper U35 < proper U36 < proper U42 < proper U62 < proper U72 < proper U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (45) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) Induction Base: U13(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) Induction Step: U13(gen_tt:mark:0':ok3_0(+(1, +(n36110_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1) mark(U13(gen_tt:mark:0':ok3_0(+(1, n36110_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). ---------------------------------------- (46) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_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: U14, active, U15, U16, U22, U23, U32, U33, U34, U35, U36, U42, U62, U72, U82, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U14 < active U15 < active U16 < active U22 < active U23 < active U32 < active U33 < active U34 < active U35 < active U36 < active U42 < active U62 < active U72 < active U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U14 < proper U15 < proper U16 < proper U22 < proper U23 < proper U32 < proper U33 < proper U34 < proper U35 < proper U36 < proper U42 < proper U62 < proper U72 < proper U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (47) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) Induction Base: U14(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) Induction Step: U14(gen_tt:mark:0':ok3_0(+(1, +(n44973_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1) mark(U14(gen_tt:mark:0':ok3_0(+(1, n44973_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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_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: U15, active, U16, U22, U23, U32, U33, U34, U35, U36, U42, U62, U72, U82, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U15 < active U16 < active U22 < active U23 < active U32 < active U33 < active U34 < active U35 < active U36 < active U42 < active U62 < active U72 < active U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U15 < proper U16 < proper U22 < proper U23 < proper U32 < proper U33 < proper U34 < proper U35 < proper U36 < proper U42 < proper U62 < proper U72 < proper U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (49) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) Induction Base: U15(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b)) Induction Step: U15(gen_tt:mark:0':ok3_0(+(1, +(n54445_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(U15(gen_tt:mark:0':ok3_0(+(1, n54445_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). ---------------------------------------- (50) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_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: U16, active, U22, U23, U32, U33, U34, U35, U36, U42, U62, U72, U82, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U16 < active U22 < active U23 < active U32 < active U33 < active U34 < active U35 < active U36 < active U42 < active U62 < active U72 < active U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U16 < proper U22 < proper U23 < proper U32 < proper U33 < proper U34 < proper U35 < proper U36 < proper U42 < proper U62 < proper U72 < proper U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (51) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) Induction Base: U16(gen_tt:mark:0':ok3_0(+(1, 0))) Induction Step: U16(gen_tt:mark:0':ok3_0(+(1, +(n60660_0, 1)))) ->_R^Omega(1) mark(U16(gen_tt:mark:0':ok3_0(+(1, n60660_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). ---------------------------------------- (52) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_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, U23, U32, U33, U34, U35, U36, U42, U62, U72, U82, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U22 < active U23 < active U32 < active U33 < active U34 < active U35 < active U36 < active U42 < active U62 < active U72 < active U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U22 < proper U23 < proper U32 < proper U33 < proper U34 < proper U35 < proper U36 < proper U42 < proper U62 < proper U72 < proper U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (53) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) Induction Base: U22(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b)) Induction Step: U22(gen_tt:mark:0':ok3_0(+(1, +(n63159_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(U22(gen_tt:mark:0':ok3_0(+(1, n63159_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). ---------------------------------------- (54) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_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: U23, active, U32, U33, U34, U35, U36, U42, U62, U72, U82, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U23 < active U32 < active U33 < active U34 < active U35 < active U36 < active U42 < active U62 < active U72 < active U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U23 < proper U32 < proper U33 < proper U34 < proper U35 < proper U36 < proper U42 < proper U62 < proper U72 < proper U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (55) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) Induction Base: U23(gen_tt:mark:0':ok3_0(+(1, 0))) Induction Step: U23(gen_tt:mark:0':ok3_0(+(1, +(n69888_0, 1)))) ->_R^Omega(1) mark(U23(gen_tt:mark:0':ok3_0(+(1, n69888_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). ---------------------------------------- (56) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_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, U34, U35, U36, U42, U62, U72, U82, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U32 < active U33 < active U34 < active U35 < active U36 < active U42 < active U62 < active U72 < active U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U32 < proper U33 < proper U34 < proper U35 < proper U36 < proper U42 < proper U62 < proper U72 < proper U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (57) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) Induction Base: U32(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) Induction Step: U32(gen_tt:mark:0':ok3_0(+(1, +(n72638_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1) mark(U32(gen_tt:mark:0':ok3_0(+(1, n72638_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). ---------------------------------------- (58) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_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, U34, U35, U36, U42, U62, U72, U82, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U33 < active U34 < active U35 < active U36 < active U42 < active U62 < active U72 < active U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U33 < proper U34 < proper U35 < proper U36 < proper U42 < proper U62 < proper U72 < proper U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (59) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_0) Induction Base: U33(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) Induction Step: U33(gen_tt:mark:0':ok3_0(+(1, +(n84273_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1) mark(U33(gen_tt:mark:0':ok3_0(+(1, n84273_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). ---------------------------------------- (60) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_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: U34, active, U35, U36, U42, U62, U72, U82, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U34 < active U35 < active U36 < active U42 < active U62 < active U72 < active U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U34 < proper U35 < proper U36 < proper U42 < proper U62 < proper U72 < proper U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (61) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U34(gen_tt:mark:0':ok3_0(+(1, n96517_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n96517_0) Induction Base: U34(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) Induction Step: U34(gen_tt:mark:0':ok3_0(+(1, +(n96517_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1) mark(U34(gen_tt:mark:0':ok3_0(+(1, n96517_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). ---------------------------------------- (62) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_0) U34(gen_tt:mark:0':ok3_0(+(1, n96517_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n96517_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: U35, active, U36, U42, U62, U72, U82, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U35 < active U36 < active U42 < active U62 < active U72 < active U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U35 < proper U36 < proper U42 < proper U62 < proper U72 < proper U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (63) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U35(gen_tt:mark:0':ok3_0(+(1, n109370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n109370_0) Induction Base: U35(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b)) Induction Step: U35(gen_tt:mark:0':ok3_0(+(1, +(n109370_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(U35(gen_tt:mark:0':ok3_0(+(1, n109370_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). ---------------------------------------- (64) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_0) U34(gen_tt:mark:0':ok3_0(+(1, n96517_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n96517_0) U35(gen_tt:mark:0':ok3_0(+(1, n109370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n109370_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: U36, active, U42, U62, U72, U82, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U36 < active U42 < active U62 < active U72 < active U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U36 < proper U42 < proper U62 < proper U72 < proper U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (65) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U36(gen_tt:mark:0':ok3_0(+(1, n117825_0))) -> *4_0, rt in Omega(n117825_0) Induction Base: U36(gen_tt:mark:0':ok3_0(+(1, 0))) Induction Step: U36(gen_tt:mark:0':ok3_0(+(1, +(n117825_0, 1)))) ->_R^Omega(1) mark(U36(gen_tt:mark:0':ok3_0(+(1, n117825_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). ---------------------------------------- (66) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_0) U34(gen_tt:mark:0':ok3_0(+(1, n96517_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n96517_0) U35(gen_tt:mark:0':ok3_0(+(1, n109370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n109370_0) U36(gen_tt:mark:0':ok3_0(+(1, n117825_0))) -> *4_0, rt in Omega(n117825_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: U42, active, U62, U72, U82, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U42 < active U62 < active U72 < active U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U42 < proper U62 < proper U72 < proper U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (67) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U42(gen_tt:mark:0':ok3_0(+(1, n121423_0))) -> *4_0, rt in Omega(n121423_0) Induction Base: U42(gen_tt:mark:0':ok3_0(+(1, 0))) Induction Step: U42(gen_tt:mark:0':ok3_0(+(1, +(n121423_0, 1)))) ->_R^Omega(1) mark(U42(gen_tt:mark:0':ok3_0(+(1, n121423_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). ---------------------------------------- (68) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_0) U34(gen_tt:mark:0':ok3_0(+(1, n96517_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n96517_0) U35(gen_tt:mark:0':ok3_0(+(1, n109370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n109370_0) U36(gen_tt:mark:0':ok3_0(+(1, n117825_0))) -> *4_0, rt in Omega(n117825_0) U42(gen_tt:mark:0':ok3_0(+(1, n121423_0))) -> *4_0, rt in Omega(n121423_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: U62, active, U72, U82, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U62 < active U72 < active U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U62 < proper U72 < proper U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (69) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U62(gen_tt:mark:0':ok3_0(+(1, n125122_0))) -> *4_0, rt in Omega(n125122_0) Induction Base: U62(gen_tt:mark:0':ok3_0(+(1, 0))) Induction Step: U62(gen_tt:mark:0':ok3_0(+(1, +(n125122_0, 1)))) ->_R^Omega(1) mark(U62(gen_tt:mark:0':ok3_0(+(1, n125122_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). ---------------------------------------- (70) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_0) U34(gen_tt:mark:0':ok3_0(+(1, n96517_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n96517_0) U35(gen_tt:mark:0':ok3_0(+(1, n109370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n109370_0) U36(gen_tt:mark:0':ok3_0(+(1, n117825_0))) -> *4_0, rt in Omega(n117825_0) U42(gen_tt:mark:0':ok3_0(+(1, n121423_0))) -> *4_0, rt in Omega(n121423_0) U62(gen_tt:mark:0':ok3_0(+(1, n125122_0))) -> *4_0, rt in Omega(n125122_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: U72, active, U82, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U72 < active U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U72 < proper