KILLED
proof of input_kk15a3t6kh.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(4) CdtProblem
    (5) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxRelTRS
    (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) typed CpxTrs
    (11) OrderProof [LOWER BOUND(ID), 9 ms]
    (12) typed CpxTrs
    (13) RewriteLemmaProof [LOWER BOUND(ID), 566 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), 38 ms]
            (20) BOUNDS(1, INF)
(21) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(22) CpxTRS
(23) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(24) CdtProblem
    (25) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CdtProblem
    (27) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CdtProblem
    (29) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) CdtProblem
    (31) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) CpxRelTRS
        (33) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
        (34) CpxTRS
        (35) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (36) CpxWeightedTrs
            (37) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
            (38) CpxTypedWeightedTrs
            (39) CompletionProof [UPPER BOUND(ID), 0 ms]
            (40) CpxTypedWeightedCompleteTrs
                (41) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
                (42) CpxTypedWeightedCompleteTrs
                (43) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (44) CpxRNTS
                (45) InliningProof [UPPER BOUND(ID), 158 ms]
                (46) CpxRNTS
                (47) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
                (48) CpxRNTS
                (49) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
                (50) CpxRNTS
                (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (52) CpxRNTS
                (53) IntTrsBoundProof [UPPER BOUND(ID), 77 ms]
                (54) CpxRNTS
                (55) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
                (56) CpxRNTS
                (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (58) CpxRNTS
                (59) IntTrsBoundProof [UPPER BOUND(ID), 77 ms]
                (60) CpxRNTS
                (61) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
                (62) CpxRNTS
                (63) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (64) CpxRNTS
                (65) IntTrsBoundProof [UPPER BOUND(ID), 1513 ms]
                (66) CpxRNTS
                (67) IntTrsBoundProof [UPPER BOUND(ID), 649 ms]
                (68) CpxRNTS
            (69) CompletionProof [UPPER BOUND(ID), 0 ms]
            (70) CpxTypedWeightedCompleteTrs
                (71) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (72) CpxRNTS
    (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
        (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (76) CdtProblem
        (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (78) CdtProblem
        (79) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
        (80) CdtProblem
        (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (82) CdtProblem
        (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (84) CdtProblem
        (85) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (86) CdtProblem
        (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (88) CdtProblem
        (89) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
        (90) CdtProblem
        (91) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (92) CdtProblem
        (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (94) CdtProblem
        (95) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (96) CdtProblem
        (97) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (98) CdtProblem
        (99) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (100) CdtProblem
        (101) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (102) CdtProblem
        (103) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 1246 ms]
        (104) CdtProblem
        (105) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 1554 ms]
        (106) CdtProblem
        (107) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (108) CdtProblem
(109) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(110) CpxWeightedTrs
    (111) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CpxTypedWeightedTrs
    (113) CompletionProof [UPPER BOUND(ID), 0 ms]
    (114) CpxTypedWeightedCompleteTrs
        (115) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (116) CpxTypedWeightedCompleteTrs
        (117) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (118) CpxRNTS
        (119) InliningProof [UPPER BOUND(ID), 205 ms]
        (120) CpxRNTS
        (121) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (122) CpxRNTS
        (123) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (124) CpxRNTS
        (125) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (126) CpxRNTS
        (127) IntTrsBoundProof [UPPER BOUND(ID), 187 ms]
        (128) CpxRNTS
        (129) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
        (130) CpxRNTS
        (131) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (132) CpxRNTS
        (133) IntTrsBoundProof [UPPER BOUND(ID), 8 ms]
        (134) CpxRNTS
        (135) IntTrsBoundProof [UPPER BOUND(ID), 12 ms]
        (136) CpxRNTS
        (137) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (138) CpxRNTS
        (139) IntTrsBoundProof [UPPER BOUND(ID), 3517 ms]
        (140) CpxRNTS
        (141) IntTrsBoundProof [UPPER BOUND(ID), 449 ms]
        (142) CpxRNTS
    (143) CompletionProof [UPPER BOUND(ID), 0 ms]
    (144) CpxTypedWeightedCompleteTrs
        (145) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (146) CpxRNTS


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U11(tt, L) -> U12(tt, activate(L))
   U12(tt, L) -> s(length(activate(L)))
   length(nil) -> 0
   length(cons(N, L)) -> U11(tt, activate(L))
   zeros -> n__zeros
   activate(n__zeros) -> zeros
   activate(X) -> X

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   zeros -> cons(0', n__zeros)
   U11(tt, L) -> U12(tt, activate(L))
   U12(tt, L) -> s(length(activate(L)))
   length(nil) -> 0'
   length(cons(N, L)) -> U11(tt, activate(L))
   zeros -> n__zeros
   activate(n__zeros) -> zeros
   activate(X) -> X

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(3) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   U11(tt, z0) -> U12(tt, activate(z0))
   U12(tt, z0) -> s(length(activate(z0)))
   length(nil) -> 0
   length(cons(z0, z1)) -> U11(tt, activate(z1))
   activate(n__zeros) -> zeros
   activate(z0) -> z0
Tuples:
   ZEROS -> c
   ZEROS -> c1
   U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0))
   U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0))
   LENGTH(nil) -> c4
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)), ACTIVATE(z1))
   ACTIVATE(n__zeros) -> c6(ZEROS)
   ACTIVATE(z0) -> c7
S tuples:
   ZEROS -> c
   ZEROS -> c1
   U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0))
   U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0))
   LENGTH(nil) -> c4
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)), ACTIVATE(z1))
   ACTIVATE(n__zeros) -> c6(ZEROS)
   ACTIVATE(z0) -> c7
K tuples:none
Defined Rule Symbols:   zeros, U11_2, U12_2, length_1, activate_1

Defined Pair Symbols:   ZEROS, U11'_2, U12'_2, LENGTH_1, ACTIVATE_1

Compound Symbols:   c, c1, c2_2, c3_2, c4, c5_2, c6_1, c7


----------------------------------------

(5) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   ZEROS -> c
   ZEROS -> c1
   U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0))
   U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0))
   LENGTH(nil) -> c4
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)), ACTIVATE(z1))
   ACTIVATE(n__zeros) -> c6(ZEROS)
   ACTIVATE(z0) -> c7

The (relative) TRS S consists of the following rules:

   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   U11(tt, z0) -> U12(tt, activate(z0))
   U12(tt, z0) -> s(length(activate(z0)))
   length(nil) -> 0
   length(cons(z0, z1)) -> U11(tt, activate(z1))
   activate(n__zeros) -> zeros
   activate(z0) -> z0

Rewrite Strategy: INNERMOST
----------------------------------------

(7) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   ZEROS -> c
   ZEROS -> c1
   U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0))
   U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0))
   LENGTH(nil) -> c4
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)), ACTIVATE(z1))
   ACTIVATE(n__zeros) -> c6(ZEROS)
   ACTIVATE(z0) -> c7

The (relative) TRS S consists of the following rules:

   zeros -> cons(0', n__zeros)
   zeros -> n__zeros
   U11(tt, z0) -> U12(tt, activate(z0))
   U12(tt, z0) -> s(length(activate(z0)))
   length(nil) -> 0'
   length(cons(z0, z1)) -> U11(tt, activate(z1))
   activate(n__zeros) -> zeros
   activate(z0) -> z0

Rewrite Strategy: INNERMOST
----------------------------------------

(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(10)
Obligation:
Innermost TRS:
Rules:
ZEROS -> c
ZEROS -> c1
U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0))
U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0))
LENGTH(nil) -> c4
LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)), ACTIVATE(z1))
ACTIVATE(n__zeros) -> c6(ZEROS)
ACTIVATE(z0) -> c7
zeros -> cons(0', n__zeros)
zeros -> n__zeros
U11(tt, z0) -> U12(tt, activate(z0))
U12(tt, z0) -> s(length(activate(z0)))
length(nil) -> 0'
length(cons(z0, z1)) -> U11(tt, activate(z1))
activate(n__zeros) -> zeros
activate(z0) -> z0

Types:
ZEROS :: c:c1
c :: c:c1
c1 :: c:c1
U11' :: tt -> nil:cons:n__zeros -> c2
tt :: tt
c2 :: c3 -> c6:c7 -> c2
U12' :: tt -> nil:cons:n__zeros -> c3
activate :: nil:cons:n__zeros -> nil:cons:n__zeros
ACTIVATE :: nil:cons:n__zeros -> c6:c7
c3 :: c4:c5 -> c6:c7 -> c3
LENGTH :: nil:cons:n__zeros -> c4:c5
nil :: nil:cons:n__zeros
c4 :: c4:c5
cons :: 0':s -> nil:cons:n__zeros -> nil:cons:n__zeros
c5 :: c2 -> c6:c7 -> c4:c5
n__zeros :: nil:cons:n__zeros
c6 :: c:c1 -> c6:c7
c7 :: c6:c7
zeros :: nil:cons:n__zeros
0' :: 0':s
U11 :: tt -> nil:cons:n__zeros -> 0':s
U12 :: tt -> nil:cons:n__zeros -> 0':s
s :: 0':s -> 0':s
length :: nil:cons:n__zeros -> 0':s
hole_c:c11_8 :: c:c1
hole_c22_8 :: c2
hole_tt3_8 :: tt
hole_nil:cons:n__zeros4_8 :: nil:cons:n__zeros
hole_c35_8 :: c3
hole_c6:c76_8 :: c6:c7
hole_c4:c57_8 :: c4:c5
hole_0':s8_8 :: 0':s
gen_nil:cons:n__zeros9_8 :: Nat -> nil:cons:n__zeros
gen_0':s10_8 :: Nat -> 0':s

