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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).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, 968 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), 455 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), 121 ms] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 114 ms] (22) typed CpxTrs (23) RewriteLemmaProof [LOWER BOUND(ID), 116 ms] (24) typed CpxTrs (25) RewriteLemmaProof [LOWER BOUND(ID), 109 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(zeros) -> mark(cons(0, zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(and(tt, X)) -> mark(X) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0) -> ok(0) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil) -> ok(nil) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(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 cons: active, proper, cons The following defined symbols can occur below the 1th argument of cons: active, proper, cons The following defined symbols can occur below the 0th argument of top: active, proper, cons The following defined symbols can occur below the 0th argument of proper: active, proper, cons The following defined symbols can occur below the 0th argument of active: active, proper, cons Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: active(U11(tt, L)) -> mark(s(length(L))) active(and(tt, X)) -> mark(X) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1, X2)) -> and(active(X1), X2) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(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: active(zeros) -> mark(cons(0, zeros)) active(cons(X1, X2)) -> cons(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0) -> ok(0) proper(tt) -> ok(tt) proper(nil) -> ok(nil) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(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: active(zeros) -> mark(cons(0, zeros)) active(cons(X1, X2)) -> cons(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0) -> ok(0) proper(tt) -> ok(tt) proper(nil) -> ok(nil) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(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 5. 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] transitions: zeros0() -> 0 mark0(0) -> 0 00() -> 0 ok0(0) -> 0 tt0() -> 0 nil0() -> 0 active0(0) -> 1 cons0(0, 0) -> 2 U110(0, 0) -> 3 s0(0) -> 4 length0(0) -> 5 and0(0, 0) -> 6 proper0(0) -> 7 isNat0(0) -> 8 isNatList0(0) -> 9 isNatIList0(0) -> 10 top0(0) -> 11 01() -> 13 zeros1() -> 14 cons1(13, 14) -> 12 mark1(12) -> 1 cons1(0, 0) -> 15 mark1(15) -> 2 U111(0, 0) -> 16 mark1(16) -> 3 s1(0) -> 17 mark1(17) -> 4 length1(0) -> 18 mark1(18) -> 5 and1(0, 0) -> 19 mark1(19) -> 6 zeros1() -> 20 ok1(20) -> 7 01() -> 21 ok1(21) -> 7 tt1() -> 22 ok1(22) -> 7 nil1() -> 23 ok1(23) -> 7 cons1(0, 0) -> 24 ok1(24) -> 2 U111(0, 0) -> 25 ok1(25) -> 3 s1(0) -> 26 ok1(26) -> 4 length1(0) -> 27 ok1(27) -> 5 and1(0, 0) -> 28 ok1(28) -> 6 isNat1(0) -> 29 ok1(29) -> 8 isNatList1(0) -> 30 ok1(30) -> 9 isNatIList1(0) -> 31 ok1(31) -> 10 proper1(0) -> 32 top1(32) -> 11 active1(0) -> 33 top1(33) -> 11 mark1(12) -> 33 mark1(15) -> 15 mark1(15) -> 24 mark1(16) -> 16 mark1(16) -> 25 mark1(17) -> 17 mark1(17) -> 26 mark1(18) -> 18 mark1(18) -> 27 mark1(19) -> 19 mark1(19) -> 28 ok1(20) -> 32 ok1(21) -> 32 ok1(22) -> 32 ok1(23) -> 32 ok1(24) -> 15 ok1(24) -> 24 ok1(25) -> 16 ok1(25) -> 25 ok1(26) -> 17 ok1(26) -> 26 ok1(27) -> 18 ok1(27) -> 27 ok1(28) -> 19 ok1(28) -> 28 ok1(29) -> 29 ok1(30) -> 30 ok1(31) -> 31 proper2(12) -> 34 top2(34) -> 11 active2(20) -> 35 top2(35) -> 11 active2(21) -> 35 active2(22) -> 35 active2(23) -> 35 02() -> 37 zeros2() -> 38 cons2(37, 38) -> 36 mark2(36) -> 35 proper2(13) -> 39 proper2(14) -> 40 cons2(39, 40) -> 34 zeros2() -> 41 ok2(41) -> 40 02() -> 42 ok2(42) -> 39 proper3(36) -> 43 top3(43) -> 11 proper3(37) -> 44 proper3(38) -> 45 cons3(44, 45) -> 43 cons3(42, 41) -> 46 ok3(46) -> 34 zeros3() -> 47 ok3(47) -> 45 03() -> 48 ok3(48) -> 44 active3(46) -> 49 top3(49) -> 11 cons4(48, 47) -> 50 ok4(50) -> 43 active4(42) -> 51 cons4(51, 41) -> 49 active4(50) -> 52 top4(52) -> 11 active5(48) -> 53 cons5(53, 47) -> 52 ---------------------------------------- (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(zeros) -> mark(cons(0', zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil) -> ok(nil) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(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(zeros) -> mark(cons(0', zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil) -> ok(nil) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok ---------------------------------------- (11) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: active, cons, s, length, isNatList, isNat, and, isNatIList, U11, proper, top They will be analysed ascendingly in the following order: cons < active s < active length < active isNatList < active isNat < active and < active isNatIList < active U11 < active active < top cons < proper s < proper length < proper isNatList < proper isNat < proper and < proper isNatIList < proper U11 < proper proper < top ---------------------------------------- (12) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil) -> ok(nil) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: cons, active, s, length, isNatList, isNat, and, isNatIList, U11, proper, top They will be analysed ascendingly in the following order: cons < active s < active length < active isNatList < active isNat < active and < active isNatIList < active U11 < active active < top cons < proper s < proper length < proper isNatList < proper isNat < proper and < proper isNatIList < proper U11 < proper proper < top ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) Induction Base: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) Induction