U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (71) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U72(gen_tt:mark:0':ok3_0(+(1, n128922_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n128922_0) Induction Base: U72(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b)) Induction Step: U72(gen_tt:mark:0':ok3_0(+(1, +(n128922_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(U72(gen_tt:mark:0':ok3_0(+(1, n128922_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). ---------------------------------------- (72) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_0) U34(gen_tt:mark:0':ok3_0(+(1, n96517_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n96517_0) U35(gen_tt:mark:0':ok3_0(+(1, n109370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n109370_0) U36(gen_tt:mark:0':ok3_0(+(1, n117825_0))) -> *4_0, rt in Omega(n117825_0) U42(gen_tt:mark:0':ok3_0(+(1, n121423_0))) -> *4_0, rt in Omega(n121423_0) U62(gen_tt:mark:0':ok3_0(+(1, n125122_0))) -> *4_0, rt in Omega(n125122_0) U72(gen_tt:mark:0':ok3_0(+(1, n128922_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n128922_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: U82, active, U83, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U82 < active U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U82 < proper U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (73) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U82(gen_tt:mark:0':ok3_0(+(1, n138307_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n138307_0) Induction Base: U82(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) Induction Step: U82(gen_tt:mark:0':ok3_0(+(1, +(n138307_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1) mark(U82(gen_tt:mark:0':ok3_0(+(1, n138307_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). ---------------------------------------- (74) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_0) U34(gen_tt:mark:0':ok3_0(+(1, n96517_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n96517_0) U35(gen_tt:mark:0':ok3_0(+(1, n109370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n109370_0) U36(gen_tt:mark:0':ok3_0(+(1, n117825_0))) -> *4_0, rt in Omega(n117825_0) U42(gen_tt:mark:0':ok3_0(+(1, n121423_0))) -> *4_0, rt in Omega(n121423_0) U62(gen_tt:mark:0':ok3_0(+(1, n125122_0))) -> *4_0, rt in Omega(n125122_0) U72(gen_tt:mark:0':ok3_0(+(1, n128922_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n128922_0) U82(gen_tt:mark:0':ok3_0(+(1, n138307_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n138307_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: U83, active, U84, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U83 < active U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U83 < proper U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (75) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U83(gen_tt:mark:0':ok3_0(+(1, n153638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n153638_0) Induction Base: U83(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) Induction Step: U83(gen_tt:mark:0':ok3_0(+(1, +(n153638_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1) mark(U83(gen_tt:mark:0':ok3_0(+(1, n153638_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). ---------------------------------------- (76) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_0) U34(gen_tt:mark:0':ok3_0(+(1, n96517_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n96517_0) U35(gen_tt:mark:0':ok3_0(+(1, n109370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n109370_0) U36(gen_tt:mark:0':ok3_0(+(1, n117825_0))) -> *4_0, rt in Omega(n117825_0) U42(gen_tt:mark:0':ok3_0(+(1, n121423_0))) -> *4_0, rt in Omega(n121423_0) U62(gen_tt:mark:0':ok3_0(+(1, n125122_0))) -> *4_0, rt in Omega(n125122_0) U72(gen_tt:mark:0':ok3_0(+(1, n128922_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n128922_0) U82(gen_tt:mark:0':ok3_0(+(1, n138307_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n138307_0) U83(gen_tt:mark:0':ok3_0(+(1, n153638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n153638_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: U84, active, s, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U84 < active s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U84 < proper s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (77) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U84(gen_tt:mark:0':ok3_0(+(1, n169578_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n169578_0) Induction Base: U84(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) Induction Step: U84(gen_tt:mark:0':ok3_0(+(1, +(n169578_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1) mark(U84(gen_tt:mark:0':ok3_0(+(1, n169578_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). ---------------------------------------- (78) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_0) U34(gen_tt:mark:0':ok3_0(+(1, n96517_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n96517_0) U35(gen_tt:mark:0':ok3_0(+(1, n109370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n109370_0) U36(gen_tt:mark:0':ok3_0(+(1, n117825_0))) -> *4_0, rt in Omega(n117825_0) U42(gen_tt:mark:0':ok3_0(+(1, n121423_0))) -> *4_0, rt in Omega(n121423_0) U62(gen_tt:mark:0':ok3_0(+(1, n125122_0))) -> *4_0, rt in Omega(n125122_0) U72(gen_tt:mark:0':ok3_0(+(1, n128922_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n128922_0) U82(gen_tt:mark:0':ok3_0(+(1, n138307_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n138307_0) U83(gen_tt:mark:0':ok3_0(+(1, n153638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n153638_0) U84(gen_tt:mark:0':ok3_0(+(1, n169578_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n169578_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, U92, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: s < active U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top s < proper U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (79) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: s(gen_tt:mark:0':ok3_0(+(1, n186127_0))) -> *4_0, rt in Omega(n186127_0) Induction Base: s(gen_tt:mark:0':ok3_0(+(1, 0))) Induction Step: s(gen_tt:mark:0':ok3_0(+(1, +(n186127_0, 1)))) ->_R^Omega(1) mark(s(gen_tt:mark:0':ok3_0(+(1, n186127_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). ---------------------------------------- (80) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_0) U34(gen_tt:mark:0':ok3_0(+(1, n96517_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n96517_0) U35(gen_tt:mark:0':ok3_0(+(1, n109370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n109370_0) U36(gen_tt:mark:0':ok3_0(+(1, n117825_0))) -> *4_0, rt