----------------------------------------

(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
LENGTH, length
----------------------------------------

(12)
Obligation:
Innermost TRS:
Rules:
ZEROS -> c
ZEROS -> c1
U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0))
U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0))
LENGTH(nil) -> c4
LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)), ACTIVATE(z1))
ACTIVATE(n__zeros) -> c6(ZEROS)
ACTIVATE(z0) -> c7
zeros -> cons(0', n__zeros)
zeros -> n__zeros
U11(tt, z0) -> U12(tt, activate(z0))
U12(tt, z0) -> s(length(activate(z0)))
length(nil) -> 0'
length(cons(z0, z1)) -> U11(tt, activate(z1))
activate(n__zeros) -> zeros
activate(z0) -> z0

Types:
ZEROS :: c:c1
c :: c:c1
c1 :: c:c1
U11' :: tt -> nil:cons:n__zeros -> c2
tt :: tt
c2 :: c3 -> c6:c7 -> c2
U12' :: tt -> nil:cons:n__zeros -> c3
activate :: nil:cons:n__zeros -> nil:cons:n__zeros
ACTIVATE :: nil:cons:n__zeros -> c6:c7
c3 :: c4:c5 -> c6:c7 -> c3
LENGTH :: nil:cons:n__zeros -> c4:c5
nil :: nil:cons:n__zeros
c4 :: c4:c5
cons :: 0':s -> nil:cons:n__zeros -> nil:cons:n__zeros
c5 :: c2 -> c6:c7 -> c4:c5
n__zeros :: nil:cons:n__zeros
c6 :: c:c1 -> c6:c7
c7 :: c6:c7
zeros :: nil:cons:n__zeros
0' :: 0':s
U11 :: tt -> nil:cons:n__zeros -> 0':s
U12 :: tt -> nil:cons:n__zeros -> 0':s
s :: 0':s -> 0':s
length :: nil:cons:n__zeros -> 0':s
hole_c:c11_8 :: c:c1
hole_c22_8 :: c2
hole_tt3_8 :: tt
hole_nil:cons:n__zeros4_8 :: nil:cons:n__zeros
hole_c35_8 :: c3
hole_c6:c76_8 :: c6:c7
hole_c4:c57_8 :: c4:c5
hole_0':s8_8 :: 0':s
gen_nil:cons:n__zeros9_8 :: Nat -> nil:cons:n__zeros
gen_0':s10_8 :: Nat -> 0':s


Generator Equations:
gen_nil:cons:n__zeros9_8(0) <=> nil
gen_nil:cons:n__zeros9_8(+(x, 1)) <=> cons(0', gen_nil:cons:n__zeros9_8(x))
gen_0':s10_8(0) <=> 0'
gen_0':s10_8(+(x, 1)) <=> s(gen_0':s10_8(x))


The following defined symbols remain to be analysed:
LENGTH, length
----------------------------------------

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LENGTH(gen_nil:cons:n__zeros9_8(n12_8)) -> *11_8, rt in Omega(n12_8)

Induction Base:
LENGTH(gen_nil:cons:n__zeros9_8(0))

Induction Step:
LENGTH(gen_nil:cons:n__zeros9_8(+(n12_8, 1))) ->_R^Omega(1)
c5(U11'(tt, activate(gen_nil:cons:n__zeros9_8(n12_8))), ACTIVATE(gen_nil:cons:n__zeros9_8(n12_8))) ->_R^Omega(0)
c5(U11'(tt, gen_nil:cons:n__zeros9_8(n12_8)), ACTIVATE(gen_nil:cons:n__zeros9_8(n12_8))) ->_R^Omega(1)
c5(c2(U12'(tt, activate(gen_nil:cons:n__zeros9_8(n12_8))), ACTIVATE(gen_nil:cons:n__zeros9_8(n12_8))), ACTIVATE(gen_nil:cons:n__zeros9_8(n12_8))) ->_R^Omega(0)
c5(c2(U12'(tt, gen_nil:cons:n__zeros9_8(n12_8)), ACTIVATE(gen_nil:cons:n__zeros9_8(n12_8))), ACTIVATE(gen_nil:cons:n__zeros9_8(n12_8))) ->_R^Omega(1)
c5(c2(c3(LENGTH(activate(gen_nil:cons:n__zeros9_8(n12_8))), ACTIVATE(gen_nil:cons:n__zeros9_8(n12_8))), ACTIVATE(gen_nil:cons:n__zeros9_8(n12_8))), ACTIVATE(gen_nil:cons:n__zeros9_8(n12_8))) ->_R^Omega(0)
c5(c2(c3(LENGTH(gen_nil:cons:n__zeros9_8(n12_8)), ACTIVATE(gen_nil:cons:n__zeros9_8(n12_8))), ACTIVATE(gen_nil:cons:n__zeros9_8(n12_8))), ACTIVATE(gen_nil:cons:n__zeros9_8(n12_8))) ->_IH
c5(c2(c3(*11_8, ACTIVATE(gen_nil:cons:n__zeros9_8(n12_8))), ACTIVATE(gen_nil:cons:n__zeros9_8(n12_8))), ACTIVATE(gen_nil:cons:n__zeros9_8(n12_8))) ->_R^Omega(1)
c5(c2(c3(*11_8, c7), ACTIVATE(gen_nil:cons:n__zeros9_8(n12_8))), ACTIVATE(gen_nil:cons:n__zeros9_8(n12_8))) ->_R^Omega(1)
c5(c2(c3(*11_8, c7), c7), ACTIVATE(gen_nil:cons:n__zeros9_8(n12_8))) ->_R^Omega(1)
c5(c2(c3(*11_8, c7), c7), c7)

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:

Innermost TRS:
Rules:
ZEROS -> c
ZEROS -> c1
U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0))
U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0))
LENGTH(nil) -> c4
LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)), ACTIVATE(z1))
ACTIVATE(n__zeros) -> c6(ZEROS)
ACTIVATE(z0) -> c7
zeros -> cons(0', n__zeros)
zeros -> n__zeros
U11(tt, z0) -> U12(tt, activate(z0))
U12(tt, z0) -> s(length(activate(z0)))
length(nil) -> 0'
length(cons(z0, z1)) -> U11(tt, activate(z1))
activate(n__zeros) -> zeros
activate(z0) -> z0

Types:
ZEROS :: c:c1
c :: c:c1
c1 :: c:c1
U11' :: tt -> nil:cons:n__zeros -> c2
tt :: tt
c2 :: c3 -> c6:c7 -> c2
U12' :: tt -> nil:cons:n__zeros -> c3
activate :: nil:cons:n__zeros -> nil:cons:n__zeros
ACTIVATE :: nil:cons:n__zeros -> c6:c7
c3 :: c4:c5 -> c6:c7 -> c3
LENGTH :: nil:cons:n__zeros -> c4:c5
nil :: nil:cons:n__zeros
c4 :: c4:c5
cons :: 0':s -> nil:cons:n__zeros -> nil:cons:n__zeros
c5 :: c2 -> c6:c7 -> c4:c5
n__zeros :: nil:cons:n__zeros
c6 :: c:c1 -> c6:c7
c7 :: c6:c7
zeros :: nil:cons:n__zeros
0' :: 0':s
U11 :: tt -> nil:cons:n__zeros -> 0':s
U12 :: tt -> nil:cons:n__zeros -> 0':s
s :: 0':s -> 0':s
length :: nil:cons:n__zeros -> 0':s
hole_c:c11_8 :: c:c1
hole_c22_8 :: c2
hole_tt3_8 :: tt
hole_nil:cons:n__zeros4_8 :: nil:cons:n__zeros
hole_c35_8 :: c3
hole_c6:c76_8 :: c6:c7
hole_c4:c57_8 :: c4:c5
hole_0':s8_8 :: 0':s
gen_nil:cons:n__zeros9_8 :: Nat -> nil:cons:n__zeros
gen_0':s10_8 :: Nat -> 0':s


Generator Equations:
gen_nil:cons:n__zeros9_8(0) <=> nil
gen_nil:cons:n__zeros9_8(+(x, 1)) <=> cons(0', gen_nil:cons:n__zeros9_8(x))
gen_0':s10_8(0) <=> 0'
gen_0':s10_8(+(x, 1)) <=> s(gen_0':s10_8(x))


The following defined symbols remain to be analysed:
LENGTH, length
----------------------------------------

(16) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(17)
BOUNDS(n^1, INF)

----------------------------------------

(18)
Obligation:
Innermost TRS:
Rules:
ZEROS -> c
ZEROS -> c1
U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0))
U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0))
LENGTH(nil) -> c4
LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)), ACTIVATE(z1))
ACTIVATE(n__zeros) -> c6(ZEROS)
ACTIVATE(z0) -> c7
zeros -> cons(0', n__zeros)
zeros -> n__zeros
U11(tt, z0) -> U12(tt, activate(z0))
U12(tt, z0) -> s(length(activate(z0)))
length(nil) -> 0'
length(cons(z0, z1)) -> U11(tt, activate(z1))
activate(n__zeros) -> zeros
activate(z0) -> z0

