KILLED
proof of input_w80Di0i0cj.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) typed CpxTrs
    (13) OrderProof [LOWER BOUND(ID), 11 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 491 ms]
    (16) typed CpxTrs
    (17) RewriteLemmaProof [LOWER BOUND(ID), 152 ms]
    (18) typed CpxTrs
    (19) RewriteLemmaProof [LOWER BOUND(ID), 336 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 1565 ms]
    (22) BEST
        (23) proven lower bound
            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
            (25) BOUNDS(n^1, INF)
        (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 536 ms]
            (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 1231 ms]
            (30) BOUNDS(1, INF)
(31) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(32) CdtProblem
    (33) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) CdtProblem
    (35) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (36) CdtProblem
    (37) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CpxRelTRS
        (41) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
        (42) CpxTRS
        (43) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (44) CpxWeightedTrs
            (45) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
            (46) CpxTypedWeightedTrs
            (47) CompletionProof [UPPER BOUND(ID), 0 ms]
            (48) CpxTypedWeightedCompleteTrs
                (49) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
                (50) CpxTypedWeightedCompleteTrs
                (51) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (52) CpxRNTS
                (53) InliningProof [UPPER BOUND(ID), 475 ms]
                (54) CpxRNTS
                (55) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
                (56) CpxRNTS
                (57) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
                (58) CpxRNTS
                (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (60) CpxRNTS
                (61) IntTrsBoundProof [UPPER BOUND(ID), 6 ms]
                (62) CpxRNTS
                (63) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
                (64) CpxRNTS
                (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (66) CpxRNTS
                (67) IntTrsBoundProof [UPPER BOUND(ID), 71 ms]
                (68) CpxRNTS
                (69) IntTrsBoundProof [UPPER BOUND(ID), 14 ms]
                (70) CpxRNTS
                (71) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (72) CpxRNTS
                (73) IntTrsBoundProof [UPPER BOUND(ID), 148 ms]
                (74) CpxRNTS
                (75) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
                (76) CpxRNTS
                (77) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (78) CpxRNTS
                (79) IntTrsBoundProof [UPPER BOUND(ID), 337 ms]
                (80) CpxRNTS
                (81) IntTrsBoundProof [UPPER BOUND(ID), 25 ms]
                (82) CpxRNTS
                (83) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (84) CpxRNTS
                (85) IntTrsBoundProof [UPPER BOUND(ID), 2161 ms]
                (86) CpxRNTS
                (87) IntTrsBoundProof [UPPER BOUND(ID), 811 ms]
                (88) CpxRNTS
            (89) CompletionProof [UPPER BOUND(ID), 0 ms]
            (90) CpxTypedWeightedCompleteTrs
                (91) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (92) CpxRNTS
    (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
        (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms]
        (96) CdtProblem
        (97) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
        (98) CdtProblem
        (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (100) CdtProblem
        (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (102) CdtProblem
        (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (104) CdtProblem
        (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (106) CdtProblem
        (107) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (108) CdtProblem
        (109) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
        (110) CdtProblem
        (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (112) CdtProblem
        (113) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (114) CdtProblem
        (115) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
        (116) CdtProblem
        (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (118) CdtProblem
        (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (120) CdtProblem
        (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (122) CdtProblem
        (123) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
        (124) CdtProblem
        (125) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (126) CdtProblem
        (127) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (128) CdtProblem
        (129) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (130) CdtProblem
        (131) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (132) CdtProblem
        (133) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (134) CdtProblem
        (135) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (136) CdtProblem
        (137) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (138) CdtProblem
        (139) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (140) CdtProblem
        (141) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (142) CdtProblem
        (143) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (144) CdtProblem
        (145) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (146) CdtProblem
        (147) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (148) CdtProblem
        (149) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (150) CdtProblem
        (151) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (152) CdtProblem
        (153) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (154) CdtProblem
        (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (156) CdtProblem
        (157) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (158) CdtProblem
        (159) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (160) CdtProblem
        (161) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (162) CdtProblem
        (163) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 1895 ms]
        (164) CdtProblem
        (165) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (166) CdtProblem
(167) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(168) CpxWeightedTrs
    (169) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (170) CpxTypedWeightedTrs
    (171) CompletionProof [UPPER BOUND(ID), 0 ms]
    (172) CpxTypedWeightedCompleteTrs
        (173) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (174) CpxTypedWeightedCompleteTrs
        (175) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (176) CpxRNTS
        (177) InliningProof [UPPER BOUND(ID), 373 ms]
        (178) CpxRNTS
        (179) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (180) CpxRNTS
        (181) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (182) CpxRNTS
        (183) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (184) CpxRNTS
        (185) IntTrsBoundProof [UPPER BOUND(ID), 27 ms]
        (186) CpxRNTS
        (187) IntTrsBoundProof [UPPER BOUND(ID), 14 ms]
        (188) CpxRNTS
        (189) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (190) CpxRNTS
        (191) IntTrsBoundProof [UPPER BOUND(ID), 225 ms]
        (192) CpxRNTS
        (193) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
        (194) CpxRNTS
        (195) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (196) CpxRNTS
        (197) IntTrsBoundProof [UPPER BOUND(ID), 169 ms]
        (198) CpxRNTS
        (199) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
        (200) CpxRNTS
        (201) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (202) CpxRNTS
        (203) IntTrsBoundProof [UPPER BOUND(ID), 195 ms]
        (204) CpxRNTS
        (205) IntTrsBoundProof [UPPER BOUND(ID), 6 ms]
        (206) CpxRNTS
        (207) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (208) CpxRNTS
        (209) IntTrsBoundProof [UPPER BOUND(ID), 3732 ms]
        (210) CpxRNTS
        (211) IntTrsBoundProof [UPPER BOUND(ID), 709 ms]
        (212) CpxRNTS
    (213) CompletionProof [UPPER BOUND(ID), 0 ms]
    (214) CpxTypedWeightedCompleteTrs
        (215) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (216) 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:

   from(X) -> cons(X, n__from(s(X)))
   length(n__nil) -> 0
   length(n__cons(X, Y)) -> s(length1(activate(Y)))
   length1(X) -> length(activate(X))
   from(X) -> n__from(X)
   nil -> n__nil
   cons(X1, X2) -> n__cons(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(n__nil) -> nil
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   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:

   from(X) -> cons(X, n__from(s(X)))
   length(n__nil) -> 0'
   length(n__cons(X, Y)) -> s(length1(activate(Y)))
   length1(X) -> length(activate(X))
   from(X) -> n__from(X)
   nil -> n__nil
   cons(X1, X2) -> n__cons(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(n__nil) -> nil
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(X) -> X

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

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

(4)
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:

   from(X) -> cons(X, n__from(s(X)))
   length(n__nil) -> 0
   length(n__cons(X, Y)) -> s(length1(activate(Y)))
   length1(X) -> length(activate(X))
   from(X) -> n__from(X)
   nil -> n__nil
   cons(X1, X2) -> n__cons(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(n__nil) -> nil
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(X) -> X

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   length(n__nil) -> 0
   length(n__cons(z0, z1)) -> s(length1(activate(z1)))
   length1(z0) -> length(activate(z0))
   nil -> n__nil
   cons(z0, z1) -> n__cons(z0, z1)
   activate(n__from(z0)) -> from(z0)
   activate(n__nil) -> nil
   activate(n__cons(z0, z1)) -> cons(z0, z1)
   activate(z0) -> z0
Tuples:
   FROM(z0) -> c(CONS(z0, n__from(s(z0))))
   FROM(z0) -> c1
   LENGTH(n__nil) -> c2
   LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)), ACTIVATE(z1))
   LENGTH1(z0) -> c4(LENGTH(activate(z0)), ACTIVATE(z0))
   NIL -> c5
   CONS(z0, z1) -> c6
   ACTIVATE(n__from(z0)) -> c7(FROM(z0))
   ACTIVATE(n__nil) -> c8(NIL)
   ACTIVATE(n__cons(z0, z1)) -> c9(CONS(z0, z1))
   ACTIVATE(z0) -> c10
S tuples:
   FROM(z0) -> c(CONS(z0, n__from(s(z0))))
   FROM(z0) -> c1
   LENGTH(n__nil) -> c2
   LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)), ACTIVATE(z1))
   LENGTH1(z0) -> c4(LENGTH(activate(z0)), ACTIVATE(z0))
   NIL -> c5
   CONS(z0, z1) -> c6
   ACTIVATE(n__from(z0)) -> c7(FROM(z0))
   ACTIVATE(n__nil) -> c8(NIL)
   ACTIVATE(n__cons(z0, z1)) -> c9(CONS(z0, z1))
   ACTIVATE(z0) -> c10
K tuples:none
Defined Rule Symbols:   from_1, length_1, length1_1, nil, cons_2, activate_1

Defined Pair Symbols:   FROM_1, LENGTH_1, LENGTH1_1, NIL, CONS_2, ACTIVATE_1

Compound Symbols:   c_1, c1, c2, c3_2, c4_2, c5, c6, c7_1, c8_1, c9_1, c10


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

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

(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:

   FROM(z0) -> c(CONS(z0, n__from(s(z0))))
   FROM(z0) -> c1
   LENGTH(n__nil) -> c2
   LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)), ACTIVATE(z1))
   LENGTH1(z0) -> c4(LENGTH(activate(z0)), ACTIVATE(z0))
   NIL -> c5
   CONS(z0, z1) -> c6
   ACTIVATE(n__from(z0)) -> c7(FROM(z0))
   ACTIVATE(n__nil) -> c8(NIL)
   ACTIVATE(n__cons(z0, z1)) -> c9(CONS(z0, z1))
   ACTIVATE(z0) -> c10

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

   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   length(n__nil) -> 0
   length(n__cons(z0, z1)) -> s(length1(activate(z1)))
   length1(z0) -> length(activate(z0))
   nil -> n__nil
   cons(z0, z1) -> n__cons(z0, z1)
   activate(n__from(z0)) -> from(z0)
   activate(n__nil) -> nil
   activate(n__cons(z0, z1)) -> cons(z0, z1)
   activate(z0) -> z0

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

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

(10)
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:

   FROM(z0) -> c(CONS(z0, n__from(s(z0))))
   FROM(z0) -> c1
   LENGTH(n__nil) -> c2
   LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)), ACTIVATE(z1))
   LENGTH1(z0) -> c4(LENGTH(activate(z0)), ACTIVATE(z0))
   NIL -> c5
   CONS(z0, z1) -> c6
   ACTIVATE(n__from(z0)) -> c7(FROM(z0))
   ACTIVATE(n__nil) -> c8(NIL)
   ACTIVATE(n__cons(z0, z1)) -> c9(CONS(z0, z1))
   ACTIVATE(z0) -> c10

