/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) CpxTrsMatchBoundsTAProof [FINISHED, 70 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), 388 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), 132 ms] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 128 ms] (22) typed CpxTrs (23) RewriteLemmaProof [LOWER BOUND(ID), 125 ms] (24) typed CpxTrs (25) RewriteLemmaProof [LOWER BOUND(ID), 126 ms] (26) 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, N)) -> mark(N) active(U21(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(and(isNat(V1), isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N, 0)) -> mark(U11(isNat(N), N)) active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U21(X1, X2, X3)) -> U21(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) -> mark(U11(X1, X2)) U21(mark(X1), X2, X3) -> mark(U21(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)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U21(X1, X2, X3)) -> U21(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(isNat(X)) -> isNat(proper(X)) proper(0) -> ok(0) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(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)) isNat(ok(X)) -> ok(isNat(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, N)) -> mark(N) active(U21(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(and(isNat(V1), isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N, 0)) -> mark(U11(isNat(N), N)) active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U21(X1, X2, X3)) -> U21(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)) -> U11(proper(X1), proper(X2)) proper(U21(X1, X2, X3)) -> U21(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(isNat(X)) -> isNat(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) -> mark(U11(X1, X2)) U21(mark(X1), X2, X3) -> mark(U21(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(U11(X1, X2)) U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(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)) isNat(ok(X)) -> ok(isNat(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) -> mark(U11(X1, X2)) U21(mark(X1), X2, X3) -> mark(U21(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(U11(X1, X2)) U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(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)) isNat(ok(X)) -> ok(isNat(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] transitions: mark0(0) -> 0 tt0() -> 0 ok0(0) -> 0 00() -> 0 active0(0) -> 0 U110(0, 0) -> 1 U210(0, 0, 0) -> 2 s0(0) -> 3 plus0(0, 0) -> 4 and0(0, 0) -> 5 proper0(0) -> 6 isNat0(0) -> 7 top0(0) -> 8 U111(0, 0) -> 9 mark1(9) -> 1 U211(0, 0, 0) -> 10 mark1(10) -> 2 s1(0) -> 11 mark1(11) -> 3 plus1(0, 0) -> 12 mark1(12) -> 4 and1(0, 0) -> 13 mark1(13) -> 5 tt1() -> 14 ok1(14) -> 6 01() -> 15 ok1(15) -> 6 U111(0, 0) -> 16 ok1(16) -> 1 U211(0, 0, 0) -> 17 ok1(17) -> 2 s1(0) -> 18 ok1(18) -> 3 plus1(0, 0) -> 19 ok1(19) -> 4 and1(0, 0) -> 20 ok1(20) -> 5 isNat1(0) -> 21 ok1(21) -> 7 proper1(0) -> 22 top1(22) -> 8 active1(0) -> 23 top1(23) -> 8 mark1(9) -> 9 mark1(9) -> 16 mark1(10) -> 10 mark1(10) -> 17 mark1(11) -> 11 mark1(11) -> 18 mark1(12) -> 12 mark1(12) -> 19 mark1(13) -> 13 mark1(13) -> 20 ok1(14) -> 22 ok1(15) -> 22 ok1(16) -> 9 ok1(16) -> 16 ok1(17) -> 10 ok1(17) -> 17 ok1(18) -> 11 ok1(18) -> 18 ok1(19) -> 12 ok1(19) -> 19 ok1(20) -> 13 ok1(20) -> 20 ok1(21) -> 21 active2(14) -> 24 top2(24) -> 8 active2(15) -> 24 ---------------------------------------- (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, N)) -> mark(N) active(U21(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(and(isNat(V1), isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N, 0')) -> mark(U11(isNat(N), N)) active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U21(X1, X2, X3)) -> U21(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) -> mark(U11(X1, X2)) U21(mark(X1), X2, X3) -> mark(U21(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)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U21(X1, X2, X3)) -> U21(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(isNat(X)) -> isNat(proper(X)) proper(0') -> ok(0') U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(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)) isNat(ok(X)) -> ok(isNat(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, N)) -> mark(N) active(U21(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(and(isNat(V1), isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N, 0')) -> mark(U11(isNat(N), N)) active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U21(X1, X2, X3)) -> U21(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) -> mark(U11(X1, X2)) U21(mark(X1), X2, X3) -> mark(U21(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)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U21(X1, X2, X3)) -> U21(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(isNat(X)) -> isNat(proper(X)) proper(0') -> ok(0') U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(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)) isNat(ok(X)) -> ok(isNat(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 :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U21 :: 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 isNat :: 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 ---------------------------------------- (11) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: active, s, plus, and, isNat, U11, U21, proper, top They will be analysed ascendingly in the following order: s < active plus < active and < active isNat < active U11 < active U21 < active active < top s < proper plus < proper and < proper isNat < proper U11 < proper U21 < proper proper < top ---------------------------------------- (12) Obligation: TRS: Rules: active(U11(tt, N)) -> mark(N) active(U21(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(and(isNat(V1), isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N, 