/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) CpxTrsMatchBoundsTAProof [FINISHED, 216 ms] (6) BOUNDS(1, n^1) (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTRS (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) typed CpxTrs (11) OrderProof [LOWER BOUND(ID), 0 ms] (12) typed CpxTrs (13) RewriteLemmaProof [LOWER BOUND(ID), 514 ms] (14) BEST (15) proven lower bound (16) LowerBoundPropagationProof [FINISHED, 0 ms] (17) BOUNDS(n^1, INF) (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 106 ms] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 112 ms] (22) typed CpxTrs (23) RewriteLemmaProof [LOWER BOUND(ID), 35 ms] (24) typed CpxTrs (25) RewriteLemmaProof [LOWER BOUND(ID), 93 ms] (26) typed CpxTrs (27) RewriteLemmaProof [LOWER BOUND(ID), 157 ms] (28) typed CpxTrs (29) RewriteLemmaProof [LOWER BOUND(ID), 113 ms] (30) typed CpxTrs (31) RewriteLemmaProof [LOWER BOUND(ID), 116 ms] (32) typed CpxTrs (33) RewriteLemmaProof [LOWER BOUND(ID), 105 ms] (34) typed CpxTrs (35) RewriteLemmaProof [LOWER BOUND(ID), 150 ms] (36) typed CpxTrs ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(s(plus(N, M))) active(and(tt, X)) -> mark(X) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N, 0)) -> mark(U31(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(0) -> ok(0) proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) The following defined symbols can occur below the 0th argument of top: proper, active The following defined symbols can occur below the 0th argument of proper: proper, active The following defined symbols can occur below the 0th argument of active: proper, active Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(s(plus(N, M))) active(and(tt, X)) -> mark(X) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N, 0)) -> mark(U31(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNatKind(X)) -> isNatKind(proper(X)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(tt) -> ok(tt) proper(0) -> ok(0) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(tt) -> ok(tt) proper(0) -> ok(0) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (5) CpxTrsMatchBoundsTAProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] transitions: mark0(0) -> 0 tt0() -> 0 ok0(0) -> 0 00() -> 0 active0(0) -> 0 U110(0, 0, 0) -> 1 U120(0, 0) -> 2 U130(0) -> 3 U210(0, 0) -> 4 U220(0) -> 5 U310(0, 0) -> 6 U410(0, 0, 0) -> 7 s0(0) -> 8 plus0(0, 0) -> 9 and0(0, 0) -> 10 proper0(0) -> 11 isNat0(0) -> 12 isNatKind0(0) -> 13 top0(0) -> 14 U111(0, 0, 0) -> 15 mark1(15) -> 1 U121(0, 0) -> 16 mark1(16) -> 2 U131(0) -> 17 mark1(17) -> 3 U211(0, 0) -> 18 mark1(18) -> 4 U221(0) -> 19 mark1(19) -> 5 U311(0, 0) -> 20 mark1(20) -> 6 U411(0, 0, 0) -> 21 mark1(21) -> 7 s1(0) -> 22 mark1(22) -> 8 plus1(0, 0) -> 23 mark1(23) -> 9 and1(0, 0) -> 24 mark1(24) -> 10 tt1() -> 25 ok1(25) -> 11 01() -> 26 ok1(26) -> 11 U111(0, 0, 0) -> 27 ok1(27) -> 1 U121(0, 0) -> 28 ok1(28) -> 2 isNat1(0) -> 29 ok1(29) -> 12 U131(0) -> 30 ok1(30) -> 3 U211(0, 0) -> 31 ok1(31) -> 4 U221(0) -> 32 ok1(32) -> 5 U311(0, 0) -> 33 ok1(33) -> 6 U411(0, 0, 0) -> 34 ok1(34) -> 7 s1(0) -> 35 ok1(35) -> 8 plus1(0, 0) -> 36 ok1(36) -> 9 and1(0, 0) -> 37 ok1(37) -> 10 isNatKind1(0) -> 38 ok1(38) -> 13 proper1(0) -> 39 top1(39) -> 14 active1(0) -> 40 top1(40) -> 14 mark1(15) -> 15 mark1(15) -> 27 mark1(16) -> 16 mark1(16) -> 28 mark1(17) -> 17 mark1(17) -> 30 mark1(18) -> 18 mark1(18) -> 31 mark1(19) -> 19 mark1(19) -> 32 mark1(20) -> 20 mark1(20) -> 33 mark1(21) -> 21 mark1(21) -> 34 mark1(22) -> 22 mark1(22) -> 35 mark1(23) -> 23 mark1(23) -> 36 mark1(24) -> 24 mark1(24) -> 37 ok1(25) -> 39 ok1(26) -> 39 ok1(27) -> 15 ok1(27) -> 27 ok1(28) -> 16 ok1(28) -> 28 ok1(29) -> 29 ok1(30) -> 17 ok1(30) -> 30 ok1(31) -> 18 ok1(31) -> 31 ok1(32) -> 19 ok1(32) -> 32 ok1(33) -> 20 ok1(33) -> 33 ok1(34) -> 21 ok1(34) -> 34 ok1(35) -> 22 ok1(35) -> 35 ok1(36) -> 23 ok1(36) -> 36 ok1(37) -> 24 ok1(37) -> 37 ok1(38) -> 38 active2(25) -> 41 top2(41) -> 14 active2(26) -> 41 ---------------------------------------- (6) BOUNDS(1, n^1) ---------------------------------------- (7) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (8) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(s(plus(N, M))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N, 0')) -> mark(U31(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(0') -> ok(0') proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: TRS: Rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(s(plus(N, M))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N, 0')) -> mark(U31(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(0') -> ok(0') proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok ---------------------------------------- (11) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: active, U12, isNat, U13, U22, s, plus, U11, and, isNatKind, U21, U31, U41, proper, top They will be analysed ascendingly in the following order: U12 < active isNat < active U13 < active U22 < active s < active plus < active U11 < active and < active isNatKind < active U21 < active U31 < active U41 < active active < top U12 < proper isNat < proper U13 < proper U22 < proper s < proper plus < proper U11 < proper and < proper isNatKind < proper U21 < proper U31 < proper U41 < proper proper < top ---------------------------------------- (12) Obligation: TRS: Rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(s(plus(N, M))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N, 0')) -> mark(U31(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(0') -> ok(0') proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: U12, active, isNat, U13, U22, s, plus, U11, and, isNatKind, U21, U31, U41, proper, top They will be analysed ascendingly in the following order: U12 < active isNat < active U13 < active U22 < active s < active plus < active U11 < active and < active isNatKind < active U21 < active U31 < active U41 < active active < top U12 < proper isNat < proper U13 < proper U22 < proper s < proper plus < proper U11 < proper and < proper isNatKind < proper U21 < proper U31 < proper U41 < proper proper < top ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) Induction Base: U12(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b)) Induction Step: U12(gen_tt:mark:0':ok3_0(+(1, +(n5_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (14) Complex Obligation (BEST) ---------------------------------------- (15) Obligation: Proved the lower bound n^1 for the following obligation: TRS: Rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(s(plus(N, M))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N, 0')) -> mark(U31(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(0') -> ok(0') proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: U12, active, isNat, U13, U22, s, plus, U11, and, isNatKind, U21, U31, U41, proper, top They will be analysed ascendingly in the following order: U12 < active isNat < active U13 < active U22 < active s < active plus < active U11 < active and < active isNatKind < active U21 < active U31 < active U41 < active active < top U12 < proper isNat < proper U13 < proper U22 < proper s < proper plus < proper U11 < proper and < proper isNatKind < proper U21 < proper U31 < proper U41 < proper proper < top ---------------------------------------- (16) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (17) BOUNDS(n^1, INF) ---------------------------------------- (18) Obligation: TRS: Rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(s(plus(N, M))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N, 0')) -> mark(U31(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(0') -> ok(0') proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: isNat, active, U13, U22, s, plus, U11, and, isNatKind, U21, U31, U41, proper, top They will be analysed ascendingly in the following order: isNat < active U13 < active U22 < active s < active plus < active U11 < active and < active isNatKind < active U21 < active U31 < active U41 < active active < top isNat < proper U13 < proper U22 < proper s < proper plus < proper U11 < proper and < proper isNatKind < proper U21 < proper U31 < proper U41 < proper proper < top ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U13(gen_tt:mark:0':ok3_0(+(1, n1373_0))) -> *4_0, rt in Omega(n1373_0) Induction Base: U13(gen_tt:mark:0':ok3_0(+(1, 0))) Induction Step: U13(gen_tt:mark:0':ok3_0(+(1, +(n1373_0, 1)))) ->_R^Omega(1) mark(U13(gen_tt:mark:0':ok3_0(+(1, n1373_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (20) Obligation: TRS: Rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(s(plus(N, M))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N, 0')) -> mark(U31(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(0') -> ok(0') proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1373_0))) -> *4_0, rt in Omega(n1373_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, s, plus, U11, and, isNatKind, U21, U31, U41, proper, top They will be analysed ascendingly in the following order: U22 < active s < active plus < active U11 < active and < active isNatKind < active U21 < active U31 < active U41 < active active < top U22 < proper s < proper plus < proper U11 < proper and < proper isNatKind < proper U21 < proper U31 < proper U41 < proper proper < top ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U22(gen_tt:mark:0':ok3_0(+(1, n1988_0))) -> *4_0, rt