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

   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   length(n__nil) -> 0'
   length(n__cons(z0, z1)) -> s(length1(activate(z1)))
   length1(z0) -> length(activate(z0))
   nil -> n__nil
   cons(z0, z1) -> n__cons(z0, z1)
   activate(n__from(z0)) -> from(z0)
   activate(n__nil) -> nil
   activate(n__cons(z0, z1)) -> cons(z0, z1)
   activate(z0) -> z0

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

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

(12)
Obligation:
Innermost TRS:
Rules:
FROM(z0) -> c(CONS(z0, n__from(s(z0))))
FROM(z0) -> c1
LENGTH(n__nil) -> c2
LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)), ACTIVATE(z1))
LENGTH1(z0) -> c4(LENGTH(activate(z0)), ACTIVATE(z0))
NIL -> c5
CONS(z0, z1) -> c6
ACTIVATE(n__from(z0)) -> c7(FROM(z0))
ACTIVATE(n__nil) -> c8(NIL)
ACTIVATE(n__cons(z0, z1)) -> c9(CONS(z0, z1))
ACTIVATE(z0) -> c10
from(z0) -> cons(z0, n__from(s(z0)))
from(z0) -> n__from(z0)
length(n__nil) -> 0'
length(n__cons(z0, z1)) -> s(length1(activate(z1)))
length1(z0) -> length(activate(z0))
nil -> n__nil
cons(z0, z1) -> n__cons(z0, z1)
activate(n__from(z0)) -> from(z0)
activate(n__nil) -> nil
activate(n__cons(z0, z1)) -> cons(z0, z1)
activate(z0) -> z0

Types:
FROM :: s:0' -> c:c1
c :: c6 -> c:c1
CONS :: s:0' -> n__from:n__nil:n__cons -> c6
n__from :: s:0' -> n__from:n__nil:n__cons
s :: s:0' -> s:0'
c1 :: c:c1
LENGTH :: n__from:n__nil:n__cons -> c2:c3
n__nil :: n__from:n__nil:n__cons
c2 :: c2:c3
n__cons :: s:0' -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
c3 :: c4 -> c7:c8:c9:c10 -> c2:c3
LENGTH1 :: n__from:n__nil:n__cons -> c4
activate :: n__from:n__nil:n__cons -> n__from:n__nil:n__cons
ACTIVATE :: n__from:n__nil:n__cons -> c7:c8:c9:c10
c4 :: c2:c3 -> c7:c8:c9:c10 -> c4
NIL :: c5
c5 :: c5
c6 :: c6
c7 :: c:c1 -> c7:c8:c9:c10
c8 :: c5 -> c7:c8:c9:c10
c9 :: c6 -> c7:c8:c9:c10
c10 :: c7:c8:c9:c10
from :: s:0' -> n__from:n__nil:n__cons
cons :: s:0' -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
length :: n__from:n__nil:n__cons -> s:0'
0' :: s:0'
length1 :: n__from:n__nil:n__cons -> s:0'
nil :: n__from:n__nil:n__cons
hole_c:c11_11 :: c:c1
hole_s:0'2_11 :: s:0'
hole_c63_11 :: c6
hole_n__from:n__nil:n__cons4_11 :: n__from:n__nil:n__cons
hole_c2:c35_11 :: c2:c3
hole_c46_11 :: c4
hole_c7:c8:c9:c107_11 :: c7:c8:c9:c10
hole_c58_11 :: c5
gen_s:0'9_11 :: Nat -> s:0'
gen_n__from:n__nil:n__cons10_11 :: Nat -> n__from:n__nil:n__cons

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

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
LENGTH, LENGTH1, length, length1

They will be analysed ascendingly in the following order:
LENGTH = LENGTH1
length = length1

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

(14)
Obligation:
Innermost TRS:
Rules:
FROM(z0) -> c(CONS(z0, n__from(s(z0))))
FROM(z0) -> c1
LENGTH(n__nil) -> c2
LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)), ACTIVATE(z1))
LENGTH1(z0) -> c4(LENGTH(activate(z0)), ACTIVATE(z0))
NIL -> c5
CONS(z0, z1) -> c6
ACTIVATE(n__from(z0)) -> c7(FROM(z0))
ACTIVATE(n__nil) -> c8(NIL)
ACTIVATE(n__cons(z0, z1)) -> c9(CONS(z0, z1))
ACTIVATE(z0) -> c10
from(z0) -> cons(z0, n__from(s(z0)))
from(z0) -> n__from(z0)
length(n__nil) -> 0'
length(n__cons(z0, z1)) -> s(length1(activate(z1)))
length1(z0) -> length(activate(z0))
nil -> n__nil
cons(z0, z1) -> n__cons(z0, z1)
activate(n__from(z0)) -> from(z0)
activate(n__nil) -> nil
activate(n__cons(z0, z1)) -> cons(z0, z1)
activate(z0) -> z0

Types:
FROM :: s:0' -> c:c1
c :: c6 -> c:c1
CONS :: s:0' -> n__from:n__nil:n__cons -> c6
n__from :: s:0' -> n__from:n__nil:n__cons
s :: s:0' -> s:0'
c1 :: c:c1
LENGTH :: n__from:n__nil:n__cons -> c2:c3
n__nil :: n__from:n__nil:n__cons
c2 :: c2:c3
n__cons :: s:0' -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
c3 :: c4 -> c7:c8:c9:c10 -> c2:c3
LENGTH1 :: n__from:n__nil:n__cons -> c4
activate :: n__from:n__nil:n__cons -> n__from:n__nil:n__cons
ACTIVATE :: n__from:n__nil:n__cons -> c7:c8:c9:c10
c4 :: c2:c3 -> c7:c8:c9:c10 -> c4
NIL :: c5
c5 :: c5
c6 :: c6
c7 :: c:c1 -> c7:c8:c9:c10
c8 :: c5 -> c7:c8:c9:c10
c9 :: c6 -> c7:c8:c9:c10
c10 :: c7:c8:c9:c10
from :: s:0' -> n__from:n__nil:n__cons
cons :: s:0' -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
length :: n__from:n__nil:n__cons -> s:0'
0' :: s:0'
length1 :: n__from:n__nil:n__cons -> s:0'
nil :: n__from:n__nil:n__cons
hole_c:c11_11 :: c:c1
hole_s:0'2_11 :: s:0'
hole_c63_11 :: c6
hole_n__from:n__nil:n__cons4_11 :: n__from:n__nil:n__cons
hole_c2:c35_11 :: c2:c3
hole_c46_11 :: c4
hole_c7:c8:c9:c107_11 :: c7:c8:c9:c10
hole_c58_11 :: c5
gen_s:0'9_11 :: Nat -> s:0'
gen_n__from:n__nil:n__cons10_11 :: Nat -> n__from:n__nil:n__cons


Generator Equations:
gen_s:0'9_11(0) <=> 0'
gen_s:0'9_11(+(x, 1)) <=> s(gen_s:0'9_11(x))
gen_n__from:n__nil:n__cons10_11(0) <=> n__from(0')
gen_n__from:n__nil:n__cons10_11(+(x, 1)) <=> n__cons(0', gen_n__from:n__nil:n__cons10_11(x))


The following defined symbols remain to be analysed:
length1, LENGTH, LENGTH1, length

They will be analysed ascendingly in the following order:
LENGTH = LENGTH1
length = length1

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length1(gen_n__from:n__nil:n__cons10_11(+(1, n12_11))) -> *11_11, rt in Omega(0)

Induction Base:
length1(gen_n__from:n__nil:n__cons10_11(+(1, 0)))

Induction Step:
length1(gen_n__from:n__nil:n__cons10_11(+(1, +(n12_11, 1)))) ->_R^Omega(0)
length(activate(gen_n__from:n__nil:n__cons10_11(+(1, +(n12_11, 1))))) ->_R^Omega(0)
length(gen_n__from:n__nil:n__cons10_11(+(2, n12_11))) ->_R^Omega(0)
s(length1(activate(gen_n__from:n__nil:n__cons10_11(+(1, n12_11))))) ->_R^Omega(0)
s(length1(gen_n__from:n__nil:n__cons10_11(+(1, n12_11)))) ->_IH
s(*11_11)

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

(16)
Obligation:
Innermost TRS:
Rules:
FROM(z0) -> c(CONS(z0, n__from(s(z0))))
FROM(z0) -> c1
LENGTH(n__nil) -> c2
LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)), ACTIVATE(z1))
LENGTH1(z0) -> c4(LENGTH(activate(z0)), ACTIVATE(z0))
NIL -> c5
CONS(z0, z1) -> c6
ACTIVATE(n__from(z0)) -> c7(FROM(z0))
ACTIVATE(n__nil) -> c8(NIL)
ACTIVATE(n__cons(z0, z1)) -> c9(CONS(z0, z1))
ACTIVATE(z0) -> c10
from(z0) -> cons(z0, n__from(s(z0)))
from(z0) -> n__from(z0)
length(n__nil) -> 0'
length(n__cons(z0, z1)) -> s(length1(activate(z1)))
length1(z0) -> length(activate(z0))
nil -> n__nil
cons(z0, z1) -> n__cons(z0, z1)
activate(n__from(z0)) -> from(z0)
activate(n__nil) -> nil
activate(n__cons(z0, z1)) -> cons(z0, z1)
activate(z0) -> z0

Types:
FROM :: s:0' -> c:c1
c :: c6 -> c:c1
CONS :: s:0' -> n__from:n__nil:n__cons -> c6
n__from :: s:0' -> n__from:n__nil:n__cons
s :: s:0' -> s:0'
c1 :: c:c1
LENGTH :: n__from:n__nil:n__cons -> c2:c3
n__nil :: n__from:n__nil:n__cons
c2 :: c2:c3
n__cons :: s:0' -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
c3 :: c4 -> c7:c8:c9:c10 -> c2:c3
LENGTH1 :: n__from:n__nil:n__cons -> c4
activate :: n__from:n__nil:n__cons -> n__from:n__nil:n__cons
ACTIVATE :: n__from:n__nil:n__cons -> c7:c8:c9:c10
c4 :: c2:c3 -> c7:c8:c9:c10 -> c4
NIL :: c5
c5 :: c5
c6 :: c6
c7 :: c:c1 -> c7:c8:c9:c10
c8 :: c5 -> c7:c8:c9:c10
c9 :: c6 -> c7:c8:c9:c10
c10 :: c7:c8:c9:c10
from :: s:0' -> n__from:n__nil:n__cons
cons :: s:0' -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
length :: n__from:n__nil:n__cons -> s:0'
0' :: s:0'
length1 :: n__from:n__nil:n__cons -> s:0'
nil :: n__from:n__nil:n__cons
hole_c:c11_11 :: c:c1
hole_s:0'2_11 :: s:0'
hole_c63_11 :: c6
hole_n__from:n__nil:n__cons4_11 :: n__from:n__nil:n__cons
hole_c2:c35_11 :: c2:c3
hole_c46_11 :: c4
hole_c7:c8:c9:c107_11 :: c7:c8:c9:c10
hole_c58_11 :: c5
gen_s:0'9_11 :: Nat -> s:0'
gen_n__from:n__nil:n__cons10_11 :: Nat -> n__from:n__nil:n__cons