0')) -> mark(U11(isNat(N), N)) active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U21(X1, X2, X3)) -> U21(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) -> mark(U11(X1, X2)) U21(mark(X1), X2, X3) -> mark(U21(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)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U21(X1, X2, X3)) -> U21(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(isNat(X)) -> isNat(proper(X)) proper(0') -> ok(0') U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(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)) isNat(ok(X)) -> ok(isNat(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 :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U21 :: 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 isNat :: 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: s, active, plus, and, isNat, U11, U21, proper, top They will be analysed ascendingly in the following order: s < active plus < active and < active isNat < active U11 < active U21 < active active < top s < proper plus < proper and < proper isNat < proper U11 < proper U21 < proper proper < top ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: s(gen_tt:mark:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) Induction Base: s(gen_tt:mark:0':ok3_0(+(1, 0))) Induction Step: s(gen_tt:mark:0':ok3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1) mark(s(gen_tt:mark:0':ok3_0(+(1, n5_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). ---------------------------------------- (14) Complex Obligation (BEST) ---------------------------------------- (15) Obligation: Proved the lower bound n^1 for the following obligation: TRS: Rules: active(U11(tt, N)) -> mark(N) active(U21(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(and(isNat(V1), isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N, 0')) -> mark(U11(isNat(N), N)) active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U21(X1, X2, X3)) -> U21(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) -> mark(U11(X1, X2)) U21(mark(X1), X2, X3) -> mark(U21(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)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U21(X1, X2, X3)) -> U21(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(isNat(X)) -> isNat(proper(X)) proper(0') -> ok(0') U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(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)) isNat(ok(X)) -> ok(isNat(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 :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U21 :: 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 isNat :: 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: s, active, plus, and, isNat, U11, U21, proper, top They will be analysed ascendingly in the following order: s < active plus < active and < active isNat < active U11 < active U21 < active active < top s < proper plus < proper and < proper isNat < proper U11 < proper U21 < proper proper < top ---------------------------------------- (16) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (17) BOUNDS(n^1, INF) ---------------------------------------- (18) Obligation: TRS: Rules: active(U11(tt, N)) -> mark(N) active(U21(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(and(isNat(V1), isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N, 0')) -> mark(U11(isNat(N), N)) active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U21(X1, X2, X3)) -> U21(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) -> mark(U11(X1, X2)) U21(mark(X1), X2, X3) -> mark(U21(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)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U21(X1, X2, X3)) -> U21(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(isNat(X)) -> isNat(proper(X)) proper(0') -> ok(0') U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(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)) isNat(ok(X)) -> ok(isNat(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 :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U21 :: 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 isNat :: 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: s(gen_tt:mark:0':ok3_0(+(1, n5_0))) -> *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: plus, active, and, isNat, U11, U21, proper, top They will be analysed ascendingly in the following order: plus < active and < active isNat < active U11 < active U21 < active active < top plus < proper and < proper isNat < proper U11 < proper U21 < proper proper < top ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: plus(gen_tt:mark:0':ok3_0(+(1, n380_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n380_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, +(n380_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(plus(gen_tt:mark:0':ok3_0(+(1, n380_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). ---------------------------------------- (20) Obligation: TRS: Rules: active(U11(tt, N)) -> mark(N) active(U21(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(and(isNat(V1), isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N, 0')) -> mark(U11(isNat(N), N)) active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U21(X1, X2, X3)) -> U21(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) -> mark(U11(X1, X2)) U21(mark(X1), X2, X3) -> mark(U21(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)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U21(X1, X2, X3)) -> U21(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(isNat(X)) -> isNat(proper(X)) proper(0') -> ok(0') U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(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)) isNat(ok(X)) -> ok(isNat(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 :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U21 :: 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 isNat :: 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: s(gen_tt:mark:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) plus(gen_tt:mark:0':ok3_0(+(1, n380_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n380_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, isNat, U11, U21, proper, top They will be analysed ascendingly in the following order: and < active isNat < active U11 < active U21 < active active < top and < proper isNat < proper