in Omega(n1988_0) Induction Base: U22(gen_tt:mark:0':ok3_0(+(1, 0))) Induction Step: U22(gen_tt:mark:0':ok3_0(+(1, +(n1988_0, 1)))) ->_R^Omega(1) mark(U22(gen_tt:mark:0':ok3_0(+(1, n1988_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (22) Obligation: TRS: Rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(s(plus(N, M))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N, 0')) -> mark(U31(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(0') -> ok(0') proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1373_0))) -> *4_0, rt in Omega(n1373_0) U22(gen_tt:mark:0':ok3_0(+(1, n1988_0))) -> *4_0, rt in Omega(n1988_0) Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: s, active, plus, U11, and, isNatKind, U21, U31, U41, proper, top They will be analysed ascendingly in the following order: s < active plus < active U11 < active and < active isNatKind < active U21 < active U31 < active U41 < active active < top s < proper plus < proper U11 < proper and < proper isNatKind < proper U21 < proper U31 < proper U41 < proper proper < top ---------------------------------------- (23) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: s(gen_tt:mark:0':ok3_0(+(1, n2704_0))) -> *4_0, rt in Omega(n2704_0) Induction Base: s(gen_tt:mark:0':ok3_0(+(1, 0))) Induction Step: s(gen_tt:mark:0':ok3_0(+(1, +(n2704_0, 1)))) ->_R^Omega(1) mark(s(gen_tt:mark:0':ok3_0(+(1, n2704_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (24) Obligation: TRS: Rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(s(plus(N, M))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N, 0')) -> mark(U31(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(0') -> ok(0') proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1373_0))) -> *4_0, rt in Omega(n1373_0) U22(gen_tt:mark:0':ok3_0(+(1, n1988_0))) -> *4_0, rt in Omega(n1988_0) s(gen_tt:mark:0':ok3_0(+(1, n2704_0))) -> *4_0, rt in Omega(n2704_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, U11, and, isNatKind, U21, U31, U41, proper, top They will be analysed ascendingly in the following order: plus < active U11 < active and < active isNatKind < active U21 < active U31 < active U41 < active active < top plus < proper U11 < proper and < proper isNatKind < proper U21 < proper U31 < proper U41 < proper proper < top ---------------------------------------- (25) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: plus(gen_tt:mark:0':ok3_0(+(1, n3521_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3521_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, +(n3521_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(plus(gen_tt:mark:0':ok3_0(+(1, n3521_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). ---------------------------------------- (26) Obligation: TRS: Rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(s(plus(N, M))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N, 0')) -> mark(U31(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(0') -> ok(0') proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1373_0))) -> *4_0, rt in Omega(n1373_0) U22(gen_tt:mark:0':ok3_0(+(1, n1988_0))) -> *4_0, rt in Omega(n1988_0) s(gen_tt:mark:0':ok3_0(+(1, n2704_0))) -> *4_0, rt in Omega(n2704_0) plus(gen_tt:mark:0':ok3_0(+(1, n3521_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3521_0) Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: U11, active, and, isNatKind, U21, U31, U41, proper, top They will be analysed ascendingly in the following order: U11 < active and < active isNatKind < active U21 < active U31 < active U41 < active active < top U11 < proper and < proper isNatKind < proper U21 < proper U31 < proper U41 < proper proper < top ---------------------------------------- (27) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U11(gen_tt:mark:0':ok3_0(+(1, n6009_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n6009_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, +(n6009_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, n6009_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). ---------------------------------------- (28) Obligation: TRS: Rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(s(plus(N, M))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N, 0')) -> mark(U31(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(0') -> ok(0') proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1373_0))) -> *4_0, rt in Omega(n1373_0) U22(gen_tt:mark:0':ok3_0(+(1, n1988_0))) -> *4_0, rt in Omega(n1988_0) s(gen_tt:mark:0':ok3_0(+(1, n2704_0))) -> *4_0, rt in