Types:
ZEROS :: c:c1
c :: c:c1
c1 :: c:c1
U11' :: tt -> nil:cons:n__zeros -> c2
tt :: tt
c2 :: c3 -> c6:c7 -> c2
U12' :: tt -> nil:cons:n__zeros -> c3
activate :: nil:cons:n__zeros -> nil:cons:n__zeros
ACTIVATE :: nil:cons:n__zeros -> c6:c7
c3 :: c4:c5 -> c6:c7 -> c3
LENGTH :: nil:cons:n__zeros -> c4:c5
nil :: nil:cons:n__zeros
c4 :: c4:c5
cons :: 0':s -> nil:cons:n__zeros -> nil:cons:n__zeros
c5 :: c2 -> c6:c7 -> c4:c5
n__zeros :: nil:cons:n__zeros
c6 :: c:c1 -> c6:c7
c7 :: c6:c7
zeros :: nil:cons:n__zeros
0' :: 0':s
U11 :: tt -> nil:cons:n__zeros -> 0':s
U12 :: tt -> nil:cons:n__zeros -> 0':s
s :: 0':s -> 0':s
length :: nil:cons:n__zeros -> 0':s
hole_c:c11_8 :: c:c1
hole_c22_8 :: c2
hole_tt3_8 :: tt
hole_nil:cons:n__zeros4_8 :: nil:cons:n__zeros
hole_c35_8 :: c3
hole_c6:c76_8 :: c6:c7
hole_c4:c57_8 :: c4:c5
hole_0':s8_8 :: 0':s
gen_nil:cons:n__zeros9_8 :: Nat -> nil:cons:n__zeros
gen_0':s10_8 :: Nat -> 0':s


Lemmas:
LENGTH(gen_nil:cons:n__zeros9_8(n12_8)) -> *11_8, rt in Omega(n12_8)


Generator Equations:
gen_nil:cons:n__zeros9_8(0) <=> nil
gen_nil:cons:n__zeros9_8(+(x, 1)) <=> cons(0', gen_nil:cons:n__zeros9_8(x))
gen_0':s10_8(0) <=> 0'
gen_0':s10_8(+(x, 1)) <=> s(gen_0':s10_8(x))


The following defined symbols remain to be analysed:
length
----------------------------------------

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length(gen_nil:cons:n__zeros9_8(n717_8)) -> gen_0':s10_8(n717_8), rt in Omega(0)

Induction Base:
length(gen_nil:cons:n__zeros9_8(0)) ->_R^Omega(0)
0'

Induction Step:
length(gen_nil:cons:n__zeros9_8(+(n717_8, 1))) ->_R^Omega(0)
U11(tt, activate(gen_nil:cons:n__zeros9_8(n717_8))) ->_R^Omega(0)
U11(tt, gen_nil:cons:n__zeros9_8(n717_8)) ->_R^Omega(0)
U12(tt, activate(gen_nil:cons:n__zeros9_8(n717_8))) ->_R^Omega(0)
U12(tt, gen_nil:cons:n__zeros9_8(n717_8)) ->_R^Omega(0)
s(length(activate(gen_nil:cons:n__zeros9_8(n717_8)))) ->_R^Omega(0)
s(length(gen_nil:cons:n__zeros9_8(n717_8))) ->_IH
s(gen_0':s10_8(c718_8))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(20)
BOUNDS(1, INF)

----------------------------------------

(21) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(22)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros)
   U11(tt, L) -> U12(tt, activate(L))
   U12(tt, L) -> s(length(activate(L)))
   length(nil) -> 0
   length(cons(N, L)) -> U11(tt, activate(L))
   zeros -> n__zeros
   activate(n__zeros) -> zeros
   activate(X) -> X

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(23) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   U11(tt, z0) -> U12(tt, activate(z0))
   U12(tt, z0) -> s(length(activate(z0)))
   length(nil) -> 0
   length(cons(z0, z1)) -> U11(tt, activate(z1))
   activate(n__zeros) -> zeros
   activate(z0) -> z0
Tuples:
   ZEROS -> c
   ZEROS -> c1
   U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0))
   U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0))
   LENGTH(nil) -> c4
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)), ACTIVATE(z1))
   ACTIVATE(n__zeros) -> c6(ZEROS)
   ACTIVATE(z0) -> c7
S tuples:
   ZEROS -> c
   ZEROS -> c1
   U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0))
   U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0))
   LENGTH(nil) -> c4
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)), ACTIVATE(z1))
   ACTIVATE(n__zeros) -> c6(ZEROS)
   ACTIVATE(z0) -> c7
K tuples:none
Defined Rule Symbols:   zeros, U11_2, U12_2, length_1, activate_1

Defined Pair Symbols:   ZEROS, U11'_2, U12'_2, LENGTH_1, ACTIVATE_1

Compound Symbols:   c, c1, c2_2, c3_2, c4, c5_2, c6_1, c7


----------------------------------------

(25) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing nodes:
   ACTIVATE(n__zeros) -> c6(ZEROS)
   ACTIVATE(z0) -> c7
   ZEROS -> c
   ZEROS -> c1
   LENGTH(nil) -> c4

----------------------------------------

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   U11(tt, z0) -> U12(tt, activate(z0))
   U12(tt, z0) -> s(length(activate(z0)))
   length(nil) -> 0
   length(cons(z0, z1)) -> U11(tt, activate(z1))
   activate(n__zeros) -> zeros
   activate(z0) -> z0
Tuples:
   U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0))
   U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0))
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)), ACTIVATE(z1))
S tuples:
   U11'(tt, z0) -> c2(U12'(tt, activate(z0)), ACTIVATE(z0))
   U12'(tt, z0) -> c3(LENGTH(activate(z0)), ACTIVATE(z0))
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:   zeros, U11_2, U12_2, length_1, activate_1

Defined Pair Symbols:   U11'_2, U12'_2, LENGTH_1

Compound Symbols:   c2_2, c3_2, c5_2


----------------------------------------

(27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing tuple parts
----------------------------------------

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
   U11(tt, z0) -> U12(tt, activate(z0))
   U12(tt, z0) -> s(length(activate(z0)))
   length(nil) -> 0
   length(cons(z0, z1)) -> U11(tt, activate(z1))
   activate(n__zeros) -> zeros
   activate(z0) -> z0
Tuples:
   U11'(tt, z0) -> c2(U12'(tt, activate(z0)))
   U12'(tt, z0) -> c3(LENGTH(activate(z0)))
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)))
S tuples:
   U11'(tt, z0) -> c2(U12'(tt, activate(z0)))
   U12'(tt, z0) -> c3(LENGTH(activate(z0)))
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)))
K tuples:none
Defined Rule Symbols:   zeros, U11_2, U12_2, length_1, activate_1

Defined Pair Symbols:   U11'_2, U12'_2, LENGTH_1

Compound Symbols:   c2_1, c3_1, c5_1


----------------------------------------

(29) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   U11(tt, z0) -> U12(tt, activate(z0))
   U12(tt, z0) -> s(length(activate(z0)))
   length(nil) -> 0
   length(cons(z0, z1)) -> U11(tt, activate(z1))

----------------------------------------

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
Tuples:
   U11'(tt, z0) -> c2(U12'(tt, activate(z0)))
   U12'(tt, z0) -> c3(LENGTH(activate(z0)))
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)))
S tuples:
   U11'(tt, z0) -> c2(U12'(tt, activate(z0)))
   U12'(tt, z0) -> c3(LENGTH(activate(z0)))
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)))
K tuples:none
Defined Rule Symbols:   activate_1, zeros

Defined Pair Symbols:   U11'_2, U12'_2, LENGTH_1

Compound Symbols:   c2_1, c3_1, c5_1


----------------------------------------

(31) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(32)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   U11'(tt, z0) -> c2(U12'(tt, activate(z0)))
   U12'(tt, z0) -> c3(LENGTH(activate(z0)))
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)))

The (relative) TRS S consists of the following rules:

   activate(n__zeros) -> zeros
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros

Rewrite Strategy: INNERMOST
----------------------------------------

(33) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(34)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   U11'(tt, z0) -> c2(U12'(tt, activate(z0)))
   U12'(tt, z0) -> c3(LENGTH(activate(z0)))
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)))
   activate(n__zeros) -> zeros
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(35) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(36)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   U11'(tt, z0) -> c2(U12'(tt, activate(z0))) [1]
   U12'(tt, z0) -> c3(LENGTH(activate(z0))) [1]
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1))) [1]
   activate(n__zeros) -> zeros [0]
   activate(z0) -> z0 [0]
   zeros -> cons(0, n__zeros) [0]
   zeros -> n__zeros [0]

Rewrite Strategy: INNERMOST
----------------------------------------

(37) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(38)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   U11'(tt, z0) -> c2(U12'(tt, activate(z0))) [1]
   U12'(tt, z0) -> c3(LENGTH(activate(z0))) [1]
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1))) [1]
   activate(n__zeros) -> zeros [0]
   activate(z0) -> z0 [0]
   zeros -> cons(0, n__zeros) [0]
   zeros -> n__zeros [0]

The TRS has the following type information:
U11' :: tt -> cons:n__zeros -> c2
tt :: tt
c2 :: c3 -> c2
U12' :: tt -> cons:n__zeros -> c3
activate :: cons:n__zeros -> cons:n__zeros
c3 :: c5 -> c3
LENGTH :: cons:n__zeros -> c5
cons :: 0 -> cons:n__zeros -> cons:n__zeros
c5 :: c2 -> c5
n__zeros :: cons:n__zeros
zeros :: cons:n__zeros
0 :: 0

Rewrite Strategy: INNERMOST
----------------------------------------

(39) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   U11'_2
   U12'_2
   LENGTH_1

(c) The following functions are completely defined:

   activate_1
   zeros

Due to the following rules being added:

   activate(v0) -> n__zeros [0]
   zeros -> n__zeros [0]

And the following fresh constants:    const, const1, const2

----------------------------------------

(40)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   U11'(tt, z0) -> c2(U12'(tt, activate(z0))) [1]
   U12'(tt, z0) -> c3(LENGTH(activate(z0))) [1]
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1))) [1]
   activate(n__zeros) -> zeros [0]
   activate(z0) -> z0 [0]
   zeros -> cons(0, n__zeros) [0]
   zeros -> n__zeros [0]
   activate(v0) -> n__zeros [0]
   zeros -> n__zeros [0]