Step: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n5_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1) mark(cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil: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(zeros) -> mark(cons(0', zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil) -> ok(nil) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: cons, active, s, length, isNatList, isNat, and, isNatIList, U11, proper, top They will be analysed ascendingly in the following order: cons < active s < active length < active isNatList < active isNat < active and < active isNatIList < active U11 < active active < top cons < proper s < proper length < proper isNatList < proper isNat < proper and < proper isNatIList < proper U11 < proper proper < top ---------------------------------------- (16) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (17) BOUNDS(n^1, INF) ---------------------------------------- (18) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil) -> ok(nil) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: s, active, length, isNatList, isNat, and, isNatIList, U11, proper, top They will be analysed ascendingly in the following order: s < active length < active isNatList < active isNat < active and < active isNatIList < active U11 < active active < top s < proper length < proper isNatList < proper isNat < proper and < proper isNatIList < proper U11 < proper proper < top ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1070_0))) -> *4_0, rt in Omega(n1070_0) Induction Base: s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0))) Induction Step: s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n1070_0, 1)))) ->_R^Omega(1) mark(s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1070_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(zeros) -> mark(cons(0', zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil) -> ok(nil) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1070_0))) -> *4_0, rt in Omega(n1070_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: length, active, isNatList, isNat, and, isNatIList, U11, proper, top They will be analysed ascendingly in the following order: length < active isNatList < active isNat < active and < active isNatIList < active U11 < active active < top length < proper isNatList < proper isNat < proper and < proper isNatIList < proper U11 < proper proper < top ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1622_0))) -> *4_0, rt in Omega(n1622_0) Induction Base: length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0))) Induction Step: length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n1622_0, 1)))) ->_R^Omega(1) mark(length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1622_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(zeros) -> mark(cons(0', zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil) -> ok(nil) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1070_0))) -> *4_0, rt in Omega(n1070_0) length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1622_0))) -> *4_0, rt in Omega(n1622_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: isNatList, active, isNat, and, isNatIList, U11, proper, top They will be analysed ascendingly in the following order: isNatList < active isNat < active and < active isNatIList < active U11 < active active < top isNatList < proper isNat < proper and < proper isNatIList < proper U11 < proper proper < top ---------------------------------------- (23) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2305_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n2305_0) Induction Base: and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) Induction Step: and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n2305_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1) mark(and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2305_0)), gen_zeros:0':mark:tt:nil: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(zeros) -> mark(cons(0', zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil) -> ok(nil) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1070_0))) -> *4_0, rt in Omega(n1070_0) length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1622_0))) -> *4_0, rt in Omega(n1622_0) and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2305_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n2305_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: isNatIList, active, U11, proper, top They will be analysed ascendingly in the following order: isNatIList < active U11 < active active < top isNatIList < proper U11 < proper proper < top ---------------------------------------- (25) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4112_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4112_0) Induction Base: U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) Induction Step: U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n4112_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1) mark(U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4112_0)), gen_zeros:0':mark:tt:nil: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(zeros) -> mark(cons(0', zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(nil) -> ok(nil) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1070_0))) -> *4_0, rt in Omega(n1070_0) length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1622_0))) -> *4_0, rt in Omega(n1622_0) and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2305_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n2305_0) U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4112_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4112_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil: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