in Omega(n117825_0) U42(gen_tt:mark:0':ok3_0(+(1, n121423_0))) -> *4_0, rt in Omega(n121423_0) U62(gen_tt:mark:0':ok3_0(+(1, n125122_0))) -> *4_0, rt in Omega(n125122_0) U72(gen_tt:mark:0':ok3_0(+(1, n128922_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n128922_0) U82(gen_tt:mark:0':ok3_0(+(1, n138307_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n138307_0) U83(gen_tt:mark:0':ok3_0(+(1, n153638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n153638_0) U84(gen_tt:mark:0':ok3_0(+(1, n169578_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n169578_0) s(gen_tt:mark:0':ok3_0(+(1, n186127_0))) -> *4_0, rt in Omega(n186127_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: U92, active, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U92 < active U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U92 < proper U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (81) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U92(gen_tt:mark:0':ok3_0(+(1, n190775_0))) -> *4_0, rt in Omega(n190775_0) Induction Base: U92(gen_tt:mark:0':ok3_0(+(1, 0))) Induction Step: U92(gen_tt:mark:0':ok3_0(+(1, +(n190775_0, 1)))) ->_R^Omega(1) mark(U92(gen_tt:mark:0':ok3_0(+(1, n190775_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). ---------------------------------------- (82) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_0) U34(gen_tt:mark:0':ok3_0(+(1, n96517_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n96517_0) U35(gen_tt:mark:0':ok3_0(+(1, n109370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n109370_0) U36(gen_tt:mark:0':ok3_0(+(1, n117825_0))) -> *4_0, rt in Omega(n117825_0) U42(gen_tt:mark:0':ok3_0(+(1, n121423_0))) -> *4_0, rt in Omega(n121423_0) U62(gen_tt:mark:0':ok3_0(+(1, n125122_0))) -> *4_0, rt in Omega(n125122_0) U72(gen_tt:mark:0':ok3_0(+(1, n128922_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n128922_0) U82(gen_tt:mark:0':ok3_0(+(1, n138307_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n138307_0) U83(gen_tt:mark:0':ok3_0(+(1, n153638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n153638_0) U84(gen_tt:mark:0':ok3_0(+(1, n169578_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n169578_0) s(gen_tt:mark:0':ok3_0(+(1, n186127_0))) -> *4_0, rt in Omega(n186127_0) U92(gen_tt:mark:0':ok3_0(+(1, n190775_0))) -> *4_0, rt in Omega(n190775_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, U21, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U11 < active U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U11 < proper U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (83) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U11(gen_tt:mark:0':ok3_0(+(1, n195524_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n195524_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, +(n195524_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, n195524_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). ---------------------------------------- (84) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_0) U34(gen_tt:mark:0':ok3_0(+(1, n96517_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n96517_0) U35(gen_tt:mark:0':ok3_0(+(1, n109370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n109370_0) U36(gen_tt:mark:0':ok3_0(+(1, n117825_0))) -> *4_0, rt in Omega(n117825_0) U42(gen_tt:mark:0':ok3_0(+(1, n121423_0))) -> *4_0, rt in Omega(n121423_0) U62(gen_tt:mark:0':ok3_0(+(1, n125122_0))) -> *4_0, rt in Omega(n125122_0) U72(gen_tt:mark:0':ok3_0(+(1, n128922_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n128922_0) U82(gen_tt:mark:0':ok3_0(+(1, n138307_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n138307_0) U83(gen_tt:mark:0':ok3_0(+(1, n153638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n153638_0) U84(gen_tt:mark:0':ok3_0(+(1, n169578_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n169578_0) s(gen_tt:mark:0':ok3_0(+(1, n186127_0))) -> *4_0, rt in Omega(n186127_0) U92(gen_tt:mark:0':ok3_0(+(1, n190775_0))) -> *4_0, rt in Omega(n190775_0) U11(gen_tt:mark:0':ok3_0(+(1, n195524_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n195524_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: U21, active, U31, U41, U51, U61, U71, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U21 < active U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U21 < proper U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (85) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U21(gen_tt:mark:0':ok3_0(+(1, n213312_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n213312_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, +(n213312_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(U21(gen_tt:mark:0':ok3_0(+(1, n213312_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). ---------------------------------------- (86) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_0) U34(gen_tt:mark:0':ok3_0(+(1, n96517_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n96517_0) U35(gen_tt:mark:0':ok3_0(+(1, n109370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n109370_0) U36(gen_tt:mark:0':ok3_0(+(1, n117825_0))) -> *4_0, rt in Omega(n117825_0) U42(gen_tt:mark:0':ok3_0(+(1, n121423_0))) -> *4_0, rt in Omega(n121423_0) U62(gen_tt:mark:0':ok3_0(+(1, n125122_0))) -> *4_0, rt in Omega(n125122_0) U72(gen_tt:mark:0':ok3_0(+(1, n128922_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n128922_0) U82(gen_tt:mark:0':ok3_0(+(1, n138307_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n138307_0) U83(gen_tt:mark:0':ok3_0(+(1, n153638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n153638_0) U84(gen_tt:mark:0':ok3_0(+(1, n169578_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n169578_0) s(gen_tt:mark:0':ok3_0(+(1, n186127_0))) -> *4_0, rt in Omega(n186127_0) U92(gen_tt:mark:0':ok3_0(+(1, n190775_0))) -> *4_0, rt in Omega(n190775_0) U11(gen_tt:mark:0':ok3_0(+(1, n195524_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n195524_0) U21(gen_tt:mark:0':ok3_0(+(1, n213312_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n213312_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, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U31 < active U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U31 < proper