The TRS has the following type information:
U11' :: tt -> cons:n__zeros -> c2
tt :: tt
c2 :: c3 -> c2
U12' :: tt -> cons:n__zeros -> c3
activate :: cons:n__zeros -> cons:n__zeros
c3 :: c5 -> c3
LENGTH :: cons:n__zeros -> c5
cons :: 0 -> cons:n__zeros -> cons:n__zeros
c5 :: c2 -> c5
n__zeros :: cons:n__zeros
zeros :: cons:n__zeros
0 :: 0
const :: c2
const1 :: c3
const2 :: c5

Rewrite Strategy: INNERMOST
----------------------------------------

(41) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(42)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   U11'(tt, n__zeros) -> c2(U12'(tt, zeros)) [1]
   U11'(tt, z0) -> c2(U12'(tt, z0)) [1]
   U11'(tt, z0) -> c2(U12'(tt, n__zeros)) [1]
   U12'(tt, n__zeros) -> c3(LENGTH(zeros)) [1]
   U12'(tt, z0) -> c3(LENGTH(z0)) [1]
   U12'(tt, z0) -> c3(LENGTH(n__zeros)) [1]
   LENGTH(cons(z0, n__zeros)) -> c5(U11'(tt, zeros)) [1]
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, z1)) [1]
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, n__zeros)) [1]
   activate(n__zeros) -> zeros [0]
   activate(z0) -> z0 [0]
   zeros -> cons(0, n__zeros) [0]
   zeros -> n__zeros [0]
   activate(v0) -> n__zeros [0]
   zeros -> n__zeros [0]

The TRS has the following type information:
U11' :: tt -> cons:n__zeros -> c2
tt :: tt
c2 :: c3 -> c2
U12' :: tt -> cons:n__zeros -> c3
activate :: cons:n__zeros -> cons:n__zeros
c3 :: c5 -> c3
LENGTH :: cons:n__zeros -> c5
cons :: 0 -> cons:n__zeros -> cons:n__zeros
c5 :: c2 -> c5
n__zeros :: cons:n__zeros
zeros :: cons:n__zeros
0 :: 0
const :: c2
const1 :: c3
const2 :: c5

Rewrite Strategy: INNERMOST
----------------------------------------

(43) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   tt => 0
   n__zeros => 0
   0 => 0
   const => 0
   const1 => 0
   const2 => 0

----------------------------------------

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 1 }-> 1 + U11'(0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, zeros) :|: z0 >= 0, z = 1 + z0 + 0
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   U11'(z, z') -{ 1 }-> 1 + U12'(0, z0) :|: z0 >= 0, z = 0, z' = z0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, zeros) :|: z = 0, z' = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z0 >= 0, z = 0, z' = z0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(z0) :|: z0 >= 0, z = 0, z' = z0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(zeros) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z0 >= 0, z = 0, z' = z0
   activate(z) -{ 0 }-> z0 :|: z = z0, z0 >= 0
   activate(z) -{ 0 }-> zeros :|: z = 0
   activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   zeros -{ 0 }-> 0 :|: 
   zeros -{ 0 }-> 1 + 0 + 0 :|: 


----------------------------------------

(45) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   zeros -{ 0 }-> 1 + 0 + 0 :|: 
   zeros -{ 0 }-> 0 :|: 

----------------------------------------

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 1 }-> 1 + U11'(0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z0 >= 0, z = 1 + z0 + 0
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 1 + 0 + 0) :|: z0 >= 0, z = 1 + z0 + 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, z0) :|: z0 >= 0, z = 0, z' = z0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z0 >= 0, z = 0, z' = z0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z = 0, z' = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(z0) :|: z0 >= 0, z = 0, z' = z0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z0 >= 0, z = 0, z' = z0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 0 }-> z0 :|: z = z0, z0 >= 0
   activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 1 + 0 + 0 :|: z = 0
   zeros -{ 0 }-> 0 :|: 
   zeros -{ 0 }-> 1 + 0 + 0 :|: 


----------------------------------------

(47) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 1 }-> 1 + U11'(0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 1 + 0 + 0) :|: z - 1 >= 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, z') :|: z' >= 0, z = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z' >= 0, z = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z = 0, z' = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(z') :|: z' >= 0, z = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z' >= 0, z = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 1 + 0 + 0 :|: z = 0
   zeros -{ 0 }-> 0 :|: 
   zeros -{ 0 }-> 1 + 0 + 0 :|: 


----------------------------------------

(49) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { activate }
   { zeros }
   { LENGTH, U11', U12' }

----------------------------------------

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 1 }-> 1 + U11'(0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 1 + 0 + 0) :|: z - 1 >= 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, z') :|: z' >= 0, z = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z' >= 0, z = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z = 0, z' = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(z') :|: z' >= 0, z = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z' >= 0, z = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 1 + 0 + 0 :|: z = 0
   zeros -{ 0 }-> 0 :|: 
   zeros -{ 0 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {activate}, {zeros}, {LENGTH,U11',U12'}

----------------------------------------

(51) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(52)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 1 }-> 1 + U11'(0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 1 + 0 + 0) :|: z - 1 >= 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, z') :|: z' >= 0, z = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z' >= 0, z = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z = 0, z' = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(z') :|: z' >= 0, z = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z' >= 0, z = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 1 + 0 + 0 :|: z = 0
   zeros -{ 0 }-> 0 :|: 
   zeros -{ 0 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {activate}, {zeros}, {LENGTH,U11',U12'}

----------------------------------------

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z

----------------------------------------

(54)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 1 }-> 1 + U11'(0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 1 + 0 + 0) :|: z - 1 >= 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, z') :|: z' >= 0, z = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z' >= 0, z = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z = 0, z' = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(z') :|: z' >= 0, z = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z' >= 0, z = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 1 + 0 + 0 :|: z = 0
   zeros -{ 0 }-> 0 :|: 
   zeros -{ 0 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {activate}, {zeros}, {LENGTH,U11',U12'}
Previous analysis results are:
  activate: runtime: ?, size: O(n^1) [1 + z]

----------------------------------------

(55) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(56)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 1 }-> 1 + U11'(0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 1 + 0 + 0) :|: z - 1 >= 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, z') :|: z' >= 0, z = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z' >= 0, z = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z = 0, z' = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(z') :|: z' >= 0, z = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z' >= 0, z = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 1 + 0 + 0 :|: z = 0
   zeros -{ 0 }-> 0 :|: 
   zeros -{ 0 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {zeros}, {LENGTH,U11',U12'}
Previous analysis results are:
  activate: runtime: O(1) [0], size: O(n^1) [1 + z]

----------------------------------------

(57) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(58)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 1 }-> 1 + U11'(0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 1 + 0 + 0) :|: z - 1 >= 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, z') :|: z' >= 0, z = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z' >= 0, z = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z = 0, z' = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(z') :|: z' >= 0, z = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z' >= 0, z = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 1 + 0 + 0 :|: z = 0
   zeros -{ 0 }-> 0 :|: 
   zeros -{ 0 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {zeros}, {LENGTH,U11',U12'}
Previous analysis results are:
  activate: runtime: O(1) [0], size: O(n^1) [1 + z]

----------------------------------------

(59) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: zeros
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(60)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 1 }-> 1 + U11'(0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 1 + 0 + 0) :|: z - 1 >= 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, z') :|: z' >= 0, z = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z' >= 0, z = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z = 0, z' = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(z') :|: z' >= 0, z = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z' >= 0, z = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 1 + 0 + 0 :|: z = 0
   zeros -{ 0 }-> 0 :|: 
   zeros -{ 0 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {zeros}, {LENGTH,U11',U12'}
Previous analysis results are:
  activate: runtime: O(1) [0], size: O(n^1) [1 + z]
  zeros: runtime: ?, size: O(1) [1]

----------------------------------------

(61) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: zeros
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(62)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 1 }-> 1 + U11'(0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 1 + 0 + 0) :|: z - 1 >= 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, z') :|: z' >= 0, z = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z' >= 0, z = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z = 0, z' = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(z') :|: z' >= 0, z = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z' >= 0, z = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 1 + 0 + 0 :|: z = 0
   zeros -{ 0 }-> 0 :|: 
   zeros -{ 0 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {LENGTH,U11',U12'}
Previous analysis results are:
  activate: runtime: O(1) [0], size: O(n^1) [1 + z]
  zeros: runtime: O(1) [0], size: O(1) [1]

----------------------------------------

(63) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(64)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 1 }-> 1 + U11'(0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 1 + 0 + 0) :|: z - 1 >= 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, z') :|: z' >= 0, z = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z' >= 0, z = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z = 0, z' = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(z') :|: z' >= 0, z = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z' >= 0, z = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 1 + 0 + 0 :|: z = 0
   zeros -{ 0 }-> 0 :|: 
   zeros -{ 0 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {LENGTH,U11',U12'}
Previous analysis results are:
  activate: runtime: O(1) [0], size: O(n^1) [1 + z]
  zeros: runtime: O(1) [0], size: O(1) [1]

----------------------------------------

(65) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: LENGTH
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed SIZE bound using CoFloCo for: U11'
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