Lemmas:
length1(gen_n__from:n__nil:n__cons10_11(+(1, n12_11))) -> *11_11, rt in Omega(0)


Generator Equations:
gen_s:0'9_11(0) <=> 0'
gen_s:0'9_11(+(x, 1)) <=> s(gen_s:0'9_11(x))
gen_n__from:n__nil:n__cons10_11(0) <=> n__from(0')
gen_n__from:n__nil:n__cons10_11(+(x, 1)) <=> n__cons(0', gen_n__from:n__nil:n__cons10_11(x))


The following defined symbols remain to be analysed:
length, LENGTH, LENGTH1

They will be analysed ascendingly in the following order:
LENGTH = LENGTH1
length = length1

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length(gen_n__from:n__nil:n__cons10_11(+(1, n3980_11))) -> *11_11, rt in Omega(0)

Induction Base:
length(gen_n__from:n__nil:n__cons10_11(+(1, 0)))

Induction Step:
length(gen_n__from:n__nil:n__cons10_11(+(1, +(n3980_11, 1)))) ->_R^Omega(0)
s(length1(activate(gen_n__from:n__nil:n__cons10_11(+(1, n3980_11))))) ->_R^Omega(0)
s(length1(gen_n__from:n__nil:n__cons10_11(+(1, n3980_11)))) ->_R^Omega(0)
s(length(activate(gen_n__from:n__nil:n__cons10_11(+(1, n3980_11))))) ->_R^Omega(0)
s(length(gen_n__from:n__nil:n__cons10_11(+(1, n3980_11)))) ->_IH
s(*11_11)

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

(18)
Obligation:
Innermost TRS:
Rules:
FROM(z0) -> c(CONS(z0, n__from(s(z0))))
FROM(z0) -> c1
LENGTH(n__nil) -> c2
LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)), ACTIVATE(z1))
LENGTH1(z0) -> c4(LENGTH(activate(z0)), ACTIVATE(z0))
NIL -> c5
CONS(z0, z1) -> c6
ACTIVATE(n__from(z0)) -> c7(FROM(z0))
ACTIVATE(n__nil) -> c8(NIL)
ACTIVATE(n__cons(z0, z1)) -> c9(CONS(z0, z1))
ACTIVATE(z0) -> c10
from(z0) -> cons(z0, n__from(s(z0)))
from(z0) -> n__from(z0)
length(n__nil) -> 0'
length(n__cons(z0, z1)) -> s(length1(activate(z1)))
length1(z0) -> length(activate(z0))
nil -> n__nil
cons(z0, z1) -> n__cons(z0, z1)
activate(n__from(z0)) -> from(z0)
activate(n__nil) -> nil
activate(n__cons(z0, z1)) -> cons(z0, z1)
activate(z0) -> z0

Types:
FROM :: s:0' -> c:c1
c :: c6 -> c:c1
CONS :: s:0' -> n__from:n__nil:n__cons -> c6
n__from :: s:0' -> n__from:n__nil:n__cons
s :: s:0' -> s:0'
c1 :: c:c1
LENGTH :: n__from:n__nil:n__cons -> c2:c3
n__nil :: n__from:n__nil:n__cons
c2 :: c2:c3
n__cons :: s:0' -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
c3 :: c4 -> c7:c8:c9:c10 -> c2:c3
LENGTH1 :: n__from:n__nil:n__cons -> c4
activate :: n__from:n__nil:n__cons -> n__from:n__nil:n__cons
ACTIVATE :: n__from:n__nil:n__cons -> c7:c8:c9:c10
c4 :: c2:c3 -> c7:c8:c9:c10 -> c4
NIL :: c5
c5 :: c5
c6 :: c6
c7 :: c:c1 -> c7:c8:c9:c10
c8 :: c5 -> c7:c8:c9:c10
c9 :: c6 -> c7:c8:c9:c10
c10 :: c7:c8:c9:c10
from :: s:0' -> n__from:n__nil:n__cons
cons :: s:0' -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
length :: n__from:n__nil:n__cons -> s:0'
0' :: s:0'
length1 :: n__from:n__nil:n__cons -> s:0'
nil :: n__from:n__nil:n__cons
hole_c:c11_11 :: c:c1
hole_s:0'2_11 :: s:0'
hole_c63_11 :: c6
hole_n__from:n__nil:n__cons4_11 :: n__from:n__nil:n__cons
hole_c2:c35_11 :: c2:c3
hole_c46_11 :: c4
hole_c7:c8:c9:c107_11 :: c7:c8:c9:c10
hole_c58_11 :: c5
gen_s:0'9_11 :: Nat -> s:0'
gen_n__from:n__nil:n__cons10_11 :: Nat -> n__from:n__nil:n__cons


Lemmas:
length1(gen_n__from:n__nil:n__cons10_11(+(1, n12_11))) -> *11_11, rt in Omega(0)
length(gen_n__from:n__nil:n__cons10_11(+(1, n3980_11))) -> *11_11, rt in Omega(0)


Generator Equations:
gen_s:0'9_11(0) <=> 0'
gen_s:0'9_11(+(x, 1)) <=> s(gen_s:0'9_11(x))
gen_n__from:n__nil:n__cons10_11(0) <=> n__from(0')
gen_n__from:n__nil:n__cons10_11(+(x, 1)) <=> n__cons(0', gen_n__from:n__nil:n__cons10_11(x))


The following defined symbols remain to be analysed:
length1, LENGTH, LENGTH1

They will be analysed ascendingly in the following order:
LENGTH = LENGTH1
length = length1

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length1(gen_n__from:n__nil:n__cons10_11(+(1, n8390_11))) -> *11_11, rt in Omega(0)

Induction Base:
length1(gen_n__from:n__nil:n__cons10_11(+(1, 0)))

Induction Step:
length1(gen_n__from:n__nil:n__cons10_11(+(1, +(n8390_11, 1)))) ->_R^Omega(0)
length(activate(gen_n__from:n__nil:n__cons10_11(+(1, +(n8390_11, 1))))) ->_R^Omega(0)
length(gen_n__from:n__nil:n__cons10_11(+(2, n8390_11))) ->_R^Omega(0)
s(length1(activate(gen_n__from:n__nil:n__cons10_11(+(1, n8390_11))))) ->_R^Omega(0)
s(length1(gen_n__from:n__nil:n__cons10_11(+(1, n8390_11)))) ->_IH
s(*11_11)

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

(20)
Obligation:
Innermost TRS:
Rules:
FROM(z0) -> c(CONS(z0, n__from(s(z0))))
FROM(z0) -> c1
LENGTH(n__nil) -> c2
LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)), ACTIVATE(z1))
LENGTH1(z0) -> c4(LENGTH(activate(z0)), ACTIVATE(z0))
NIL -> c5
CONS(z0, z1) -> c6
ACTIVATE(n__from(z0)) -> c7(FROM(z0))
ACTIVATE(n__nil) -> c8(NIL)
ACTIVATE(n__cons(z0, z1)) -> c9(CONS(z0, z1))
ACTIVATE(z0) -> c10
from(z0) -> cons(z0, n__from(s(z0)))
from(z0) -> n__from(z0)
length(n__nil) -> 0'
length(n__cons(z0, z1)) -> s(length1(activate(z1)))
length1(z0) -> length(activate(z0))
nil -> n__nil
cons(z0, z1) -> n__cons(z0, z1)
activate(n__from(z0)) -> from(z0)
activate(n__nil) -> nil
activate(n__cons(z0, z1)) -> cons(z0, z1)
activate(z0) -> z0

Types:
FROM :: s:0' -> c:c1
c :: c6 -> c:c1
CONS :: s:0' -> n__from:n__nil:n__cons -> c6
n__from :: s:0' -> n__from:n__nil:n__cons
s :: s:0' -> s:0'
c1 :: c:c1
LENGTH :: n__from:n__nil:n__cons -> c2:c3
n__nil :: n__from:n__nil:n__cons
c2 :: c2:c3
n__cons :: s:0' -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
c3 :: c4 -> c7:c8:c9:c10 -> c2:c3
LENGTH1 :: n__from:n__nil:n__cons -> c4
activate :: n__from:n__nil:n__cons -> n__from:n__nil:n__cons
ACTIVATE :: n__from:n__nil:n__cons -> c7:c8:c9:c10
c4 :: c2:c3 -> c7:c8:c9:c10 -> c4
NIL :: c5
c5 :: c5
c6 :: c6
c7 :: c:c1 -> c7:c8:c9:c10
c8 :: c5 -> c7:c8:c9:c10
c9 :: c6 -> c7:c8:c9:c10
c10 :: c7:c8:c9:c10
from :: s:0' -> n__from:n__nil:n__cons
cons :: s:0' -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
length :: n__from:n__nil:n__cons -> s:0'
0' :: s:0'
length1 :: n__from:n__nil:n__cons -> s:0'
nil :: n__from:n__nil:n__cons
hole_c:c11_11 :: c:c1
hole_s:0'2_11 :: s:0'
hole_c63_11 :: c6
hole_n__from:n__nil:n__cons4_11 :: n__from:n__nil:n__cons
hole_c2:c35_11 :: c2:c3
hole_c46_11 :: c4
hole_c7:c8:c9:c107_11 :: c7:c8:c9:c10
hole_c58_11 :: c5
gen_s:0'9_11 :: Nat -> s:0'
gen_n__from:n__nil:n__cons10_11 :: Nat -> n__from:n__nil:n__cons


Lemmas:
length1(gen_n__from:n__nil:n__cons10_11(+(1, n8390_11))) -> *11_11, rt in Omega(0)
length(gen_n__from:n__nil:n__cons10_11(+(1, n3980_11))) -> *11_11, rt in Omega(0)


Generator Equations:
gen_s:0'9_11(0) <=> 0'
gen_s:0'9_11(+(x, 1)) <=> s(gen_s:0'9_11(x))
gen_n__from:n__nil:n__cons10_11(0) <=> n__from(0')
gen_n__from:n__nil:n__cons10_11(+(x, 1)) <=> n__cons(0', gen_n__from:n__nil:n__cons10_11(x))


The following defined symbols remain to be analysed:
LENGTH1, LENGTH

They will be analysed ascendingly in the following order:
LENGTH = LENGTH1

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LENGTH1(gen_n__from:n__nil:n__cons10_11(+(1, n16176_11))) -> *11_11, rt in Omega(n16176_11)

Induction Base:
LENGTH1(gen_n__from:n__nil:n__cons10_11(+(1, 0)))