U11 < proper U21 < proper proper < top ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: and(gen_tt:mark:0':ok3_0(+(1, n1726_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n1726_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, +(n1726_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(and(gen_tt:mark:0':ok3_0(+(1, n1726_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). ---------------------------------------- (22) Obligation: TRS: Rules: active(U11(tt, N)) -> mark(N) active(U21(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(and(isNat(V1), isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N, 0')) -> mark(U11(isNat(N), N)) active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U21(X1, X2, X3)) -> U21(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) -> mark(U11(X1, X2)) U21(mark(X1), X2, X3) -> mark(U21(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)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U21(X1, X2, X3)) -> U21(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(isNat(X)) -> isNat(proper(X)) proper(0') -> ok(0') U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(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)) isNat(ok(X)) -> ok(isNat(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 :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U21 :: 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 isNat :: 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: s(gen_tt:mark:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) plus(gen_tt:mark:0':ok3_0(+(1, n380_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n380_0) and(gen_tt:mark:0':ok3_0(+(1, n1726_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n1726_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, U11, U21, proper, top They will be analysed ascendingly in the following order: isNat < active U11 < active U21 < active active < top isNat < proper U11 < proper U21 < proper proper < top ---------------------------------------- (23) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U11(gen_tt:mark:0':ok3_0(+(1, n3196_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3196_0) Induction Base: U11(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b)) Induction Step: U11(gen_tt:mark:0':ok3_0(+(1, +(n3196_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1) mark(U11(gen_tt:mark:0':ok3_0(+(1, n3196_0)), gen_tt:mark:0':ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (24) Obligation: TRS: Rules: active(U11(tt, N)) -> mark(N) active(U21(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(and(isNat(V1), isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N, 0')) -> mark(U11(isNat(N), N)) active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U21(X1, X2, X3)) -> U21(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) -> mark(U11(X1, X2)) U21(mark(X1), X2, X3) -> mark(U21(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)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U21(X1, X2, X3)) -> U21(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(isNat(X)) -> isNat(proper(X)) proper(0') -> ok(0') U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(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)) isNat(ok(X)) -> ok(isNat(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 :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U21 :: 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 isNat :: 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: s(gen_tt:mark:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) plus(gen_tt:mark:0':ok3_0(+(1, n380_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n380_0) and(gen_tt:mark:0':ok3_0(+(1, n1726_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n1726_0) U11(gen_tt:mark:0':ok3_0(+(1, n3196_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3196_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, proper, top They will be analysed ascendingly in the following order: U21 < active active < top U21 < proper proper < top ---------------------------------------- (25) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U21(gen_tt:mark:0':ok3_0(+(1, n4955_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n4955_0) Induction Base: U21(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) Induction Step: U21(gen_tt:mark:0':ok3_0(+(1, +(n4955_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1) mark(U21(gen_tt:mark:0':ok3_0(+(1, n4955_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). ---------------------------------------- (26) Obligation: TRS: Rules: active(U11(tt, N)) -> mark(N) active(U21(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(and(isNat(V1), isNat(V2))) active(isNat(s(V1))) -> mark(isNat(V1)) active(plus(N, 0')) -> mark(U11(isNat(N), N)) active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N)) active(U11(X1, X2)) -> U11(active(X1), X2) active(U21(X1, X2, X3)) -> U21(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) -> mark(U11(X1, X2)) U21(mark(X1), X2, X3) -> mark(U21(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)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(U21(X1, X2, X3)) -> U21(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(isNat(X)) -> isNat(proper(X)) proper(0') -> ok(0') U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(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)) isNat(ok(X)) -> ok(isNat(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 :: tt:mark:0':ok mark :: tt:mark:0':ok -> tt:mark:0':ok U21 :: 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 isNat :: 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: s(gen_tt:mark:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0) plus(gen_tt:mark:0':ok3_0(+(1, n380_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n380_0) and(gen_tt:mark:0':ok3_0(+(1, n1726_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n1726_0) U11(gen_tt:mark:0':ok3_0(+(1, n3196_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3196_0) U21(gen_tt:mark:0':ok3_0(+(1, n4955_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n4955_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