Computed SIZE bound using CoFloCo for: U12'
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(66)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 1 }-> 1 + U11'(0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 1 + 0 + 0) :|: z - 1 >= 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, z') :|: z' >= 0, z = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z' >= 0, z = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z = 0, z' = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(z') :|: z' >= 0, z = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z' >= 0, z = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 1 + 0 + 0 :|: z = 0
   zeros -{ 0 }-> 0 :|: 
   zeros -{ 0 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {LENGTH,U11',U12'}
Previous analysis results are:
  activate: runtime: O(1) [0], size: O(n^1) [1 + z]
  zeros: runtime: O(1) [0], size: O(1) [1]
  LENGTH: runtime: ?, size: O(1) [0]
  U11': runtime: ?, size: O(1) [2]
  U12': runtime: ?, size: O(1) [1]

----------------------------------------

(67) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: LENGTH
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

----------------------------------------

(68)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 1 }-> 1 + U11'(0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + U11'(0, 1 + 0 + 0) :|: z - 1 >= 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, z') :|: z' >= 0, z = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z' >= 0, z = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 0) :|: z = 0, z' = 0
   U11'(z, z') -{ 1 }-> 1 + U12'(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(z') :|: z' >= 0, z = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z' >= 0, z = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(0) :|: z = 0, z' = 0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 1 + 0 + 0 :|: z = 0
   zeros -{ 0 }-> 0 :|: 
   zeros -{ 0 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {LENGTH,U11',U12'}
Previous analysis results are:
  activate: runtime: O(1) [0], size: O(n^1) [1 + z]
  zeros: runtime: O(1) [0], size: O(1) [1]
  LENGTH: runtime: INF, size: O(1) [0]
  U11': runtime: ?, size: O(1) [2]
  U12': runtime: ?, size: O(1) [1]

----------------------------------------

(69) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   activate(v0) -> null_activate [0]
   zeros -> null_zeros [0]
   LENGTH(v0) -> null_LENGTH [0]

And the following fresh constants:    null_activate, null_zeros, null_LENGTH, const, const1

----------------------------------------

(70)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   U11'(tt, z0) -> c2(U12'(tt, activate(z0))) [1]
   U12'(tt, z0) -> c3(LENGTH(activate(z0))) [1]
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1))) [1]
   activate(n__zeros) -> zeros [0]
   activate(z0) -> z0 [0]
   zeros -> cons(0, n__zeros) [0]
   zeros -> n__zeros [0]
   activate(v0) -> null_activate [0]
   zeros -> null_zeros [0]
   LENGTH(v0) -> null_LENGTH [0]

The TRS has the following type information:
U11' :: tt -> cons:n__zeros:null_activate:null_zeros -> c2
tt :: tt
c2 :: c3 -> c2
U12' :: tt -> cons:n__zeros:null_activate:null_zeros -> c3
activate :: cons:n__zeros:null_activate:null_zeros -> cons:n__zeros:null_activate:null_zeros
c3 :: c5:null_LENGTH -> c3
LENGTH :: cons:n__zeros:null_activate:null_zeros -> c5:null_LENGTH
cons :: 0 -> cons:n__zeros:null_activate:null_zeros -> cons:n__zeros:null_activate:null_zeros
c5 :: c2 -> c5:null_LENGTH
n__zeros :: cons:n__zeros:null_activate:null_zeros
zeros :: cons:n__zeros:null_activate:null_zeros
0 :: 0
null_activate :: cons:n__zeros:null_activate:null_zeros
null_zeros :: cons:n__zeros:null_activate:null_zeros
null_LENGTH :: c5:null_LENGTH
const :: c2
const1 :: c3

Rewrite Strategy: INNERMOST
----------------------------------------

(71) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   tt => 0
   n__zeros => 0
   0 => 0
   null_activate => 0
   null_zeros => 0
   null_LENGTH => 0
   const => 0
   const1 => 0

----------------------------------------

(72)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   LENGTH(z) -{ 1 }-> 1 + U11'(0, activate(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   U11'(z, z') -{ 1 }-> 1 + U12'(0, activate(z0)) :|: z0 >= 0, z = 0, z' = z0
   U12'(z, z') -{ 1 }-> 1 + LENGTH(activate(z0)) :|: z0 >= 0, z = 0, z' = z0
   activate(z) -{ 0 }-> z0 :|: z = z0, z0 >= 0
   activate(z) -{ 0 }-> zeros :|: z = 0
   activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   zeros -{ 0 }-> 0 :|: 
   zeros -{ 0 }-> 1 + 0 + 0 :|: 

Only complete derivations are relevant for the runtime complexity.

----------------------------------------

(73) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace U11'(tt, z0) -> c2(U12'(tt, activate(z0))) by 
   U11'(tt, n__zeros) -> c2(U12'(tt, zeros))
   U11'(tt, z0) -> c2(U12'(tt, z0))

----------------------------------------

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
Tuples:
   U12'(tt, z0) -> c3(LENGTH(activate(z0)))
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)))
   U11'(tt, n__zeros) -> c2(U12'(tt, zeros))
   U11'(tt, z0) -> c2(U12'(tt, z0))
S tuples:
   U12'(tt, z0) -> c3(LENGTH(activate(z0)))
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)))
   U11'(tt, n__zeros) -> c2(U12'(tt, zeros))
   U11'(tt, z0) -> c2(U12'(tt, z0))
K tuples:none
Defined Rule Symbols:   activate_1, zeros

Defined Pair Symbols:   U12'_2, LENGTH_1, U11'_2

Compound Symbols:   c3_1, c5_1, c2_1


----------------------------------------

(75) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace U12'(tt, z0) -> c3(LENGTH(activate(z0))) by 
   U12'(tt, n__zeros) -> c3(LENGTH(zeros))
   U12'(tt, z0) -> c3(LENGTH(z0))

----------------------------------------

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
Tuples:
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)))
   U11'(tt, n__zeros) -> c2(U12'(tt, zeros))
   U11'(tt, z0) -> c2(U12'(tt, z0))
   U12'(tt, n__zeros) -> c3(LENGTH(zeros))
   U12'(tt, z0) -> c3(LENGTH(z0))
S tuples:
   LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1)))
   U11'(tt, n__zeros) -> c2(U12'(tt, zeros))
   U11'(tt, z0) -> c2(U12'(tt, z0))
   U12'(tt, n__zeros) -> c3(LENGTH(zeros))
   U12'(tt, z0) -> c3(LENGTH(z0))
K tuples:none
Defined Rule Symbols:   activate_1, zeros

Defined Pair Symbols:   LENGTH_1, U11'_2, U12'_2

Compound Symbols:   c5_1, c2_1, c3_1


----------------------------------------

(77) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LENGTH(cons(z0, z1)) -> c5(U11'(tt, activate(z1))) by 
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros))
   LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0))

----------------------------------------

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__zeros) -> zeros
   activate(z0) -> z0
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
Tuples:
   U11'(tt, n__zeros) -> c2(U12'(tt, zeros))
   U11'(tt, z0) -> c2(U12'(tt, z0))
   U12'(tt, n__zeros) -> c3(LENGTH(zeros))
   U12'(tt, z0) -> c3(LENGTH(z0))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros))
   LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0))
S tuples:
   U11'(tt, n__zeros) -> c2(U12'(tt, zeros))
   U11'(tt, z0) -> c2(U12'(tt, z0))
   U12'(tt, n__zeros) -> c3(LENGTH(zeros))
   U12'(tt, z0) -> c3(LENGTH(z0))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros))
   LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0))
K tuples:none
Defined Rule Symbols:   activate_1, zeros

Defined Pair Symbols:   U11'_2, U12'_2, LENGTH_1

Compound Symbols:   c2_1, c3_1, c5_1


----------------------------------------

(79) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   activate(n__zeros) -> zeros
   activate(z0) -> z0

----------------------------------------

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
Tuples:
   U11'(tt, n__zeros) -> c2(U12'(tt, zeros))
   U11'(tt, z0) -> c2(U12'(tt, z0))
   U12'(tt, n__zeros) -> c3(LENGTH(zeros))
   U12'(tt, z0) -> c3(LENGTH(z0))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros))
   LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0))
S tuples:
   U11'(tt, n__zeros) -> c2(U12'(tt, zeros))
   U11'(tt, z0) -> c2(U12'(tt, z0))
   U12'(tt, n__zeros) -> c3(LENGTH(zeros))
   U12'(tt, z0) -> c3(LENGTH(z0))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros))
   LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0))
K tuples:none
Defined Rule Symbols:   zeros

Defined Pair Symbols:   U11'_2, U12'_2, LENGTH_1

Compound Symbols:   c2_1, c3_1, c5_1


----------------------------------------

(81) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace U11'(tt, n__zeros) -> c2(U12'(tt, zeros)) by 
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))

----------------------------------------

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
Tuples:
   U11'(tt, z0) -> c2(U12'(tt, z0))
   U12'(tt, n__zeros) -> c3(LENGTH(zeros))
   U12'(tt, z0) -> c3(LENGTH(z0))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros))
   LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0))
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
S tuples:
   U11'(tt, z0) -> c2(U12'(tt, z0))
   U12'(tt, n__zeros) -> c3(LENGTH(zeros))
   U12'(tt, z0) -> c3(LENGTH(z0))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros))
   LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0))
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
K tuples:none
Defined Rule Symbols:   zeros

Defined Pair Symbols:   U11'_2, U12'_2, LENGTH_1

Compound Symbols:   c2_1, c3_1, c5_1


----------------------------------------

(83) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace U12'(tt, n__zeros) -> c3(LENGTH(zeros)) by 
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   U12'(tt, n__zeros) -> c3(LENGTH(n__zeros))