Induction Step:
LENGTH1(gen_n__from:n__nil:n__cons10_11(+(1, +(n16176_11, 1)))) ->_R^Omega(1)
c4(LENGTH(activate(gen_n__from:n__nil:n__cons10_11(+(1, +(n16176_11, 1))))), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(1, +(n16176_11, 1))))) ->_R^Omega(0)
c4(LENGTH(gen_n__from:n__nil:n__cons10_11(+(2, n16176_11))), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(2, n16176_11)))) ->_R^Omega(1)
c4(c3(LENGTH1(activate(gen_n__from:n__nil:n__cons10_11(+(1, n16176_11)))), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(1, n16176_11)))), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(2, n16176_11)))) ->_R^Omega(0)
c4(c3(LENGTH1(gen_n__from:n__nil:n__cons10_11(+(1, n16176_11))), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(1, n16176_11)))), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(2, n16176_11)))) ->_IH
c4(c3(*11_11, ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(1, n16176_11)))), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(2, n16176_11)))) ->_R^Omega(1)
c4(c3(*11_11, c10), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(2, n16176_11)))) ->_R^Omega(1)
c4(c3(*11_11, c10), c10)

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

(22)
Complex Obligation (BEST)

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

(23)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
FROM(z0) -> c(CONS(z0, n__from(s(z0))))
FROM(z0) -> c1
LENGTH(n__nil) -> c2
LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)), ACTIVATE(z1))
LENGTH1(z0) -> c4(LENGTH(activate(z0)), ACTIVATE(z0))
NIL -> c5
CONS(z0, z1) -> c6
ACTIVATE(n__from(z0)) -> c7(FROM(z0))
ACTIVATE(n__nil) -> c8(NIL)
ACTIVATE(n__cons(z0, z1)) -> c9(CONS(z0, z1))
ACTIVATE(z0) -> c10
from(z0) -> cons(z0, n__from(s(z0)))
from(z0) -> n__from(z0)
length(n__nil) -> 0'
length(n__cons(z0, z1)) -> s(length1(activate(z1)))
length1(z0) -> length(activate(z0))
nil -> n__nil
cons(z0, z1) -> n__cons(z0, z1)
activate(n__from(z0)) -> from(z0)
activate(n__nil) -> nil
activate(n__cons(z0, z1)) -> cons(z0, z1)
activate(z0) -> z0

Types:
FROM :: s:0' -> c:c1
c :: c6 -> c:c1
CONS :: s:0' -> n__from:n__nil:n__cons -> c6
n__from :: s:0' -> n__from:n__nil:n__cons
s :: s:0' -> s:0'
c1 :: c:c1
LENGTH :: n__from:n__nil:n__cons -> c2:c3
n__nil :: n__from:n__nil:n__cons
c2 :: c2:c3
n__cons :: s:0' -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
c3 :: c4 -> c7:c8:c9:c10 -> c2:c3
LENGTH1 :: n__from:n__nil:n__cons -> c4
activate :: n__from:n__nil:n__cons -> n__from:n__nil:n__cons
ACTIVATE :: n__from:n__nil:n__cons -> c7:c8:c9:c10
c4 :: c2:c3 -> c7:c8:c9:c10 -> c4
NIL :: c5
c5 :: c5
c6 :: c6
c7 :: c:c1 -> c7:c8:c9:c10
c8 :: c5 -> c7:c8:c9:c10
c9 :: c6 -> c7:c8:c9:c10
c10 :: c7:c8:c9:c10
from :: s:0' -> n__from:n__nil:n__cons
cons :: s:0' -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
length :: n__from:n__nil:n__cons -> s:0'
0' :: s:0'
length1 :: n__from:n__nil:n__cons -> s:0'
nil :: n__from:n__nil:n__cons
hole_c:c11_11 :: c:c1
hole_s:0'2_11 :: s:0'
hole_c63_11 :: c6
hole_n__from:n__nil:n__cons4_11 :: n__from:n__nil:n__cons
hole_c2:c35_11 :: c2:c3
hole_c46_11 :: c4
hole_c7:c8:c9:c107_11 :: c7:c8:c9:c10
hole_c58_11 :: c5
gen_s:0'9_11 :: Nat -> s:0'
gen_n__from:n__nil:n__cons10_11 :: Nat -> n__from:n__nil:n__cons


Lemmas:
length1(gen_n__from:n__nil:n__cons10_11(+(1, n8390_11))) -> *11_11, rt in Omega(0)
length(gen_n__from:n__nil:n__cons10_11(+(1, n3980_11))) -> *11_11, rt in Omega(0)


Generator Equations:
gen_s:0'9_11(0) <=> 0'
gen_s:0'9_11(+(x, 1)) <=> s(gen_s:0'9_11(x))
gen_n__from:n__nil:n__cons10_11(0) <=> n__from(0')
gen_n__from:n__nil:n__cons10_11(+(x, 1)) <=> n__cons(0', gen_n__from:n__nil:n__cons10_11(x))


The following defined symbols remain to be analysed:
LENGTH1, LENGTH

They will be analysed ascendingly in the following order:
LENGTH = LENGTH1

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

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

(25)
BOUNDS(n^1, INF)

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

(26)
Obligation:
Innermost TRS:
Rules:
FROM(z0) -> c(CONS(z0, n__from(s(z0))))
FROM(z0) -> c1
LENGTH(n__nil) -> c2
LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)), ACTIVATE(z1))
LENGTH1(z0) -> c4(LENGTH(activate(z0)), ACTIVATE(z0))
NIL -> c5
CONS(z0, z1) -> c6
ACTIVATE(n__from(z0)) -> c7(FROM(z0))
ACTIVATE(n__nil) -> c8(NIL)
ACTIVATE(n__cons(z0, z1)) -> c9(CONS(z0, z1))
ACTIVATE(z0) -> c10
from(z0) -> cons(z0, n__from(s(z0)))
from(z0) -> n__from(z0)
length(n__nil) -> 0'
length(n__cons(z0, z1)) -> s(length1(activate(z1)))
length1(z0) -> length(activate(z0))
nil -> n__nil
cons(z0, z1) -> n__cons(z0, z1)
activate(n__from(z0)) -> from(z0)
activate(n__nil) -> nil
activate(n__cons(z0, z1)) -> cons(z0, z1)
activate(z0) -> z0

Types:
FROM :: s:0' -> c:c1
c :: c6 -> c:c1
CONS :: s:0' -> n__from:n__nil:n__cons -> c6
n__from :: s:0' -> n__from:n__nil:n__cons
s :: s:0' -> s:0'
c1 :: c:c1
LENGTH :: n__from:n__nil:n__cons -> c2:c3
n__nil :: n__from:n__nil:n__cons
c2 :: c2:c3
n__cons :: s:0' -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
c3 :: c4 -> c7:c8:c9:c10 -> c2:c3
LENGTH1 :: n__from:n__nil:n__cons -> c4
activate :: n__from:n__nil:n__cons -> n__from:n__nil:n__cons
ACTIVATE :: n__from:n__nil:n__cons -> c7:c8:c9:c10
c4 :: c2:c3 -> c7:c8:c9:c10 -> c4
NIL :: c5
c5 :: c5
c6 :: c6
c7 :: c:c1 -> c7:c8:c9:c10
c8 :: c5 -> c7:c8:c9:c10
c9 :: c6 -> c7:c8:c9:c10
c10 :: c7:c8:c9:c10
from :: s:0' -> n__from:n__nil:n__cons
cons :: s:0' -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
length :: n__from:n__nil:n__cons -> s:0'
0' :: s:0'
length1 :: n__from:n__nil:n__cons -> s:0'
nil :: n__from:n__nil:n__cons
hole_c:c11_11 :: c:c1
hole_s:0'2_11 :: s:0'
hole_c63_11 :: c6
hole_n__from:n__nil:n__cons4_11 :: n__from:n__nil:n__cons
hole_c2:c35_11 :: c2:c3
hole_c46_11 :: c4
hole_c7:c8:c9:c107_11 :: c7:c8:c9:c10
hole_c58_11 :: c5
gen_s:0'9_11 :: Nat -> s:0'
gen_n__from:n__nil:n__cons10_11 :: Nat -> n__from:n__nil:n__cons


Lemmas:
length1(gen_n__from:n__nil:n__cons10_11(+(1, n8390_11))) -> *11_11, rt in Omega(0)
length(gen_n__from:n__nil:n__cons10_11(+(1, n3980_11))) -> *11_11, rt in Omega(0)
LENGTH1(gen_n__from:n__nil:n__cons10_11(+(1, n16176_11))) -> *11_11, rt in Omega(n16176_11)


Generator Equations:
gen_s:0'9_11(0) <=> 0'
gen_s:0'9_11(+(x, 1)) <=> s(gen_s:0'9_11(x))
gen_n__from:n__nil:n__cons10_11(0) <=> n__from(0')
gen_n__from:n__nil:n__cons10_11(+(x, 1)) <=> n__cons(0', gen_n__from:n__nil:n__cons10_11(x))


The following defined symbols remain to be analysed:
LENGTH

They will be analysed ascendingly in the following order:
LENGTH = LENGTH1

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LENGTH(gen_n__from:n__nil:n__cons10_11(+(1, n36375_11))) -> *11_11, rt in Omega(n36375_11)

Induction Base:
LENGTH(gen_n__from:n__nil:n__cons10_11(+(1, 0)))

Induction Step:
LENGTH(gen_n__from:n__nil:n__cons10_11(+(1, +(n36375_11, 1)))) ->_R^Omega(1)
c3(LENGTH1(activate(gen_n__from:n__nil:n__cons10_11(+(1, n36375_11)))), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(1, n36375_11)))) ->_R^Omega(0)
c3(LENGTH1(gen_n__from:n__nil:n__cons10_11(+(1, n36375_11))), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(1, n36375_11)))) ->_R^Omega(1)
c3(c4(LENGTH(activate(gen_n__from:n__nil:n__cons10_11(+(1, n36375_11)))), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(1, n36375_11)))), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(1, n36375_11)))) ->_R^Omega(0)
c3(c4(LENGTH(gen_n__from:n__nil:n__cons10_11(+(1, n36375_11))), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(1, n36375_11)))), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(1, n36375_11)))) ->_IH
c3(c4(*11_11, ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(1, n36375_11)))), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(1, n36375_11)))) ->_R^Omega(1)
c3(c4(*11_11, c10), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(1, n36375_11)))) ->_R^Omega(1)
c3(c4(*11_11, c10), c10)