Omega(n2704_0) plus(gen_tt:mark:0':ok3_0(+(1, n3521_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3521_0) U11(gen_tt:mark:0':ok3_0(+(1, n6009_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n6009_0) Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: and, active, isNatKind, U21, U31, U41, proper, top They will be analysed ascendingly in the following order: and < active isNatKind < active U21 < active U31 < active U41 < active active < top and < proper isNatKind < proper U21 < proper U31 < proper U41 < proper proper < top ---------------------------------------- (29) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: and(gen_tt:mark:0':ok3_0(+(1, n10131_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n10131_0) Induction Base: and(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b)) Induction Step: and(gen_tt:mark:0':ok3_0(+(1, +(n10131_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(and(gen_tt:mark:0':ok3_0(+(1, n10131_0)), gen_tt:mark:0':ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (30) Obligation: TRS: Rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(s(plus(N, M))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N, 0')) -> mark(U31(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(0') -> ok(0') proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1373_0))) -> *4_0, rt in Omega(n1373_0) U22(gen_tt:mark:0':ok3_0(+(1, n1988_0))) -> *4_0, rt in Omega(n1988_0) s(gen_tt:mark:0':ok3_0(+(1, n2704_0))) -> *4_0, rt in Omega(n2704_0) plus(gen_tt:mark:0':ok3_0(+(1, n3521_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3521_0) U11(gen_tt:mark:0':ok3_0(+(1, n6009_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n6009_0) and(gen_tt:mark:0':ok3_0(+(1, n10131_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n10131_0) Generator Equations: gen_tt:mark:0':ok3_0(0) <=> tt gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x)) The following defined symbols remain to be analysed: isNatKind, active, U21, U31, U41, proper, top They will be analysed ascendingly in the following order: isNatKind < active U21 < active U31 < active U41 < active active < top isNatKind < proper U21 < proper U31 < proper U41 < proper proper < top ---------------------------------------- (31) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U21(gen_tt:mark:0':ok3_0(+(1, n13165_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n13165_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, +(n13165_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(U21(gen_tt:mark:0':ok3_0(+(1, n13165_0)), gen_tt:mark:0':ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (32) Obligation: TRS: Rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(s(plus(N, M))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N, 0')) -> mark(U31(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(0') -> ok(0') proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1373_0))) -> *4_0, rt in Omega(n1373_0) U22(gen_tt:mark:0':ok3_0(+(1, n1988_0))) -> *4_0, rt in Omega(n1988_0) s(gen_tt:mark:0':ok3_0(+(1, n2704_0))) -> *4_0, rt in Omega(n2704_0) plus(gen_tt:mark:0':ok3_0(+(1, n3521_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3521_0) U11(gen_tt:mark:0':ok3_0(+(1, n6009_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n6009_0) and(gen_tt:mark:0':ok3_0(+(1, n10131_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n10131_0) U21(gen_tt:mark:0':ok3_0(+(1, n13165_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n13165_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, proper, top They will be analysed ascendingly in the following order: U31 < active U41 < active active < top U31 < proper U41 < proper proper < top ---------------------------------------- (33) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U31(gen_tt:mark:0':ok3_0(+(1, n16470_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n16470_0) Induction Base: U31(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b)) Induction Step: U31(gen_tt:mark:0':ok3_0(+(1, +(n16470_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(U31(gen_tt:mark:0':ok3_0(+(1, n16470_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). ---------------------------------------- (34) Obligation: TRS: Rules: active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(s(plus(N, M))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N, 0')) -> mark(U31(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(0') -> ok(0') proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1373_0))) -> *4_0, rt in Omega(n1373_0) U22(gen_tt:mark:0':ok3_0(+(1, n1988_0))) -> *4_0, rt in Omega(n1988_0) s(gen_tt:mark:0':ok3_0(+(1, n2704_0))) -> *4_0, rt in