----------------------------------------

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
Tuples:
   U11'(tt, z0) -> c2(U12'(tt, z0))
   U12'(tt, z0) -> c3(LENGTH(z0))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros))
   LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0))
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   U12'(tt, n__zeros) -> c3(LENGTH(n__zeros))
S tuples:
   U11'(tt, z0) -> c2(U12'(tt, z0))
   U12'(tt, z0) -> c3(LENGTH(z0))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros))
   LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0))
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   U12'(tt, n__zeros) -> c3(LENGTH(n__zeros))
K tuples:none
Defined Rule Symbols:   zeros

Defined Pair Symbols:   U11'_2, U12'_2, LENGTH_1

Compound Symbols:   c2_1, c3_1, c5_1


----------------------------------------

(85) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   U12'(tt, n__zeros) -> c3(LENGTH(n__zeros))

----------------------------------------

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
Tuples:
   U11'(tt, z0) -> c2(U12'(tt, z0))
   U12'(tt, z0) -> c3(LENGTH(z0))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros))
   LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0))
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
S tuples:
   U11'(tt, z0) -> c2(U12'(tt, z0))
   U12'(tt, z0) -> c3(LENGTH(z0))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros))
   LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0))
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
K tuples:none
Defined Rule Symbols:   zeros

Defined Pair Symbols:   U11'_2, U12'_2, LENGTH_1

Compound Symbols:   c2_1, c3_1, c5_1


----------------------------------------

(87) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, zeros)) by 
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))

----------------------------------------

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros
Tuples:
   U11'(tt, z0) -> c2(U12'(tt, z0))
   U12'(tt, z0) -> c3(LENGTH(z0))
   LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0))
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
S tuples:
   U11'(tt, z0) -> c2(U12'(tt, z0))
   U12'(tt, z0) -> c3(LENGTH(z0))
   LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0))
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
K tuples:none
Defined Rule Symbols:   zeros

Defined Pair Symbols:   U11'_2, U12'_2, LENGTH_1

Compound Symbols:   c2_1, c3_1, c5_1


----------------------------------------

(89) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   zeros -> cons(0, n__zeros)
   zeros -> n__zeros

----------------------------------------

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   U11'(tt, z0) -> c2(U12'(tt, z0))
   U12'(tt, z0) -> c3(LENGTH(z0))
   LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0))
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
S tuples:
   U11'(tt, z0) -> c2(U12'(tt, z0))
   U12'(tt, z0) -> c3(LENGTH(z0))
   LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0))
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   U11'_2, U12'_2, LENGTH_1

Compound Symbols:   c2_1, c3_1, c5_1


----------------------------------------

(91) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace U12'(tt, z0) -> c3(LENGTH(z0)) by 
   U12'(tt, cons(y0, y1)) -> c3(LENGTH(cons(y0, y1)))
   U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros)))

----------------------------------------

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   U11'(tt, z0) -> c2(U12'(tt, z0))
   LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0))
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
   U12'(tt, cons(y0, y1)) -> c3(LENGTH(cons(y0, y1)))
   U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros)))
S tuples:
   U11'(tt, z0) -> c2(U12'(tt, z0))
   LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0))
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
   U12'(tt, cons(y0, y1)) -> c3(LENGTH(cons(y0, y1)))
   U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros)))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   U11'_2, LENGTH_1, U12'_2

Compound Symbols:   c2_1, c5_1, c3_1


----------------------------------------

(93) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace U11'(tt, z0) -> c2(U12'(tt, z0)) by 
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U11'(tt, cons(y0, y1)) -> c2(U12'(tt, cons(y0, y1)))
   U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros)))

----------------------------------------

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0))
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
   U12'(tt, cons(y0, y1)) -> c3(LENGTH(cons(y0, y1)))
   U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros)))
   U11'(tt, cons(y0, y1)) -> c2(U12'(tt, cons(y0, y1)))
   U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros)))
S tuples:
   LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0))
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
   U12'(tt, cons(y0, y1)) -> c3(LENGTH(cons(y0, y1)))
   U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros)))
   U11'(tt, cons(y0, y1)) -> c2(U12'(tt, cons(y0, y1)))
   U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros)))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, U11'_2, U12'_2

Compound Symbols:   c5_1, c2_1, c3_1


----------------------------------------

(95) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH(cons(x0, z0)) -> c5(U11'(tt, z0)) by 
   LENGTH(cons(z0, n__zeros)) -> c5(U11'(tt, n__zeros))
   LENGTH(cons(z0, cons(y0, y1))) -> c5(U11'(tt, cons(y0, y1)))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros)))

----------------------------------------

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
   U12'(tt, cons(y0, y1)) -> c3(LENGTH(cons(y0, y1)))
   U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros)))
   U11'(tt, cons(y0, y1)) -> c2(U12'(tt, cons(y0, y1)))
   U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros)))
   LENGTH(cons(z0, cons(y0, y1))) -> c5(U11'(tt, cons(y0, y1)))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros)))
S tuples:
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
   U12'(tt, cons(y0, y1)) -> c3(LENGTH(cons(y0, y1)))
   U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros)))
   U11'(tt, cons(y0, y1)) -> c2(U12'(tt, cons(y0, y1)))
   U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros)))
   LENGTH(cons(z0, cons(y0, y1))) -> c5(U11'(tt, cons(y0, y1)))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros)))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   U11'_2, U12'_2, LENGTH_1

Compound Symbols:   c2_1, c3_1, c5_1


----------------------------------------

(97) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace U12'(tt, cons(y0, y1)) -> c3(LENGTH(cons(y0, y1))) by 
   U12'(tt, cons(z0, n__zeros)) -> c3(LENGTH(cons(z0, n__zeros)))
   U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2))))
   U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros))))

----------------------------------------

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
   U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros)))
   U11'(tt, cons(y0, y1)) -> c2(U12'(tt, cons(y0, y1)))
   U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros)))
   LENGTH(cons(z0, cons(y0, y1))) -> c5(U11'(tt, cons(y0, y1)))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros)))
   U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2))))
   U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros))))
S tuples:
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
   U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros)))
   U11'(tt, cons(y0, y1)) -> c2(U12'(tt, cons(y0, y1)))
   U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros)))
   LENGTH(cons(z0, cons(y0, y1))) -> c5(U11'(tt, cons(y0, y1)))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros)))
   U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2))))
   U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   U11'_2, U12'_2, LENGTH_1

Compound Symbols:   c2_1, c3_1, c5_1


----------------------------------------

(99) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace U11'(tt, cons(y0, y1)) -> c2(U12'(tt, cons(y0, y1))) by 
   U11'(tt, cons(z0, n__zeros)) -> c2(U12'(tt, cons(z0, n__zeros)))
   U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2))))
   U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros))))

----------------------------------------

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
   U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros)))
   U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros)))
   LENGTH(cons(z0, cons(y0, y1))) -> c5(U11'(tt, cons(y0, y1)))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros)))
   U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2))))
   U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros))))
   U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2))))
   U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros))))
S tuples:
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
   U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros)))
   U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros)))
   LENGTH(cons(z0, cons(y0, y1))) -> c5(U11'(tt, cons(y0, y1)))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros)))
   U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2))))
   U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros))))
   U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2))))
   U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   U11'_2, U12'_2, LENGTH_1

Compound Symbols:   c2_1, c3_1, c5_1


----------------------------------------

(101) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH(cons(z0, cons(y0, y1))) -> c5(U11'(tt, cons(y0, y1))) by 
   LENGTH(cons(z0, cons(z1, n__zeros))) -> c5(U11'(tt, cons(z1, n__zeros)))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros))))

----------------------------------------

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
   U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros)))
   U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros)))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros)))
   U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2))))
   U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros))))
   U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2))))
   U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros))))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros))))
S tuples:
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
   U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros)))
   U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros)))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros)))
   U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2))))
   U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros))))
   U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2))))
   U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros))))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   U11'_2, U12'_2, LENGTH_1

Compound Symbols:   c2_1, c3_1, c5_1


----------------------------------------

(103) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros)))
We considered the (Usable) Rules:none
And the Tuples:
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
   U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros)))
   U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros)))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros)))
   U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2))))
   U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros))))
   U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2))))
   U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros))))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros))))
The order we found is given by the following interpretation:

Matrix interpretation [MATRO]:

Non-tuple symbols: 
<<<
 M( n__zeros ) =  	[[0], [0], [0]]
>>>

<<<
 M( 0 ) =  	[[2], [0], [0]]
>>>

<<<
 M( cons_2(x_1, x_2) ) =  	[[0], [0], [3]] 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1 	+ 	[[0, 0, 0], [0, 0, 2], [0, 0, 0]] 	* 	x_2
>>>

<<<
 M( tt ) =  	[[0], [0], [0]]
>>>

Tuple symbols: 
<<<
 M( LENGTH_1(x_1) ) =  	[[0], [2], [0]] 	+ 	[[0, 3, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

<<<
 M( U11'_2(x_1, x_2) ) =  	[[0], [0], [0]] 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1 	+ 	[[0, 3, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_2
>>>

<<<
 M( c5_1(x_1) ) =  	[[0], [0], [0]] 	+ 	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

<<<
 M( c3_1(x_1) ) =  	[[0], [0], [0]] 	+ 	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

<<<
 M( c2_1(x_1) ) =  	[[0], [0], [0]] 	+ 	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

<<<
 M( U12'_2(x_1, x_2) ) =  	[[0], [0], [0]] 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1 	+ 	[[0, 3, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_2
>>>



Matrix type: 

We used a basic matrix type which is not further parametrizeable.