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

(28)
Obligation:
Innermost TRS:
Rules:
FROM(z0) -> c(CONS(z0, n__from(s(z0))))
FROM(z0) -> c1
LENGTH(n__nil) -> c2
LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)), ACTIVATE(z1))
LENGTH1(z0) -> c4(LENGTH(activate(z0)), ACTIVATE(z0))
NIL -> c5
CONS(z0, z1) -> c6
ACTIVATE(n__from(z0)) -> c7(FROM(z0))
ACTIVATE(n__nil) -> c8(NIL)
ACTIVATE(n__cons(z0, z1)) -> c9(CONS(z0, z1))
ACTIVATE(z0) -> c10
from(z0) -> cons(z0, n__from(s(z0)))
from(z0) -> n__from(z0)
length(n__nil) -> 0'
length(n__cons(z0, z1)) -> s(length1(activate(z1)))
length1(z0) -> length(activate(z0))
nil -> n__nil
cons(z0, z1) -> n__cons(z0, z1)
activate(n__from(z0)) -> from(z0)
activate(n__nil) -> nil
activate(n__cons(z0, z1)) -> cons(z0, z1)
activate(z0) -> z0

Types:
FROM :: s:0' -> c:c1
c :: c6 -> c:c1
CONS :: s:0' -> n__from:n__nil:n__cons -> c6
n__from :: s:0' -> n__from:n__nil:n__cons
s :: s:0' -> s:0'
c1 :: c:c1
LENGTH :: n__from:n__nil:n__cons -> c2:c3
n__nil :: n__from:n__nil:n__cons
c2 :: c2:c3
n__cons :: s:0' -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
c3 :: c4 -> c7:c8:c9:c10 -> c2:c3
LENGTH1 :: n__from:n__nil:n__cons -> c4
activate :: n__from:n__nil:n__cons -> n__from:n__nil:n__cons
ACTIVATE :: n__from:n__nil:n__cons -> c7:c8:c9:c10
c4 :: c2:c3 -> c7:c8:c9:c10 -> c4
NIL :: c5
c5 :: c5
c6 :: c6
c7 :: c:c1 -> c7:c8:c9:c10
c8 :: c5 -> c7:c8:c9:c10
c9 :: c6 -> c7:c8:c9:c10
c10 :: c7:c8:c9:c10
from :: s:0' -> n__from:n__nil:n__cons
cons :: s:0' -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
length :: n__from:n__nil:n__cons -> s:0'
0' :: s:0'
length1 :: n__from:n__nil:n__cons -> s:0'
nil :: n__from:n__nil:n__cons
hole_c:c11_11 :: c:c1
hole_s:0'2_11 :: s:0'
hole_c63_11 :: c6
hole_n__from:n__nil:n__cons4_11 :: n__from:n__nil:n__cons
hole_c2:c35_11 :: c2:c3
hole_c46_11 :: c4
hole_c7:c8:c9:c107_11 :: c7:c8:c9:c10
hole_c58_11 :: c5
gen_s:0'9_11 :: Nat -> s:0'
gen_n__from:n__nil:n__cons10_11 :: Nat -> n__from:n__nil:n__cons


Lemmas:
length1(gen_n__from:n__nil:n__cons10_11(+(1, n8390_11))) -> *11_11, rt in Omega(0)
length(gen_n__from:n__nil:n__cons10_11(+(1, n3980_11))) -> *11_11, rt in Omega(0)
LENGTH1(gen_n__from:n__nil:n__cons10_11(+(1, n16176_11))) -> *11_11, rt in Omega(n16176_11)
LENGTH(gen_n__from:n__nil:n__cons10_11(+(1, n36375_11))) -> *11_11, rt in Omega(n36375_11)


Generator Equations:
gen_s:0'9_11(0) <=> 0'
gen_s:0'9_11(+(x, 1)) <=> s(gen_s:0'9_11(x))
gen_n__from:n__nil:n__cons10_11(0) <=> n__from(0')
gen_n__from:n__nil:n__cons10_11(+(x, 1)) <=> n__cons(0', gen_n__from:n__nil:n__cons10_11(x))


The following defined symbols remain to be analysed:
LENGTH1

They will be analysed ascendingly in the following order:
LENGTH = LENGTH1

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LENGTH1(gen_n__from:n__nil:n__cons10_11(+(1, n47941_11))) -> *11_11, rt in Omega(n47941_11)

Induction Base:
LENGTH1(gen_n__from:n__nil:n__cons10_11(+(1, 0)))

Induction Step:
LENGTH1(gen_n__from:n__nil:n__cons10_11(+(1, +(n47941_11, 1)))) ->_R^Omega(1)
c4(LENGTH(activate(gen_n__from:n__nil:n__cons10_11(+(1, +(n47941_11, 1))))), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(1, +(n47941_11, 1))))) ->_R^Omega(0)
c4(LENGTH(gen_n__from:n__nil:n__cons10_11(+(2, n47941_11))), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(2, n47941_11)))) ->_R^Omega(1)
c4(c3(LENGTH1(activate(gen_n__from:n__nil:n__cons10_11(+(1, n47941_11)))), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(1, n47941_11)))), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(2, n47941_11)))) ->_R^Omega(0)
c4(c3(LENGTH1(gen_n__from:n__nil:n__cons10_11(+(1, n47941_11))), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(1, n47941_11)))), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(2, n47941_11)))) ->_IH
c4(c3(*11_11, ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(1, n47941_11)))), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(2, n47941_11)))) ->_R^Omega(1)
c4(c3(*11_11, c10), ACTIVATE(gen_n__from:n__nil:n__cons10_11(+(2, n47941_11)))) ->_R^Omega(1)
c4(c3(*11_11, c10), c10)

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

(30)
BOUNDS(1, INF)

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

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   length(n__nil) -> 0
   length(n__cons(z0, z1)) -> s(length1(activate(z1)))
   length1(z0) -> length(activate(z0))
   nil -> n__nil
   cons(z0, z1) -> n__cons(z0, z1)
   activate(n__from(z0)) -> from(z0)
   activate(n__nil) -> nil
   activate(n__cons(z0, z1)) -> cons(z0, z1)
   activate(z0) -> z0
Tuples:
   FROM(z0) -> c(CONS(z0, n__from(s(z0))))
   FROM(z0) -> c1
   LENGTH(n__nil) -> c2
   LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)), ACTIVATE(z1))
   LENGTH1(z0) -> c4(LENGTH(activate(z0)), ACTIVATE(z0))
   NIL -> c5
   CONS(z0, z1) -> c6
   ACTIVATE(n__from(z0)) -> c7(FROM(z0))
   ACTIVATE(n__nil) -> c8(NIL)
   ACTIVATE(n__cons(z0, z1)) -> c9(CONS(z0, z1))
   ACTIVATE(z0) -> c10
S tuples:
   FROM(z0) -> c(CONS(z0, n__from(s(z0))))
   FROM(z0) -> c1
   LENGTH(n__nil) -> c2
   LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)), ACTIVATE(z1))
   LENGTH1(z0) -> c4(LENGTH(activate(z0)), ACTIVATE(z0))
   NIL -> c5
   CONS(z0, z1) -> c6
   ACTIVATE(n__from(z0)) -> c7(FROM(z0))
   ACTIVATE(n__nil) -> c8(NIL)
   ACTIVATE(n__cons(z0, z1)) -> c9(CONS(z0, z1))
   ACTIVATE(z0) -> c10
K tuples:none
Defined Rule Symbols:   from_1, length_1, length1_1, nil, cons_2, activate_1

Defined Pair Symbols:   FROM_1, LENGTH_1, LENGTH1_1, NIL, CONS_2, ACTIVATE_1

Compound Symbols:   c_1, c1, c2, c3_2, c4_2, c5, c6, c7_1, c8_1, c9_1, c10


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

(33) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 9 trailing nodes:
   ACTIVATE(n__nil) -> c8(NIL)
   ACTIVATE(n__from(z0)) -> c7(FROM(z0))
   LENGTH(n__nil) -> c2
   FROM(z0) -> c1
   ACTIVATE(n__cons(z0, z1)) -> c9(CONS(z0, z1))
   FROM(z0) -> c(CONS(z0, n__from(s(z0))))
   ACTIVATE(z0) -> c10
   NIL -> c5
   CONS(z0, z1) -> c6

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   length(n__nil) -> 0
   length(n__cons(z0, z1)) -> s(length1(activate(z1)))
   length1(z0) -> length(activate(z0))
   nil -> n__nil
   cons(z0, z1) -> n__cons(z0, z1)
   activate(n__from(z0)) -> from(z0)
   activate(n__nil) -> nil
   activate(n__cons(z0, z1)) -> cons(z0, z1)
   activate(z0) -> z0
Tuples:
   LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)), ACTIVATE(z1))
   LENGTH1(z0) -> c4(LENGTH(activate(z0)), ACTIVATE(z0))
S tuples:
   LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)), ACTIVATE(z1))
   LENGTH1(z0) -> c4(LENGTH(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:   from_1, length_1, length1_1, nil, cons_2, activate_1

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_2, c4_2


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

(35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   length(n__nil) -> 0
   length(n__cons(z0, z1)) -> s(length1(activate(z1)))
   length1(z0) -> length(activate(z0))
   nil -> n__nil
   cons(z0, z1) -> n__cons(z0, z1)
   activate(n__from(z0)) -> from(z0)
   activate(n__nil) -> nil
   activate(n__cons(z0, z1)) -> cons(z0, z1)
   activate(z0) -> z0
Tuples:
   LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)))
   LENGTH1(z0) -> c4(LENGTH(activate(z0)))
S tuples:
   LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)))
   LENGTH1(z0) -> c4(LENGTH(activate(z0)))
K tuples:none
Defined Rule Symbols:   from_1, length_1, length1_1, nil, cons_2, activate_1

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(37) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   length(n__nil) -> 0
   length(n__cons(z0, z1)) -> s(length1(activate(z1)))
   length1(z0) -> length(activate(z0))

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__from(z0)) -> from(z0)
   activate(n__nil) -> nil
   activate(n__cons(z0, z1)) -> cons(z0, z1)
   activate(z0) -> z0
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   cons(z0, z1) -> n__cons(z0, z1)
   nil -> n__nil
Tuples:
   LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)))
   LENGTH1(z0) -> c4(LENGTH(activate(z0)))
S tuples:
   LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)))
   LENGTH1(z0) -> c4(LENGTH(activate(z0)))
K tuples:none
Defined Rule Symbols:   activate_1, from_1, cons_2, nil

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

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

(40)
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:

   LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)))
   LENGTH1(z0) -> c4(LENGTH(activate(z0)))

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

   activate(n__from(z0)) -> from(z0)
   activate(n__nil) -> nil
   activate(n__cons(z0, z1)) -> cons(z0, z1)
   activate(z0) -> z0
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   cons(z0, z1) -> n__cons(z0, z1)
   nil -> n__nil

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

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

(42)
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:

   LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1)))
   LENGTH1(z0) -> c4(LENGTH(activate(z0)))
   activate(n__from(z0)) -> from(z0)
   activate(n__nil) -> nil
   activate(n__cons(z0, z1)) -> cons(z0, z1)
   activate(z0) -> z0
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   cons(z0, z1) -> n__cons(z0, z1)
   nil -> n__nil