Omega(n2704_0) plus(gen_tt:mark:0':ok3_0(+(1, n3521_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3521_0) U11(gen_tt:mark:0':ok3_0(+(1, n6009_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n6009_0) and(gen_tt:mark:0':ok3_0(+(1, n10131_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n10131_0) U21(gen_tt:mark:0':ok3_0(+(1, n13165_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n13165_0) U31(gen_tt:mark:0':ok3_0(+(1, n16470_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n16470_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, proper, top They will be analysed ascendingly in the following order: U41 < active active < top U41 < proper proper < top ---------------------------------------- (35) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U41(gen_tt:mark:0':ok3_0(+(1, n20081_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n20081_0) Induction Base: U41(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) Induction Step: U41(gen_tt:mark:0':ok3_0(+(1, +(n20081_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1) mark(U41(gen_tt:mark:0':ok3_0(+(1, n20081_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(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2)) active(U12(tt, V2)) -> mark(U13(isNat(V2))) active(U13(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNat(V1))) active(U22(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(s(plus(N, M))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNatKind(0')) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2))) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(plus(N, 0')) -> mark(U31(and(isNat(N), isNatKind(N)), N)) active(plus(N, s(M))) -> mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) active(U12(X1, X2)) -> U12(active(X1), X2) active(U13(X)) -> U13(active(X)) active(U21(X1, X2)) -> U21(active(X1), X2) active(U22(X)) -> U22(active(X)) active(U31(X1, X2)) -> U31(active(X1), X2) active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) active(s(X)) -> s(active(X)) active(plus(X1, X2)) -> plus(active(X1), X2) active(plus(X1, X2)) -> plus(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) U12(mark(X1), X2) -> mark(U12(X1, X2)) U13(mark(X)) -> mark(U13(X)) U21(mark(X1), X2) -> mark(U21(X1, X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1), X2) -> mark(U31(X1, X2)) U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) s(mark(X)) -> mark(s(X)) plus(mark(X1), X2) -> mark(plus(X1, X2)) plus(X1, mark(X2)) -> mark(plus(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) proper(tt) -> ok(tt) proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(U13(X)) -> U13(proper(X)) proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) proper(s(X)) -> s(proper(X)) proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(0') -> ok(0') proper(isNatKind(X)) -> isNatKind(proper(X)) U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) U13(ok(X)) -> ok(U13(X)) U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) U22(ok(X)) -> ok(U22(X)) U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) s(ok(X)) -> ok(s(X)) plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: tt:mark:0':ok -> tt:mark:0':ok U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok tt :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok isNat :: tt:mark:0':ok -> tt:mark:0':ok U13 :: tt:mark:0':ok -> tt:mark:0':ok U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U22 :: tt:mark:0':ok -> tt:mark:0':ok U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok s :: tt:mark:0':ok -> tt:mark:0':ok plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok 0' :: tt:mark:0':ok isNatKind :: tt:mark:0':ok -> tt:mark:0':ok proper :: tt:mark:0':ok -> tt:mark:0':ok ok :: tt:mark:0':ok -> tt:mark:0':ok top :: tt:mark:0':ok -> top hole_tt:mark:0':ok1_0 :: tt:mark:0':ok hole_top2_0 :: top gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok Lemmas: U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0) U13(gen_tt:mark:0':ok3_0(+(1, n1373_0))) -> *4_0, rt in Omega(n1373_0) U22(gen_tt:mark:0':ok3_0(+(1, n1988_0))) -> *4_0, rt in Omega(n1988_0) s(gen_tt:mark:0':ok3_0(+(1, n2704_0))) -> *4_0, rt in Omega(n2704_0) plus(gen_tt:mark:0':ok3_0(+(1, n3521_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3521_0) U11(gen_tt:mark:0':ok3_0(+(1, n6009_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n6009_0) and(gen_tt:mark:0':ok3_0(+(1, n10131_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n10131_0) U21(gen_tt:mark:0':ok3_0(+(1, n13165_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n13165_0) U31(gen_tt:mark:0':ok3_0(+(1, n16470_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n16470_0) U41(gen_tt:mark:0':ok3_0(+(1, n20081_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n20081_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