As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
----------------------------------------

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
   U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros)))
   U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros)))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros)))
   U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2))))
   U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros))))
   U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2))))
   U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros))))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros))))
S tuples:
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
   U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros)))
   U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros)))
   U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2))))
   U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros))))
   U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2))))
   U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros))))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros))))
K tuples:
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros)))
Defined Rule Symbols:none

Defined Pair Symbols:   U11'_2, U12'_2, LENGTH_1

Compound Symbols:   c2_1, c3_1, c5_1


----------------------------------------

(105) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros))))
We considered the (Usable) Rules:none
And the Tuples:
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
   U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros)))
   U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros)))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros)))
   U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2))))
   U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros))))
   U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2))))
   U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros))))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros))))
The order we found is given by the following interpretation:

Matrix interpretation [MATRO]:

Non-tuple symbols: 
<<<
 M( n__zeros ) =  	[[0], [0], [0]]
>>>

<<<
 M( 0 ) =  	[[2], [0], [0]]
>>>

<<<
 M( cons_2(x_1, x_2) ) =  	[[0], [0], [2]] 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1 	+ 	[[0, 0, 0], [0, 1, 2], [0, 0, 0]] 	* 	x_2
>>>

<<<
 M( tt ) =  	[[0], [0], [0]]
>>>

Tuple symbols: 
<<<
 M( LENGTH_1(x_1) ) =  	[[0], [0], [0]] 	+ 	[[0, 2, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

<<<
 M( U11'_2(x_1, x_2) ) =  	[[0], [0], [2]] 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1 	+ 	[[0, 2, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_2
>>>

<<<
 M( c5_1(x_1) ) =  	[[0], [0], [0]] 	+ 	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

<<<
 M( c3_1(x_1) ) =  	[[0], [0], [0]] 	+ 	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

<<<
 M( c2_1(x_1) ) =  	[[0], [0], [0]] 	+ 	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

<<<
 M( U12'_2(x_1, x_2) ) =  	[[0], [0], [2]] 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1 	+ 	[[0, 2, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_2
>>>



Matrix type: 

We used a basic matrix type which is not further parametrizeable.





As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
----------------------------------------

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
   U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros)))
   U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros)))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros)))
   U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2))))
   U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros))))
   U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2))))
   U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros))))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros))))
S tuples:
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
   U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros)))
   U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros)))
   U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2))))
   U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros))))
   U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2))))
   U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros))))
K tuples:
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros)))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros))))
Defined Rule Symbols:none

Defined Pair Symbols:   U11'_2, U12'_2, LENGTH_1

Compound Symbols:   c2_1, c3_1, c5_1


----------------------------------------

(107) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2))))
   U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros))))
   U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2))))
   U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros))))
   U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2))))
   U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros))))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros)))
   LENGTH(cons(z0, cons(z1, n__zeros))) -> c5(U11'(tt, cons(z1, n__zeros)))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros))))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros)))
   LENGTH(cons(z0, cons(z1, n__zeros))) -> c5(U11'(tt, cons(z1, n__zeros)))

----------------------------------------

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
   U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros)))
   U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros)))
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros)))
   U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2))))
   U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros))))
   U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2))))
   U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros))))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros))))
S tuples:
   U11'(tt, n__zeros) -> c2(U12'(tt, cons(0, n__zeros)))
   U11'(tt, n__zeros) -> c2(U12'(tt, n__zeros))
   U12'(tt, n__zeros) -> c3(LENGTH(cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, cons(0, n__zeros)))
   LENGTH(cons(x0, n__zeros)) -> c5(U11'(tt, n__zeros))
   U12'(tt, cons(y0, n__zeros)) -> c3(LENGTH(cons(y0, n__zeros)))
   U11'(tt, cons(y0, n__zeros)) -> c2(U12'(tt, cons(y0, n__zeros)))
K tuples:
   LENGTH(cons(z0, cons(y0, n__zeros))) -> c5(U11'(tt, cons(y0, n__zeros)))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c5(U11'(tt, cons(z1, cons(y1, y2))))
   LENGTH(cons(z0, cons(z1, cons(y1, n__zeros)))) -> c5(U11'(tt, cons(z1, cons(y1, n__zeros))))
   U11'(tt, cons(z0, cons(y1, y2))) -> c2(U12'(tt, cons(z0, cons(y1, y2))))
   U11'(tt, cons(z0, cons(y1, n__zeros))) -> c2(U12'(tt, cons(z0, cons(y1, n__zeros))))
   U12'(tt, cons(z0, cons(y1, y2))) -> c3(LENGTH(cons(z0, cons(y1, y2))))
   U12'(tt, cons(z0, cons(y1, n__zeros))) -> c3(LENGTH(cons(z0, cons(y1, n__zeros))))
Defined Rule Symbols:none

Defined Pair Symbols:   U11'_2, U12'_2, LENGTH_1

Compound Symbols:   c2_1, c3_1, c5_1


----------------------------------------

(109) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(110)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros) [1]
   U11(tt, L) -> U12(tt, activate(L)) [1]
   U12(tt, L) -> s(length(activate(L))) [1]
   length(nil) -> 0 [1]
   length(cons(N, L)) -> U11(tt, activate(L)) [1]
   zeros -> n__zeros [1]
   activate(n__zeros) -> zeros [1]
   activate(X) -> X [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(111) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(112)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros) [1]
   U11(tt, L) -> U12(tt, activate(L)) [1]
   U12(tt, L) -> s(length(activate(L))) [1]
   length(nil) -> 0 [1]
   length(cons(N, L)) -> U11(tt, activate(L)) [1]
   zeros -> n__zeros [1]
   activate(n__zeros) -> zeros [1]
   activate(X) -> X [1]

The TRS has the following type information:
zeros :: n__zeros:cons:nil
cons :: 0:s -> n__zeros:cons:nil -> n__zeros:cons:nil
0 :: 0:s
n__zeros :: n__zeros:cons:nil
U11 :: tt -> n__zeros:cons:nil -> 0:s
tt :: tt
U12 :: tt -> n__zeros:cons:nil -> 0:s
activate :: n__zeros:cons:nil -> n__zeros:cons:nil
s :: 0:s -> 0:s
length :: n__zeros:cons:nil -> 0:s
nil :: n__zeros:cons:nil

Rewrite Strategy: INNERMOST
----------------------------------------

(113) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   U11_2
   U12_2
   length_1

(c) The following functions are completely defined:

   activate_1
   zeros

Due to the following rules being added:
none

And the following fresh constants: none

----------------------------------------

(114)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros) [1]
   U11(tt, L) -> U12(tt, activate(L)) [1]
   U12(tt, L) -> s(length(activate(L))) [1]
   length(nil) -> 0 [1]
   length(cons(N, L)) -> U11(tt, activate(L)) [1]
   zeros -> n__zeros [1]
   activate(n__zeros) -> zeros [1]
   activate(X) -> X [1]

The TRS has the following type information:
zeros :: n__zeros:cons:nil
cons :: 0:s -> n__zeros:cons:nil -> n__zeros:cons:nil
0 :: 0:s
n__zeros :: n__zeros:cons:nil
U11 :: tt -> n__zeros:cons:nil -> 0:s
tt :: tt
U12 :: tt -> n__zeros:cons:nil -> 0:s
activate :: n__zeros:cons:nil -> n__zeros:cons:nil
s :: 0:s -> 0:s
length :: n__zeros:cons:nil -> 0:s
nil :: n__zeros:cons:nil

Rewrite Strategy: INNERMOST
----------------------------------------

(115) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(116)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros) [1]
   U11(tt, n__zeros) -> U12(tt, zeros) [2]
   U11(tt, L) -> U12(tt, L) [2]
   U12(tt, n__zeros) -> s(length(zeros)) [2]
   U12(tt, L) -> s(length(L)) [2]
   length(nil) -> 0 [1]
   length(cons(N, n__zeros)) -> U11(tt, zeros) [2]
   length(cons(N, L)) -> U11(tt, L) [2]
   zeros -> n__zeros [1]
   activate(n__zeros) -> zeros [1]
   activate(X) -> X [1]

The TRS has the following type information:
zeros :: n__zeros:cons:nil
cons :: 0:s -> n__zeros:cons:nil -> n__zeros:cons:nil
0 :: 0:s
n__zeros :: n__zeros:cons:nil
U11 :: tt -> n__zeros:cons:nil -> 0:s
tt :: tt
U12 :: tt -> n__zeros:cons:nil -> 0:s
activate :: n__zeros:cons:nil -> n__zeros:cons:nil
s :: 0:s -> 0:s
length :: n__zeros:cons:nil -> 0:s
nil :: n__zeros:cons:nil

Rewrite Strategy: INNERMOST
----------------------------------------

(117) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   0 => 0
   n__zeros => 0
   tt => 0
   nil => 1

----------------------------------------

(118)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z') -{ 2 }-> U12(0, L) :|: L >= 0, z = 0, z' = L
   U11(z, z') -{ 2 }-> U12(0, zeros) :|: z = 0, z' = 0
   U12(z, z') -{ 2 }-> 1 + length(L) :|: L >= 0, z = 0, z' = L
   U12(z, z') -{ 2 }-> 1 + length(zeros) :|: z = 0, z' = 0
   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> zeros :|: z = 0
   length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 2 }-> U11(0, zeros) :|: z = 1 + N + 0, N >= 0
   length(z) -{ 1 }-> 0 :|: z = 1
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 


----------------------------------------

(119) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   zeros -{ 1 }-> 1 + 0 + 0 :|: 
   zeros -{ 1 }-> 0 :|: 