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

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

(44)
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:

   LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1))) [1]
   LENGTH1(z0) -> c4(LENGTH(activate(z0))) [1]
   activate(n__from(z0)) -> from(z0) [0]
   activate(n__nil) -> nil [0]
   activate(n__cons(z0, z1)) -> cons(z0, z1) [0]
   activate(z0) -> z0 [0]
   from(z0) -> cons(z0, n__from(s(z0))) [0]
   from(z0) -> n__from(z0) [0]
   cons(z0, z1) -> n__cons(z0, z1) [0]
   nil -> n__nil [0]

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

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

(46)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1))) [1]
   LENGTH1(z0) -> c4(LENGTH(activate(z0))) [1]
   activate(n__from(z0)) -> from(z0) [0]
   activate(n__nil) -> nil [0]
   activate(n__cons(z0, z1)) -> cons(z0, z1) [0]
   activate(z0) -> z0 [0]
   from(z0) -> cons(z0, n__from(s(z0))) [0]
   from(z0) -> n__from(z0) [0]
   cons(z0, z1) -> n__cons(z0, z1) [0]
   nil -> n__nil [0]

The TRS has the following type information:
LENGTH :: n__cons:n__from:n__nil -> c3
n__cons :: s -> n__cons:n__from:n__nil -> n__cons:n__from:n__nil
c3 :: c4 -> c3
LENGTH1 :: n__cons:n__from:n__nil -> c4
activate :: n__cons:n__from:n__nil -> n__cons:n__from:n__nil
c4 :: c3 -> c4
n__from :: s -> n__cons:n__from:n__nil
from :: s -> n__cons:n__from:n__nil
n__nil :: n__cons:n__from:n__nil
nil :: n__cons:n__from:n__nil
cons :: s -> n__cons:n__from:n__nil -> n__cons:n__from:n__nil
s :: s -> s

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

(47) 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:

   LENGTH_1
   LENGTH1_1

(c) The following functions are completely defined:

   activate_1
   from_1
   cons_2
   nil

Due to the following rules being added:

   activate(v0) -> n__nil [0]
   from(v0) -> n__nil [0]
   cons(v0, v1) -> n__nil [0]
   nil -> n__nil [0]

And the following fresh constants:    const, const1, const2

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

(48)
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:

   LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1))) [1]
   LENGTH1(z0) -> c4(LENGTH(activate(z0))) [1]
   activate(n__from(z0)) -> from(z0) [0]
   activate(n__nil) -> nil [0]
   activate(n__cons(z0, z1)) -> cons(z0, z1) [0]
   activate(z0) -> z0 [0]
   from(z0) -> cons(z0, n__from(s(z0))) [0]
   from(z0) -> n__from(z0) [0]
   cons(z0, z1) -> n__cons(z0, z1) [0]
   nil -> n__nil [0]
   activate(v0) -> n__nil [0]
   from(v0) -> n__nil [0]
   cons(v0, v1) -> n__nil [0]
   nil -> n__nil [0]

The TRS has the following type information:
LENGTH :: n__cons:n__from:n__nil -> c3
n__cons :: s -> n__cons:n__from:n__nil -> n__cons:n__from:n__nil
c3 :: c4 -> c3
LENGTH1 :: n__cons:n__from:n__nil -> c4
activate :: n__cons:n__from:n__nil -> n__cons:n__from:n__nil
c4 :: c3 -> c4
n__from :: s -> n__cons:n__from:n__nil
from :: s -> n__cons:n__from:n__nil
n__nil :: n__cons:n__from:n__nil
nil :: n__cons:n__from:n__nil
cons :: s -> n__cons:n__from:n__nil -> n__cons:n__from:n__nil
s :: s -> s
const :: c3
const1 :: s
const2 :: c4

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

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

(50)
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:

   LENGTH(n__cons(z0, n__from(z0'))) -> c3(LENGTH1(from(z0'))) [1]
   LENGTH(n__cons(z0, n__nil)) -> c3(LENGTH1(nil)) [1]
   LENGTH(n__cons(z0, n__cons(z0'', z1'))) -> c3(LENGTH1(cons(z0'', z1'))) [1]
   LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(z1)) [1]
   LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(n__nil)) [1]
   LENGTH1(n__from(z01)) -> c4(LENGTH(from(z01))) [1]
   LENGTH1(n__nil) -> c4(LENGTH(nil)) [1]
   LENGTH1(n__cons(z02, z1'')) -> c4(LENGTH(cons(z02, z1''))) [1]
   LENGTH1(z0) -> c4(LENGTH(z0)) [1]
   LENGTH1(z0) -> c4(LENGTH(n__nil)) [1]
   activate(n__from(z0)) -> from(z0) [0]
   activate(n__nil) -> nil [0]
   activate(n__cons(z0, z1)) -> cons(z0, z1) [0]
   activate(z0) -> z0 [0]
   from(z0) -> cons(z0, n__from(s(z0))) [0]
   from(z0) -> n__from(z0) [0]
   cons(z0, z1) -> n__cons(z0, z1) [0]
   nil -> n__nil [0]
   activate(v0) -> n__nil [0]
   from(v0) -> n__nil [0]
   cons(v0, v1) -> n__nil [0]
   nil -> n__nil [0]

The TRS has the following type information:
LENGTH :: n__cons:n__from:n__nil -> c3
n__cons :: s -> n__cons:n__from:n__nil -> n__cons:n__from:n__nil
c3 :: c4 -> c3
LENGTH1 :: n__cons:n__from:n__nil -> c4
activate :: n__cons:n__from:n__nil -> n__cons:n__from:n__nil
c4 :: c3 -> c4
n__from :: s -> n__cons:n__from:n__nil
from :: s -> n__cons:n__from:n__nil
n__nil :: n__cons:n__from:n__nil
nil :: n__cons:n__from:n__nil
cons :: s -> n__cons:n__from:n__nil -> n__cons:n__from:n__nil
s :: s -> s
const :: c3
const1 :: s
const2 :: c4

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

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

   n__nil => 0
   const => 0
   const1 => 0
   const2 => 0

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

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

   LENGTH(z) -{ 1 }-> 1 + LENGTH1(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(nil) :|: z0 >= 0, z = 1 + z0 + 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(from(z0')) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(cons(z0'', z1')) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(z0) :|: z = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(nil) :|: z = 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(from(z01)) :|: z01 >= 0, z = 1 + z01
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(cons(z02, z1'')) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z = z0, z0 >= 0
   activate(z) -{ 0 }-> z0 :|: z = z0, z0 >= 0
   activate(z) -{ 0 }-> nil :|: z = 0
   activate(z) -{ 0 }-> from(z0) :|: z = 1 + z0, z0 >= 0
   activate(z) -{ 0 }-> cons(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   cons(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   cons(z, z') -{ 0 }-> 1 + z0 + z1 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   from(z) -{ 0 }-> cons(z0, 1 + (1 + z0)) :|: z = z0, z0 >= 0
   from(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   from(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0
   nil -{ 0 }-> 0 :|: 


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

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

   cons(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   cons(z, z') -{ 0 }-> 1 + z0 + z1 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   nil -{ 0 }-> 0 :|: 
   from(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   from(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0
   from(z) -{ 0 }-> cons(z0, 1 + (1 + z0)) :|: z = z0, z0 >= 0

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

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

   LENGTH(z) -{ 1 }-> 1 + LENGTH1(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(cons(z0'', 1 + (1 + z0''))) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, v0 >= 0, v1 >= 0, z0'' = v0, z1' = v1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z0 >= 0, z = 1 + z0 + 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, v0 >= 0, z0' = v0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0'') :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0' + z1) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, z0'' = z0', z1 >= 0, z1' = z1, z0' >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(z0) :|: z = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(cons(z0, 1 + (1 + z0))) :|: z01 >= 0, z = 1 + z01, z01 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, v0 >= 0, v1 >= 0, z02 = v0, z1'' = v1
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z = 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z01 >= 0, z = 1 + z01, v0 >= 0, z01 = v0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0) :|: z01 >= 0, z = 1 + z01, z01 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0 + z1) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, z02 = z0, z1 >= 0, z1'' = z1, z0 >= 0
   activate(z) -{ 0 }-> z0 :|: z = z0, z0 >= 0
   activate(z) -{ 0 }-> cons(z0', 1 + (1 + z0')) :|: z = 1 + z0, z0 >= 0, z0 = z0', z0' >= 0
   activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   activate(z) -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, z0 = v0, z1 = v1
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 0 :|: z = 1 + z0, z0 >= 0, v0 >= 0, z0 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z = 1 + z0, z0 >= 0, z0 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + z1' :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z0 = z0', z1' >= 0, z1 = z1', z0' >= 0
   cons(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   cons(z, z') -{ 0 }-> 1 + z0 + z1 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   from(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   from(z) -{ 0 }-> 0 :|: z = z0, z0 >= 0, v0 >= 0, v1 >= 0, z0 = v0, 1 + (1 + z0) = v1
   from(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0
   from(z) -{ 0 }-> 1 + z0' + z1 :|: z = z0, z0 >= 0, z0 = z0', z1 >= 0, 1 + (1 + z0) = z1, z0' >= 0
   nil -{ 0 }-> 0 :|: 


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

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

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

   LENGTH(z) -{ 1 }-> 1 + LENGTH1(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(cons(z0'', 1 + (1 + z0''))) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, v0 >= 0, v1 >= 0, z0'' = v0, z1' = v1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, v0 >= 0, z0' = v0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0'') :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0' + z1) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, z0'' = z0', z1 >= 0, z1' = z1, z0' >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(z) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(cons(z0, 1 + (1 + z0))) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, v0 >= 0, v1 >= 0, z02 = v0, z1'' = v1
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z = 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0 + z1) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, z02 = z0, z1 >= 0, z1'' = z1, z0 >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> cons(z0', 1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, z0 = v0, z1 = v1
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + z1' :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z0 = z0', z1' >= 0, z1 = z1', z0' >= 0
   cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cons(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0, v1 >= 0, 1 + (1 + z) = v1
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + z1 :|: z >= 0, z1 >= 0, 1 + (1 + z) = z1
   nil -{ 0 }-> 0 :|: 