U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (87) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U31(gen_tt:mark:0':ok3_0(+(1, n225035_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n225035_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, +(n225035_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, n225035_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). ---------------------------------------- (88) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_0) U34(gen_tt:mark:0':ok3_0(+(1, n96517_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n96517_0) U35(gen_tt:mark:0':ok3_0(+(1, n109370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n109370_0) U36(gen_tt:mark:0':ok3_0(+(1, n117825_0))) -> *4_0, rt in Omega(n117825_0) U42(gen_tt:mark:0':ok3_0(+(1, n121423_0))) -> *4_0, rt in Omega(n121423_0) U62(gen_tt:mark:0':ok3_0(+(1, n125122_0))) -> *4_0, rt in Omega(n125122_0) U72(gen_tt:mark:0':ok3_0(+(1, n128922_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n128922_0) U82(gen_tt:mark:0':ok3_0(+(1, n138307_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n138307_0) U83(gen_tt:mark:0':ok3_0(+(1, n153638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n153638_0) U84(gen_tt:mark:0':ok3_0(+(1, n169578_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n169578_0) s(gen_tt:mark:0':ok3_0(+(1, n186127_0))) -> *4_0, rt in Omega(n186127_0) U92(gen_tt:mark:0':ok3_0(+(1, n190775_0))) -> *4_0, rt in Omega(n190775_0) U11(gen_tt:mark:0':ok3_0(+(1, n195524_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n195524_0) U21(gen_tt:mark:0':ok3_0(+(1, n213312_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n213312_0) U31(gen_tt:mark:0':ok3_0(+(1, n225035_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n225035_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, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U41 < active U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U41 < proper U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (89) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U41(gen_tt:mark:0':ok3_0(+(1, n243894_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n243894_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, +(n243894_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(U41(gen_tt:mark:0':ok3_0(+(1, n243894_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). ---------------------------------------- (90) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_0) U34(gen_tt:mark:0':ok3_0(+(1, n96517_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n96517_0) U35(gen_tt:mark:0':ok3_0(+(1, n109370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n109370_0) U36(gen_tt:mark:0':ok3_0(+(1, n117825_0))) -> *4_0, rt in Omega(n117825_0) U42(gen_tt:mark:0':ok3_0(+(1, n121423_0))) -> *4_0, rt in Omega(n121423_0) U62(gen_tt:mark:0':ok3_0(+(1, n125122_0))) -> *4_0, rt in Omega(n125122_0) U72(gen_tt:mark:0':ok3_0(+(1, n128922_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n128922_0) U82(gen_tt:mark:0':ok3_0(+(1, n138307_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n138307_0) U83(gen_tt:mark:0':ok3_0(+(1, n153638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n153638_0) U84(gen_tt:mark:0':ok3_0(+(1, n169578_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n169578_0) s(gen_tt:mark:0':ok3_0(+(1, n186127_0))) -> *4_0, rt in Omega(n186127_0) U92(gen_tt:mark:0':ok3_0(+(1, n190775_0))) -> *4_0, rt in Omega(n190775_0) U11(gen_tt:mark:0':ok3_0(+(1, n195524_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n195524_0) U21(gen_tt:mark:0':ok3_0(+(1, n213312_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n213312_0) U31(gen_tt:mark:0':ok3_0(+(1, n225035_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n225035_0) U41(gen_tt:mark:0':ok3_0(+(1, n243894_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n243894_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, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U51 < active U61 < active U71 < active U81 < active U91 < active U101 < active active < top U51 < proper U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (91) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U51(gen_tt:mark:0':ok3_0(+(1, n256327_0))) -> *4_0, rt in Omega(n256327_0) Induction Base: U51(gen_tt:mark:0':ok3_0(+(1, 0))) Induction Step: U51(gen_tt:mark:0':ok3_0(+(1, +(n256327_0, 1)))) ->_R^Omega(1) mark(U51(gen_tt:mark:0':ok3_0(+(1, n256327_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). ---------------------------------------- (92) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_0) U34(gen_tt:mark:0':ok3_0(+(1, n96517_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n96517_0) U35(gen_tt:mark:0':ok3_0(+(1, n109370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n109370_0) U36(gen_tt:mark:0':ok3_0(+(1, n117825_0))) -> *4_0, rt in Omega(n117825_0) U42(gen_tt:mark:0':ok3_0(+(1, n121423_0))) -> *4_0, rt in Omega(n121423_0) U62(gen_tt:mark:0':ok3_0(+(1, n125122_0))) -> *4_0, rt in Omega(n125122_0) U72(gen_tt:mark:0':ok3_0(+(1, n128922_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n128922_0) U82(gen_tt:mark:0':ok3_0(+(1, n138307_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n138307_0) U83(gen_tt:mark:0':ok3_0(+(1, n153638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n153638_0) U84(gen_tt:mark:0':ok3_0(+(1, n169578_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n169578_0) s(gen_tt:mark:0':ok3_0(+(1, n186127_0))) -> *4_0, rt in Omega(n186127_0) U92(gen_tt:mark:0':ok3_0(+(1, n190775_0))) -> *4_0, rt in Omega(n190775_0) U11(gen_tt:mark:0':ok3_0(+(1, n195524_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n195524_0) U21(gen_tt:mark:0':ok3_0(+(1, n213312_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n213312_0) U31(gen_tt:mark:0':ok3_0(+(1, n225035_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n225035_0) U41(gen_tt:mark:0':ok3_0(+(1, n243894_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n243894_0) U51(gen_tt:mark:0':ok3_0(+(1, n256327_0))) -> *4_0, rt in Omega(n256327_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, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U61 < active U71 < active U81 < active U91 < active U101 < active active < top U61 < proper U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (93) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U61(gen_tt:mark:0':ok3_0(+(1, n261875_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n261875_0) Induction Base: U61(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b)) Induction Step: U61(gen_tt:mark:0':ok3_0(+(1, +(n261875_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(U61(gen_tt:mark:0':ok3_0(+(1, n261875_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). ---------------------------------------- (94) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_0) U34(gen_tt:mark:0':ok3_0(+(1, n96517_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n96517_0) U35(gen_tt:mark:0':ok3_0(+(1, n109370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n109370_0) U36(gen_tt:mark:0':ok3_0(+(1, n117825_0))) -> *4_0, rt in Omega(n117825_0) U42(gen_tt:mark:0':ok3_0(+(1, n121423_0))) -> *4_0, rt in Omega(n121423_0) U62(gen_tt:mark:0':ok3_0(+(1, n125122_0))) -> *4_0, rt in Omega(n125122_0) U72(gen_tt:mark:0':ok3_0(+(1, n128922_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n128922_0) U82(gen_tt:mark:0':ok3_0(+(1, n138307_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n138307_0) U83(gen_tt:mark:0':ok3_0(+(1, n153638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n153638_0) U84(gen_tt:mark:0':ok3_0(+(1, n169578_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n169578_0) s(gen_tt:mark:0':ok3_0(+(1, n186127_0))) -> *4_0, rt in Omega(n186127_0) U92(gen_tt:mark:0':ok3_0(+(1, n190775_0))) -> *4_0, rt in Omega(n190775_0) U11(gen_tt:mark:0':ok3_0(+(1, n195524_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n195524_0) U21(gen_tt:mark:0':ok3_0(+(1, n213312_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n213312_0) U31(gen_tt:mark:0':ok3_0(+(1, n225035_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n225035_0) U41(gen_tt:mark:0':ok3_0(+(1, n243894_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n243894_0) U51(gen_tt:mark:0':ok3_0(+(1, n256327_0))) -> *4_0, rt in Omega(n256327_0) U61(gen_tt:mark:0':ok3_0(+(1, n261875_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n261875_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, U81, U91, U101, proper, top They will be analysed ascendingly in the following order: U71 < active U81 < active U91 < active U101 < active active < top U71 < proper U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (95) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U71(gen_tt:mark:0':ok3_0(+(1, n274822_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n274822_0) Induction Base: U71(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b)) Induction Step: U71(gen_tt:mark:0':ok3_0(+(1, +(n274822_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(U71(gen_tt:mark:0':ok3_0(+(1, n274822_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). ---------------------------------------- (96) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_0) U34(gen_tt:mark:0':ok3_0(+(1, n96517_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n96517_0) U35(gen_tt:mark:0':ok3_0(+(1, n109370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n109370_0) U36(gen_tt:mark:0':ok3_0(+(1, n117825_0))) -> *4_0, rt in Omega(n117825_0) U42(gen_tt:mark:0':ok3_0(+(1, n121423_0))) -> *4_0, rt in Omega(n121423_0) U62(gen_tt:mark:0':ok3_0(+(1, n125122_0))) -> *4_0, rt in Omega(n125122_0) U72(gen_tt:mark:0':ok3_0(+(1, n128922_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n128922_0) U82(gen_tt:mark:0':ok3_0(+(1, n138307_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n138307_0) U83(gen_tt:mark:0':ok3_0(+(1, n153638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n153638_0) U84(gen_tt:mark:0':ok3_0(+(1, n169578_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n169578_0) s(gen_tt:mark:0':ok3_0(+(1, n186127_0))) -> *4_0, rt in Omega(n186127_0) U92(gen_tt:mark:0':ok3_0(+(1, n190775_0))) -> *4_0, rt in Omega(n190775_0) U11(gen_tt:mark:0':ok3_0(+(1, n195524_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n195524_0) U21(gen_tt:mark:0':ok3_0(+(1, n213312_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n213312_0) U31(gen_tt:mark:0':ok3_0(+(1, n225035_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n225035_0) U41(gen_tt:mark:0':ok3_0(+(1, n243894_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n243894_0) U51(gen_tt:mark:0':ok3_0(+(1, n256327_0))) -> *4_0, rt in Omega(n256327_0) U61(gen_tt:mark:0':ok3_0(+(1, n261875_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n261875_0) U71(gen_tt:mark:0':ok3_0(+(1, n274822_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n274822_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: U81, active, U91, U101, proper, top They will be analysed ascendingly in the following order: U81 < active U91 < active U101 < active active < top U81 < proper U91 < proper U101 < proper proper < top ---------------------------------------- (97) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U81(gen_tt:mark:0':ok3_0(+(1, n288075_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n288075_0) Induction Base: U81(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) Induction Step: U81(gen_tt:mark:0':ok3_0(+(1, +(n288075_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1) mark(U81(gen_tt:mark:0':ok3_0(+(1, n288075_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). ---------------------------------------- (98) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_0) U34(gen_tt:mark:0':ok3_0(+(1, n96517_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n96517_0) U35(gen_tt:mark:0':ok3_0(+(1, n109370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n109370_0) U36(gen_tt:mark:0':ok3_0(+(1, n117825_0))) -> *4_0, rt in Omega(n117825_0) U42(gen_tt:mark:0':ok3_0(+(1, n121423_0))) -> *4_0, rt in Omega(n121423_0) U62(gen_tt:mark:0':ok3_0(+(1, n125122_0))) -> *4_0, rt in Omega(n125122_0) U72(gen_tt:mark:0':ok3_0(+(1, n128922_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n128922_0) U82(gen_tt:mark:0':ok3_0(+(1, n138307_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n138307_0) U83(gen_tt:mark:0':ok3_0(+(1, n153638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n153638_0) U84(gen_tt:mark:0':ok3_0(+(1, n169578_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n169578_0) s(gen_tt:mark:0':ok3_0(+(1, n186127_0))) -> *4_0, rt in Omega(n186127_0) U92(gen_tt:mark:0':ok3_0(+(1, n190775_0))) -> *4_0, rt in Omega(n190775_0) U11(gen_tt:mark:0':ok3_0(+(1, n195524_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n195524_0) U21(gen_tt:mark:0':ok3_0(+(1, n213312_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n213312_0) U31(gen_tt:mark:0':ok3_0(+(1, n225035_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n225035_0) U41(gen_tt:mark:0':ok3_0(+(1, n243894_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n243894_0) U51(gen_tt:mark:0':ok3_0(+(1, n256327_0))) -> *4_0, rt in Omega(n256327_0) U61(gen_tt:mark:0':ok3_0(+(1, n261875_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n261875_0) U71(gen_tt:mark:0':ok3_0(+(1, n274822_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n274822_0) U81(gen_tt:mark:0':ok3_0(+(1, n288075_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n288075_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: U91, active, U101, proper, top They will be analysed ascendingly in the following order: U91 < active U101 < active active < top U91 < proper U101 < proper proper < top ---------------------------------------- (99) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U91(gen_tt:mark:0':ok3_0(+(1, n309244_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n309244_0) Induction Base: U91(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b)) Induction Step: U91(gen_tt:mark:0':ok3_0(+(1, +(n309244_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(U91(gen_tt:mark:0':ok3_0(+(1, n309244_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). ---------------------------------------- (100) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_0) U34(gen_tt:mark:0':ok3_0(+(1, n96517_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n96517_0) U35(gen_tt:mark:0':ok3_0(+(1, n109370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n109370_0) U36(gen_tt:mark:0':ok3_0(+(1, n117825_0))) -> *4_0, rt in Omega(n117825_0) U42(gen_tt:mark:0':ok3_0(+(1, n121423_0))) -> *4_0, rt in Omega(n121423_0) U62(gen_tt:mark:0':ok3_0(+(1, n125122_0))) -> *4_0, rt in Omega(n125122_0) U72(gen_tt:mark:0':ok3_0(+(1, n128922_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n128922_0) U82(gen_tt:mark:0':ok3_0(+(1, n138307_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n138307_0) U83(gen_tt:mark:0':ok3_0(+(1, n153638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n153638_0) U84(gen_tt:mark:0':ok3_0(+(1, n169578_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n169578_0) s(gen_tt:mark:0':ok3_0(+(1, n186127_0))) -> *4_0, rt in Omega(n186127_0) U92(gen_tt:mark:0':ok3_0(+(1, n190775_0))) -> *4_0, rt in Omega(n190775_0) U11(gen_tt:mark:0':ok3_0(+(1, n195524_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n195524_0) U21(gen_tt:mark:0':ok3_0(+(1, n213312_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n213312_0) U31(gen_tt:mark:0':ok3_0(+(1, n225035_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n225035_0) U41(gen_tt:mark:0':ok3_0(+(1, n243894_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n243894_0) U51(gen_tt:mark:0':ok3_0(+(1, n256327_0))) -> *4_0, rt in Omega(n256327_0) U61(gen_tt:mark:0':ok3_0(+(1, n261875_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n261875_0) U71(gen_tt:mark:0':ok3_0(+(1, n274822_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n274822_0) U81(gen_tt:mark:0':ok3_0(+(1, n288075_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n288075_0) U91(gen_tt:mark:0':ok3_0(+(1, n309244_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n309244_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: U101, active, proper, top They will be analysed ascendingly in the following order: U101 < active active < top U101 < proper proper < top ---------------------------------------- (101) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U101(gen_tt:mark:0':ok3_0(+(1, n323207_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n323207_0) Induction Base: U101(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) Induction Step: U101(gen_tt:mark:0':ok3_0(+(1, +(n323207_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1) mark(U101(gen_tt:mark:0':ok3_0(+(1, n323207_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). ---------------------------------------- (102) Obligation: TRS: Rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0') active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0')) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0')) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(x(X1, X2)) -> x(active(X1), X2) active(x(X1, X2)) -> x(X1, active(X2)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) active(U15(X1, X2)) -> U15(active(X1), X2) active(U16(X)) -> U16(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X1, X2)) -> U22(active(X1), X2) active(U23(X)) -> U23(active(X)) active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) active(U34(X1, X2, X3)) -> U34(active(X1), X2, X3) active(U35(X1, X2)) -> U35(active(X1), X2) active(U36(X)) -> U36(active(X)) active(U41(X1, X2)) -> U41(active(X1), X2) active(U42(X)) -> U42(active(X)) active(U51(X)) -> U51(active(X)) active(U61(X1, X2)) -> U61(active(X1), X2) active(U62(X)) -> U62(active(X)) active(U71(X1, X2)) -> U71(active(X1), X2) active(U72(X1, X2)) -> U72(active(X1), X2) active(U81(X1, X2, X3)) -> U81(active(X1), X2, X3) active(U82(X1, X2, X3)) -> U82(active(X1), X2, X3) active(U83(X1, X2, X3)) -> U83(active(X1), X2, X3) active(U84(X1, X2, X3)) -> U84(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(U91(X1, X2)) -> U91(active(X1), X2) active(U92(X)) -> U92(active(X)) U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) x(mark(X1), X2) -> mark(x(X1, X2)) x(X1, mark(X2)) -> mark(x(X1, X2)) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) U15(mark(X1), X2) -> mark(U15(X1, X2)) U16(mark(X)) -> mark(U16(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X1), X2) -> mark(U22(X1, X2)) U23(mark(X)) -> mark(U23(X)) U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) U34(mark(X1), X2, X3) -> mark(U34(X1, X2, X3)) U35(mark(X1), X2) -> mark(U35(X1, X2)) U36(mark(X)) -> mark(U36(X)) U41(mark(X1), X2) -> mark(U41(X1, X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X)) -> mark(U51(X)) U61(mark(X1), X2) -> mark(U61(X1, X2)) U62(mark(X)) -> mark(U62(X)) U71(mark(X1), X2) -> mark(U71(X1, X2)) U72(mark(X1), X2) -> mark(U72(X1, X2)) U81(mark(X1), X2, X3) -> mark(U81(X1, X2, X3)) U82(mark(X1), X2, X3) -> mark(U82(X1, X2, X3)) U83(mark(X1), X2, X3) -> mark(U83(X1, X2, X3)) U84(mark(X1), X2, X3) -> mark(U84(X1, X2, X3)) s(mark(X)) -> mark(s(X)) U91(mark(X1), X2) -> mark(U91(X1, X2)) U92(mark(X)) -> mark(U92(X)) proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) proper(isNat(X)) -> isNat(proper(X)) proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(x(X1, X2)) -> x(proper(X1), proper(X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) proper(U15(X1, X2)) -> U15(proper(X1), proper(X2)) proper(U16(X)) -> U16(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) proper(U23(X)) -> U23(proper(X)) proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) proper(U34(X1, X2, X3)) -> U34(proper(X1), proper(X2), proper(X3)) proper(U35(X1, X2)) -> U35(proper(X1), proper(X2)) proper(U36(X)) -> U36(proper(X)) proper(U41(X1, X2