----------------------------------------

(120)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z') -{ 2 }-> U12(0, L) :|: L >= 0, z = 0, z' = L
   U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0
   U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12(z, z') -{ 2 }-> 1 + length(L) :|: L >= 0, z = 0, z' = L
   U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0
   U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> U11(0, 0) :|: z = 1 + N + 0, N >= 0
   length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z = 1 + N + 0, N >= 0
   length(z) -{ 1 }-> 0 :|: z = 1
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 


----------------------------------------

(121) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(122)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z') -{ 2 }-> U12(0, z') :|: z' >= 0, z = 0
   U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0
   U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12(z, z') -{ 2 }-> 1 + length(z') :|: z' >= 0, z = 0
   U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0
   U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> U11(0, 0) :|: z - 1 >= 0
   length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z - 1 >= 0
   length(z) -{ 1 }-> 0 :|: z = 1
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 


----------------------------------------

(123) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { activate }
   { zeros }
   { length, U12, U11 }

----------------------------------------

(124)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z') -{ 2 }-> U12(0, z') :|: z' >= 0, z = 0
   U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0
   U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12(z, z') -{ 2 }-> 1 + length(z') :|: z' >= 0, z = 0
   U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0
   U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> U11(0, 0) :|: z - 1 >= 0
   length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z - 1 >= 0
   length(z) -{ 1 }-> 0 :|: z = 1
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {activate}, {zeros}, {length,U12,U11}

----------------------------------------

(125) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(126)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z') -{ 2 }-> U12(0, z') :|: z' >= 0, z = 0
   U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0
   U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12(z, z') -{ 2 }-> 1 + length(z') :|: z' >= 0, z = 0
   U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0
   U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> U11(0, 0) :|: z - 1 >= 0
   length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z - 1 >= 0
   length(z) -{ 1 }-> 0 :|: z = 1
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {activate}, {zeros}, {length,U12,U11}

----------------------------------------

(127) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z

----------------------------------------

(128)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z') -{ 2 }-> U12(0, z') :|: z' >= 0, z = 0
   U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0
   U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12(z, z') -{ 2 }-> 1 + length(z') :|: z' >= 0, z = 0
   U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0
   U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> U11(0, 0) :|: z - 1 >= 0
   length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z - 1 >= 0
   length(z) -{ 1 }-> 0 :|: z = 1
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {activate}, {zeros}, {length,U12,U11}
Previous analysis results are:
  activate: runtime: ?, size: O(n^1) [1 + z]

----------------------------------------

(129) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

----------------------------------------

(130)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z') -{ 2 }-> U12(0, z') :|: z' >= 0, z = 0
   U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0
   U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12(z, z') -{ 2 }-> 1 + length(z') :|: z' >= 0, z = 0
   U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0
   U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> U11(0, 0) :|: z - 1 >= 0
   length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z - 1 >= 0
   length(z) -{ 1 }-> 0 :|: z = 1
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {zeros}, {length,U12,U11}
Previous analysis results are:
  activate: runtime: O(1) [2], size: O(n^1) [1 + z]

----------------------------------------

(131) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(132)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z') -{ 2 }-> U12(0, z') :|: z' >= 0, z = 0
   U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0
   U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12(z, z') -{ 2 }-> 1 + length(z') :|: z' >= 0, z = 0
   U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0
   U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> U11(0, 0) :|: z - 1 >= 0
   length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z - 1 >= 0
   length(z) -{ 1 }-> 0 :|: z = 1
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {zeros}, {length,U12,U11}
Previous analysis results are:
  activate: runtime: O(1) [2], size: O(n^1) [1 + z]

----------------------------------------

(133) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: zeros
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(134)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z') -{ 2 }-> U12(0, z') :|: z' >= 0, z = 0
   U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0
   U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12(z, z') -{ 2 }-> 1 + length(z') :|: z' >= 0, z = 0
   U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0
   U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> U11(0, 0) :|: z - 1 >= 0
   length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z - 1 >= 0
   length(z) -{ 1 }-> 0 :|: z = 1
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {zeros}, {length,U12,U11}
Previous analysis results are:
  activate: runtime: O(1) [2], size: O(n^1) [1 + z]
  zeros: runtime: ?, size: O(1) [1]

----------------------------------------

(135) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: zeros
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(136)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z') -{ 2 }-> U12(0, z') :|: z' >= 0, z = 0
   U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0
   U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12(z, z') -{ 2 }-> 1 + length(z') :|: z' >= 0, z = 0
   U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0
   U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> U11(0, 0) :|: z - 1 >= 0
   length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z - 1 >= 0
   length(z) -{ 1 }-> 0 :|: z = 1
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {length,U12,U11}
Previous analysis results are:
  activate: runtime: O(1) [2], size: O(n^1) [1 + z]
  zeros: runtime: O(1) [1], size: O(1) [1]

----------------------------------------

(137) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(138)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z') -{ 2 }-> U12(0, z') :|: z' >= 0, z = 0
   U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0
   U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12(z, z') -{ 2 }-> 1 + length(z') :|: z' >= 0, z = 0
   U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0
   U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> U11(0, 0) :|: z - 1 >= 0
   length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z - 1 >= 0
   length(z) -{ 1 }-> 0 :|: z = 1
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {length,U12,U11}
Previous analysis results are:
  activate: runtime: O(1) [2], size: O(n^1) [1 + z]
  zeros: runtime: O(1) [1], size: O(1) [1]

----------------------------------------

(139) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: length
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

Computed SIZE bound using CoFloCo for: U12
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

Computed SIZE bound using CoFloCo for: U11
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

----------------------------------------

(140)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z') -{ 2 }-> U12(0, z') :|: z' >= 0, z = 0
   U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0
   U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12(z, z') -{ 2 }-> 1 + length(z') :|: z' >= 0, z = 0
   U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0
   U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> U11(0, 0) :|: z - 1 >= 0
   length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z - 1 >= 0
   length(z) -{ 1 }-> 0 :|: z = 1
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {length,U12,U11}
Previous analysis results are:
  activate: runtime: O(1) [2], size: O(n^1) [1 + z]
  zeros: runtime: O(1) [1], size: O(1) [1]
  length: runtime: ?, size: INF
  U12: runtime: ?, size: INF
  U11: runtime: ?, size: INF

----------------------------------------

(141) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: length
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

----------------------------------------

(142)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z') -{ 2 }-> U12(0, z') :|: z' >= 0, z = 0
   U11(z, z') -{ 3 }-> U12(0, 0) :|: z = 0, z' = 0
   U11(z, z') -{ 3 }-> U12(0, 1 + 0 + 0) :|: z = 0, z' = 0
   U12(z, z') -{ 2 }-> 1 + length(z') :|: z' >= 0, z = 0
   U12(z, z') -{ 3 }-> 1 + length(0) :|: z = 0, z' = 0
   U12(z, z') -{ 3 }-> 1 + length(1 + 0 + 0) :|: z = 0, z' = 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + 0 :|: z = 0
   length(z) -{ 2 }-> U11(0, L) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 3 }-> U11(0, 0) :|: z - 1 >= 0
   length(z) -{ 3 }-> U11(0, 1 + 0 + 0) :|: z - 1 >= 0
   length(z) -{ 1 }-> 0 :|: z = 1
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Function symbols to be analyzed: {length,U12,U11}
Previous analysis results are:
  activate: runtime: O(1) [2], size: O(n^1) [1 + z]
  zeros: runtime: O(1) [1], size: O(1) [1]
  length: runtime: INF, size: INF
  U12: runtime: ?, size: INF
  U11: runtime: ?, size: INF

----------------------------------------

(143) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   length(v0) -> null_length [0]

And the following fresh constants:    null_length

----------------------------------------

(144)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   zeros -> cons(0, n__zeros) [1]
   U11(tt, L) -> U12(tt, activate(L)) [1]
   U12(tt, L) -> s(length(activate(L))) [1]
   length(nil) -> 0 [1]
   length(cons(N, L)) -> U11(tt, activate(L)) [1]
   zeros -> n__zeros [1]
   activate(n__zeros) -> zeros [1]
   activate(X) -> X [1]
   length(v0) -> null_length [0]

The TRS has the following type information:
zeros :: n__zeros:cons:nil
cons :: 0:s:null_length -> n__zeros:cons:nil -> n__zeros:cons:nil
0 :: 0:s:null_length
n__zeros :: n__zeros:cons:nil
U11 :: tt -> n__zeros:cons:nil -> 0:s:null_length
tt :: tt
U12 :: tt -> n__zeros:cons:nil -> 0:s:null_length
activate :: n__zeros:cons:nil -> n__zeros:cons:nil
s :: 0:s:null_length -> 0:s:null_length
length :: n__zeros:cons:nil -> 0:s:null_length
nil :: n__zeros:cons:nil
null_length :: 0:s:null_length

Rewrite Strategy: INNERMOST
----------------------------------------

(145) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   0 => 0
   n__zeros => 0
   tt => 0
   nil => 1
   null_length => 0

----------------------------------------

(146)
Obligation:
Complexity RNTS consisting of the following rules:

   U11(z, z') -{ 1 }-> U12(0, activate(L)) :|: L >= 0, z = 0, z' = L
   U12(z, z') -{ 1 }-> 1 + length(activate(L)) :|: L >= 0, z = 0, z' = L
   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> zeros :|: z = 0
   length(z) -{ 1 }-> U11(0, activate(L)) :|: z = 1 + N + L, L >= 0, N >= 0
   length(z) -{ 1 }-> 0 :|: z = 1
   length(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   zeros -{ 1 }-> 0 :|: 
   zeros -{ 1 }-> 1 + 0 + 0 :|: 

Only complete derivations are relevant for the runtime complexity.