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

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

   { nil }
   { from }
   { cons }
   { activate }
   { LENGTH, LENGTH1 }

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

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

   LENGTH(z) -{ 1 }-> 1 + LENGTH1(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(cons(z0'', 1 + (1 + z0''))) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, v0 >= 0, v1 >= 0, z0'' = v0, z1' = v1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, v0 >= 0, z0' = v0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0'') :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0' + z1) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, z0'' = z0', z1 >= 0, z1' = z1, z0' >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(z) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(cons(z0, 1 + (1 + z0))) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, v0 >= 0, v1 >= 0, z02 = v0, z1'' = v1
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z = 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0 + z1) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, z02 = z0, z1 >= 0, z1'' = z1, z0 >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> cons(z0', 1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, z0 = v0, z1 = v1
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + z1' :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z0 = z0', z1' >= 0, z1 = z1', z0' >= 0
   cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cons(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0, v1 >= 0, 1 + (1 + z) = v1
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + z1 :|: z >= 0, z1 >= 0, 1 + (1 + z) = z1
   nil -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {nil}, {from}, {cons}, {activate}, {LENGTH,LENGTH1}

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

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

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

   LENGTH(z) -{ 1 }-> 1 + LENGTH1(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(cons(z0'', 1 + (1 + z0''))) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, v0 >= 0, v1 >= 0, z0'' = v0, z1' = v1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, v0 >= 0, z0' = v0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0'') :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0' + z1) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, z0'' = z0', z1 >= 0, z1' = z1, z0' >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(z) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(cons(z0, 1 + (1 + z0))) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, v0 >= 0, v1 >= 0, z02 = v0, z1'' = v1
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z = 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0 + z1) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, z02 = z0, z1 >= 0, z1'' = z1, z0 >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> cons(z0', 1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, z0 = v0, z1 = v1
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + z1' :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z0 = z0', z1' >= 0, z1 = z1', z0' >= 0
   cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cons(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0, v1 >= 0, 1 + (1 + z) = v1
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + z1 :|: z >= 0, z1 >= 0, 1 + (1 + z) = z1
   nil -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {nil}, {from}, {cons}, {activate}, {LENGTH,LENGTH1}

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

(61) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: nil
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 + LENGTH1(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(cons(z0'', 1 + (1 + z0''))) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, v0 >= 0, v1 >= 0, z0'' = v0, z1' = v1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, v0 >= 0, z0' = v0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0'') :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0' + z1) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, z0'' = z0', z1 >= 0, z1' = z1, z0' >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(z) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(cons(z0, 1 + (1 + z0))) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, v0 >= 0, v1 >= 0, z02 = v0, z1'' = v1
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z = 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0 + z1) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, z02 = z0, z1 >= 0, z1'' = z1, z0 >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> cons(z0', 1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, z0 = v0, z1 = v1
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + z1' :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z0 = z0', z1' >= 0, z1 = z1', z0' >= 0
   cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cons(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0, v1 >= 0, 1 + (1 + z) = v1
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + z1 :|: z >= 0, z1 >= 0, 1 + (1 + z) = z1
   nil -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {nil}, {from}, {cons}, {activate}, {LENGTH,LENGTH1}
Previous analysis results are:
  nil: runtime: ?, size: O(1) [0]

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   LENGTH(z) -{ 1 }-> 1 + LENGTH1(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(cons(z0'', 1 + (1 + z0''))) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, v0 >= 0, v1 >= 0, z0'' = v0, z1' = v1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, v0 >= 0, z0' = v0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0'') :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0' + z1) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, z0'' = z0', z1 >= 0, z1' = z1, z0' >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(z) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(cons(z0, 1 + (1 + z0))) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, v0 >= 0, v1 >= 0, z02 = v0, z1'' = v1
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z = 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0 + z1) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, z02 = z0, z1 >= 0, z1'' = z1, z0 >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> cons(z0', 1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, z0 = v0, z1 = v1
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + z1' :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z0 = z0', z1' >= 0, z1 = z1', z0' >= 0
   cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cons(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0, v1 >= 0, 1 + (1 + z) = v1
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + z1 :|: z >= 0, z1 >= 0, 1 + (1 + z) = z1
   nil -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {from}, {cons}, {activate}, {LENGTH,LENGTH1}
Previous analysis results are:
  nil: runtime: O(1) [0], size: O(1) [0]

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

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

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

   LENGTH(z) -{ 1 }-> 1 + LENGTH1(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(cons(z0'', 1 + (1 + z0''))) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, v0 >= 0, v1 >= 0, z0'' = v0, z1' = v1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, v0 >= 0, z0' = v0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0'') :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0' + z1) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, z0'' = z0', z1 >= 0, z1' = z1, z0' >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(z) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(cons(z0, 1 + (1 + z0))) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, v0 >= 0, v1 >= 0, z02 = v0, z1'' = v1
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z = 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0 + z1) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, z02 = z0, z1 >= 0, z1'' = z1, z0 >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> cons(z0', 1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, z0 = v0, z1 = v1
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + z1' :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z0 = z0', z1' >= 0, z1 = z1', z0' >= 0
   cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cons(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0, v1 >= 0, 1 + (1 + z) = v1
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + z1 :|: z >= 0, z1 >= 0, 1 + (1 + z) = z1
   nil -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {from}, {cons}, {activate}, {LENGTH,LENGTH1}
Previous analysis results are:
  nil: runtime: O(1) [0], size: O(1) [0]

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

(67) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: from
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + 2*z

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

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

   LENGTH(z) -{ 1 }-> 1 + LENGTH1(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(cons(z0'', 1 + (1 + z0''))) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, v0 >= 0, v1 >= 0, z0'' = v0, z1' = v1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, v0 >= 0, z0' = v0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0'') :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0' + z1) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, z0'' = z0', z1 >= 0, z1' = z1, z0' >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(z) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(cons(z0, 1 + (1 + z0))) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, v0 >= 0, v1 >= 0, z02 = v0, z1'' = v1
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z = 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0 + z1) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, z02 = z0, z1 >= 0, z1'' = z1, z0 >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> cons(z0', 1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, z0 = v0, z1 = v1
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + z1' :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z0 = z0', z1' >= 0, z1 = z1', z0' >= 0
   cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cons(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0, v1 >= 0, 1 + (1 + z) = v1
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + z1 :|: z >= 0, z1 >= 0, 1 + (1 + z) = z1
   nil -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {from}, {cons}, {activate}, {LENGTH,LENGTH1}
Previous analysis results are:
  nil: runtime: O(1) [0], size: O(1) [0]
  from: runtime: ?, size: O(n^1) [3 + 2*z]

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

(69) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(70)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 1 }-> 1 + LENGTH1(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(cons(z0'', 1 + (1 + z0''))) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, v0 >= 0, v1 >= 0, z0'' = v0, z1' = v1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, v0 >= 0, z0' = v0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0'') :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0' + z1) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, z0'' = z0', z1 >= 0, z1' = z1, z0' >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(z) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(cons(z0, 1 + (1 + z0))) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, v0 >= 0, v1 >= 0, z02 = v0, z1'' = v1
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z = 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0 + z1) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, z02 = z0, z1 >= 0, z1'' = z1, z0 >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> cons(z0', 1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, z0 = v0, z1 = v1
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + z1' :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z0 = z0', z1' >= 0, z1 = z1', z0' >= 0
   cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cons(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0, v1 >= 0, 1 + (1 + z) = v1
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + z1 :|: z >= 0, z1 >= 0, 1 + (1 + z) = z1
   nil -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {cons}, {activate}, {LENGTH,LENGTH1}
Previous analysis results are:
  nil: runtime: O(1) [0], size: O(1) [0]
  from: runtime: O(1) [0], size: O(n^1) [3 + 2*z]

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

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

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

   LENGTH(z) -{ 1 }-> 1 + LENGTH1(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(cons(z0'', 1 + (1 + z0''))) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, v0 >= 0, v1 >= 0, z0'' = v0, z1' = v1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, v0 >= 0, z0' = v0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0'') :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0' + z1) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, z0'' = z0', z1 >= 0, z1' = z1, z0' >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(z) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(cons(z0, 1 + (1 + z0))) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, v0 >= 0, v1 >= 0, z02 = v0, z1'' = v1
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z = 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0 + z1) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, z02 = z0, z1 >= 0, z1'' = z1, z0 >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> cons(z0', 1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, z0 = v0, z1 = v1
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + z1' :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z0 = z0', z1' >= 0, z1 = z1', z0' >= 0
   cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cons(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0, v1 >= 0, 1 + (1 + z) = v1
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + z1 :|: z >= 0, z1 >= 0, 1 + (1 + z) = z1
   nil -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {cons}, {activate}, {LENGTH,LENGTH1}
Previous analysis results are:
  nil: runtime: O(1) [0], size: O(1) [0]
  from: runtime: O(1) [0], size: O(n^1) [3 + 2*z]

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

(73) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(74)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 1 }-> 1 + LENGTH1(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(cons(z0'', 1 + (1 + z0''))) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, v0 >= 0, v1 >= 0, z0'' = v0, z1' = v1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, v0 >= 0, z0' = v0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0'') :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0' + z1) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, z0'' = z0', z1 >= 0, z1' = z1, z0' >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(z) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(cons(z0, 1 + (1 + z0))) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, v0 >= 0, v1 >= 0, z02 = v0, z1'' = v1
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z = 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0 + z1) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, z02 = z0, z1 >= 0, z1'' = z1, z0 >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> cons(z0', 1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, z0 = v0, z1 = v1
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + z1' :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z0 = z0', z1' >= 0, z1 = z1', z0' >= 0
   cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cons(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0, v1 >= 0, 1 + (1 + z) = v1
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + z1 :|: z >= 0, z1 >= 0, 1 + (1 + z) = z1
   nil -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {cons}, {activate}, {LENGTH,LENGTH1}
Previous analysis results are:
  nil: runtime: O(1) [0], size: O(1) [0]
  from: runtime: O(1) [0], size: O(n^1) [3 + 2*z]
  cons: runtime: ?, size: O(n^1) [1 + z + z']

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

(75) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(76)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 1 }-> 1 + LENGTH1(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(cons(z0'', 1 + (1 + z0''))) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, v0 >= 0, v1 >= 0, z0'' = v0, z1' = v1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, v0 >= 0, z0' = v0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0'') :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0' + z1) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, z0'' = z0', z1 >= 0, z1' = z1, z0' >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(z) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(cons(z0, 1 + (1 + z0))) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, v0 >= 0, v1 >= 0, z02 = v0, z1'' = v1
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z = 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0 + z1) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, z02 = z0, z1 >= 0, z1'' = z1, z0 >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> cons(z0', 1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, z0 = v0, z1 = v1
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + z1' :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z0 = z0', z1' >= 0, z1 = z1', z0' >= 0
   cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cons(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0, v1 >= 0, 1 + (1 + z) = v1
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + z1 :|: z >= 0, z1 >= 0, 1 + (1 + z) = z1
   nil -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {activate}, {LENGTH,LENGTH1}
Previous analysis results are:
  nil: runtime: O(1) [0], size: O(1) [0]
  from: runtime: O(1) [0], size: O(n^1) [3 + 2*z]
  cons: runtime: O(1) [0], size: O(n^1) [1 + z + z']