)) -> U41(proper(X1), proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(U51(X)) -> U51(proper(X)) proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) proper(U62(X)) -> U62(proper(X)) proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) proper(U81(X1, X2, X3)) -> U81(proper(X1), proper(X2), proper(X3)) proper(U82(X1, X2, X3)) -> U82(proper(X1), proper(X2), proper(X3)) proper(U83(X1, X2, X3)) -> U83(proper(X1), proper(X2), proper(X3)) proper(U84(X1, X2, X3)) -> U84(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) proper(U92(X)) -> U92(proper(X)) proper(0') -> ok(0') U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) isNatKind(ok(X)) -> ok(isNatKind(X)) U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) isNat(ok(X)) -> ok(isNat(X)) U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) x(ok(X1), ok(X2)) -> ok(x(X1, X2)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) U15(ok(X1), ok(X2)) -> ok(U15(X1, X2)) U16(ok(X)) -> ok(U16(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) U23(ok(X)) -> ok(U23(X)) U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) U34(ok(X1), ok(X2), ok(X3)) -> ok(U34(X1, X2, X3)) U35(ok(X1), ok(X2)) -> ok(U35(X1, X2)) U36(ok(X)) -> ok(U36(X)) U41(ok(X1), ok(X2)) -> ok(U41(X1, X2)) U42(ok(X)) -> ok(U42(X)) U51(ok(X)) -> ok(U51(X)) U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) U62(ok(X)) -> ok(U62(X)) U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) U81(ok(X1), ok(X2), ok(X3)) -> ok(U81(X1, X2, X3)) U82(ok(X1), ok(X2), ok(X3)) -> ok(U82(X1, X2, X3)) U83(ok(X1), ok(X2), ok(X3)) -> ok(U83(X1, X2, X3)) U84(ok(X1), ok(X2), ok(X3)) -> ok(U84(X1, X2, X3)) s(ok(X)) -> ok(s(X)) U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) U92(ok(X)) -> ok(U92(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U101 :: 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 U102 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok U103 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U104 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok plus :: 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 U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U16 :: 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 -> tt:mark:0':ok U23 :: 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 -> tt:mark:0':ok U33 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U34 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U35 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U36 :: tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U42 :: tt:mark:0':ok -> tt:mark:0':ok U51 :: tt:mark:0':ok -> tt:mark:0':ok U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U62 :: tt:mark:0':ok -> tt:mark:0':ok U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U72 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U81 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U82 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U83 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U84 :: 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 U91 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U92 :: tt:mark:0':ok -> tt:mark:0':ok 0' :: 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: U102(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0) U103(gen_tt:mark:0':ok3_0(+(1, n5519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5519_0) U104(gen_tt:mark:0':ok3_0(+(1, n11649_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n11649_0) plus(gen_tt:mark:0':ok3_0(+(1, n18370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n18370_0) x(gen_tt:mark:0':ok3_0(+(1, n22960_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n22960_0) U12(gen_tt:mark:0':ok3_0(+(1, n27856_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n27856_0) U13(gen_tt:mark:0':ok3_0(+(1, n36110_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36110_0) U14(gen_tt:mark:0':ok3_0(+(1, n44973_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44973_0) U15(gen_tt:mark:0':ok3_0(+(1, n54445_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n54445_0) U16(gen_tt:mark:0':ok3_0(+(1, n60660_0))) -> *4_0, rt in Omega(n60660_0) U22(gen_tt:mark:0':ok3_0(+(1, n63159_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n63159_0) U23(gen_tt:mark:0':ok3_0(+(1, n69888_0))) -> *4_0, rt in Omega(n69888_0) U32(gen_tt:mark:0':ok3_0(+(1, n72638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n72638_0) U33(gen_tt:mark:0':ok3_0(+(1, n84273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n84273_0) U34(gen_tt:mark:0':ok3_0(+(1, n96517_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n96517_0) U35(gen_tt:mark:0':ok3_0(+(1, n109370_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n109370_0) U36(gen_tt:mark:0':ok3_0(+(1, n117825_0))) -> *4_0, rt in Omega(n117825_0) U42(gen_tt:mark:0':ok3_0(+(1, n121423_0))) -> *4_0, rt in Omega(n121423_0) U62(gen_tt:mark:0':ok3_0(+(1, n125122_0))) -> *4_0, rt in Omega(n125122_0) U72(gen_tt:mark:0':ok3_0(+(1, n128922_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n128922_0) U82(gen_tt:mark:0':ok3_0(+(1, n138307_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n138307_0) U83(gen_tt:mark:0':ok3_0(+(1, n153638_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n153638_0) U84(gen_tt:mark:0':ok3_0(+(1, n169578_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n169578_0) s(gen_tt:mark:0':ok3_0(+(1, n186127_0))) -> *4_0, rt in Omega(n186127_0) U92(gen_tt:mark:0':ok3_0(+(1, n190775_0))) -> *4_0, rt in Omega(n190775_0) U11(gen_tt:mark:0':ok3_0(+(1, n195524_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n195524_0) U21(gen_tt:mark:0':ok3_0(+(1, n213312_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n213312_0) U31(gen_tt:mark:0':ok3_0(+(1, n225035_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n225035_0) U41(gen_tt:mark:0':ok3_0(+(1, n243894_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n243894_0) U51(gen_tt:mark:0':ok3_0(+(1, n256327_0))) -> *4_0, rt in Omega(n256327_0) U61(gen_tt:mark:0':ok3_0(+(1, n261875_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n261875_0) U71(gen_tt:mark:0':ok3_0(+(1, n274822_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n274822_0) U81(gen_tt:mark:0':ok3_0(+(1, n288075_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n288075_0) U91(gen_tt:mark:0':ok3_0(+(1, n309244_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n309244_0) U101(gen_tt:mark:0':ok3_0(+(1, n323207_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n323207_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