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

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

(78)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 1 }-> 1 + LENGTH1(s) :|: s >= 0, s <= z0'' + (1 + (1 + z0'')) + 1, z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, v0 >= 0, v1 >= 0, z0'' = v0, z1' = v1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, v0 >= 0, z0' = v0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0'') :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0' + z1) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, z0'' = z0', z1 >= 0, z1' = z1, z0' >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(s') :|: s' >= 0, s' <= z0 + (1 + (1 + z0)) + 1, z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(z) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, v0 >= 0, v1 >= 0, z02 = v0, z1'' = v1
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z = 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0 + z1) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, z02 = z0, z1 >= 0, z1'' = z1, z0 >= 0
   activate(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= z0' + (1 + (1 + z0')) + 1, z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, z0 = v0, z1 = v1
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + z1' :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z0 = z0', z1' >= 0, z1 = z1', z0' >= 0
   cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cons(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0, v1 >= 0, 1 + (1 + z) = v1
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + z1 :|: z >= 0, z1 >= 0, 1 + (1 + z) = z1
   nil -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {activate}, {LENGTH,LENGTH1}
Previous analysis results are:
  nil: runtime: O(1) [0], size: O(1) [0]
  from: runtime: O(1) [0], size: O(n^1) [3 + 2*z]
  cons: runtime: O(1) [0], size: O(n^1) [1 + z + z']

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

(79) 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 + 2*z

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

(80)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 1 }-> 1 + LENGTH1(s) :|: s >= 0, s <= z0'' + (1 + (1 + z0'')) + 1, z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, v0 >= 0, v1 >= 0, z0'' = v0, z1' = v1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, v0 >= 0, z0' = v0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0'') :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0' + z1) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, z0'' = z0', z1 >= 0, z1' = z1, z0' >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(s') :|: s' >= 0, s' <= z0 + (1 + (1 + z0)) + 1, z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(z) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, v0 >= 0, v1 >= 0, z02 = v0, z1'' = v1
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z = 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0 + z1) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, z02 = z0, z1 >= 0, z1'' = z1, z0 >= 0
   activate(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= z0' + (1 + (1 + z0')) + 1, z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, z0 = v0, z1 = v1
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + z1' :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z0 = z0', z1' >= 0, z1 = z1', z0' >= 0
   cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cons(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0, v1 >= 0, 1 + (1 + z) = v1
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + z1 :|: z >= 0, z1 >= 0, 1 + (1 + z) = z1
   nil -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {activate}, {LENGTH,LENGTH1}
Previous analysis results are:
  nil: runtime: O(1) [0], size: O(1) [0]
  from: runtime: O(1) [0], size: O(n^1) [3 + 2*z]
  cons: runtime: O(1) [0], size: O(n^1) [1 + z + z']
  activate: runtime: ?, size: O(n^1) [1 + 2*z]

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

(81) 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

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

(82)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 1 }-> 1 + LENGTH1(s) :|: s >= 0, s <= z0'' + (1 + (1 + z0'')) + 1, z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, v0 >= 0, v1 >= 0, z0'' = v0, z1' = v1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, v0 >= 0, z0' = v0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0'') :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0' + z1) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, z0'' = z0', z1 >= 0, z1' = z1, z0' >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(s') :|: s' >= 0, s' <= z0 + (1 + (1 + z0)) + 1, z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(z) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, v0 >= 0, v1 >= 0, z02 = v0, z1'' = v1
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z = 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0 + z1) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, z02 = z0, z1 >= 0, z1'' = z1, z0 >= 0
   activate(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= z0' + (1 + (1 + z0')) + 1, z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, z0 = v0, z1 = v1
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + z1' :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z0 = z0', z1' >= 0, z1 = z1', z0' >= 0
   cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cons(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0, v1 >= 0, 1 + (1 + z) = v1
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + z1 :|: z >= 0, z1 >= 0, 1 + (1 + z) = z1
   nil -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {LENGTH,LENGTH1}
Previous analysis results are:
  nil: runtime: O(1) [0], size: O(1) [0]
  from: runtime: O(1) [0], size: O(n^1) [3 + 2*z]
  cons: runtime: O(1) [0], size: O(n^1) [1 + z + z']
  activate: runtime: O(1) [0], size: O(n^1) [1 + 2*z]

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

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

(84)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 1 }-> 1 + LENGTH1(s) :|: s >= 0, s <= z0'' + (1 + (1 + z0'')) + 1, z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, v0 >= 0, v1 >= 0, z0'' = v0, z1' = v1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, v0 >= 0, z0' = v0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0'') :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0' + z1) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, z0'' = z0', z1 >= 0, z1' = z1, z0' >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(s') :|: s' >= 0, s' <= z0 + (1 + (1 + z0)) + 1, z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(z) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, v0 >= 0, v1 >= 0, z02 = v0, z1'' = v1
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z = 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0 + z1) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, z02 = z0, z1 >= 0, z1'' = z1, z0 >= 0
   activate(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= z0' + (1 + (1 + z0')) + 1, z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, z0 = v0, z1 = v1
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + z1' :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z0 = z0', z1' >= 0, z1 = z1', z0' >= 0
   cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cons(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0, v1 >= 0, 1 + (1 + z) = v1
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + z1 :|: z >= 0, z1 >= 0, 1 + (1 + z) = z1
   nil -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {LENGTH,LENGTH1}
Previous analysis results are:
  nil: runtime: O(1) [0], size: O(1) [0]
  from: runtime: O(1) [0], size: O(n^1) [3 + 2*z]
  cons: runtime: O(1) [0], size: O(n^1) [1 + z + z']
  activate: runtime: O(1) [0], size: O(n^1) [1 + 2*z]

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

(85) 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: LENGTH1
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

(86)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 1 }-> 1 + LENGTH1(s) :|: s >= 0, s <= z0'' + (1 + (1 + z0'')) + 1, z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, v0 >= 0, v1 >= 0, z0'' = v0, z1' = v1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, v0 >= 0, z0' = v0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0'') :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0' + z1) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, z0'' = z0', z1 >= 0, z1' = z1, z0' >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(s') :|: s' >= 0, s' <= z0 + (1 + (1 + z0)) + 1, z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(z) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, v0 >= 0, v1 >= 0, z02 = v0, z1'' = v1
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z = 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0 + z1) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, z02 = z0, z1 >= 0, z1'' = z1, z0 >= 0
   activate(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= z0' + (1 + (1 + z0')) + 1, z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, z0 = v0, z1 = v1
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + z1' :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z0 = z0', z1' >= 0, z1 = z1', z0' >= 0
   cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cons(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0, v1 >= 0, 1 + (1 + z) = v1
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + z1 :|: z >= 0, z1 >= 0, 1 + (1 + z) = z1
   nil -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {LENGTH,LENGTH1}
Previous analysis results are:
  nil: runtime: O(1) [0], size: O(1) [0]
  from: runtime: O(1) [0], size: O(n^1) [3 + 2*z]
  cons: runtime: O(1) [0], size: O(n^1) [1 + z + z']
  activate: runtime: O(1) [0], size: O(n^1) [1 + 2*z]
  LENGTH: runtime: ?, size: O(1) [0]
  LENGTH1: runtime: ?, size: O(1) [1]

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

(87) 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: ?

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

(88)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 1 }-> 1 + LENGTH1(s) :|: s >= 0, s <= z0'' + (1 + (1 + z0'')) + 1, z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, v0 >= 0, v1 >= 0, z0'' = v0, z1' = v1
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z - 1 >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(0) :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, v0 >= 0, z0' = v0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0'') :|: z0' >= 0, z = 1 + z0 + (1 + z0'), z0 >= 0, z0' = z0'', z0'' >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(1 + z0' + z1) :|: z = 1 + z0 + (1 + z0'' + z1'), z1' >= 0, z0 >= 0, z0'' >= 0, z0'' = z0', z1 >= 0, z1' = z1, z0' >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(s') :|: s' >= 0, s' <= z0 + (1 + (1 + z0)) + 1, z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(z) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, v0 >= 0, v1 >= 0, z02 = v0, z1'' = v1
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z = 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(0) :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0) :|: z - 1 >= 0, z - 1 = z0, z0 >= 0
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(1 + z0 + z1) :|: z02 >= 0, z = 1 + z02 + z1'', z1'' >= 0, z02 = z0, z1 >= 0, z1'' = z1, z0 >= 0
   activate(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= z0' + (1 + (1 + z0')) + 1, z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, v0 >= 0, v1 >= 0, z0 = v0, z1 = v1
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + z1' :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z0 = z0', z1' >= 0, z1 = z1', z0' >= 0
   cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   cons(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0, v1 >= 0, 1 + (1 + z) = v1
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + z1 :|: z >= 0, z1 >= 0, 1 + (1 + z) = z1
   nil -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {LENGTH,LENGTH1}
Previous analysis results are:
  nil: runtime: O(1) [0], size: O(1) [0]
  from: runtime: O(1) [0], size: O(n^1) [3 + 2*z]
  cons: runtime: O(1) [0], size: O(n^1) [1 + z + z']
  activate: runtime: O(1) [0], size: O(n^1) [1 + 2*z]
  LENGTH: runtime: INF, size: O(1) [0]
  LENGTH1: runtime: ?, size: O(1) [1]

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

(89) 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]
   from(v0) -> null_from [0]
   cons(v0, v1) -> null_cons [0]
   nil -> null_nil [0]
   LENGTH(v0) -> null_LENGTH [0]

And the following fresh constants:    null_activate, null_from, null_cons, null_nil, null_LENGTH, const, const1

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

(90)
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:

   LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1))) [1]
   LENGTH1(z0) -> c4(LENGTH(activate(z0))) [1]
   activate(n__from(z0)) -> from(z0) [0]
   activate(n__nil) -> nil [0]
   activate(n__cons(z0, z1)) -> cons(z0, z1) [0]
   activate(z0) -> z0 [0]
   from(z0) -> cons(z0, n__from(s(z0))) [0]
   from(z0) -> n__from(z0) [0]
   cons(z0, z1) -> n__cons(z0, z1) [0]
   nil -> n__nil [0]
   activate(v0) -> null_activate [0]
   from(v0) -> null_from [0]
   cons(v0, v1) -> null_cons [0]
   nil -> null_nil [0]
   LENGTH(v0) -> null_LENGTH [0]

The TRS has the following type information:
LENGTH :: n__cons:n__from:n__nil:null_activate:null_from:null_cons:null_nil -> c3:null_LENGTH
n__cons :: s -> n__cons:n__from:n__nil:null_activate:null_from:null_cons:null_nil -> n__cons:n__from:n__nil:null_activate:null_from:null_cons:null_nil
c3 :: c4 -> c3:null_LENGTH
LENGTH1 :: n__cons:n__from:n__nil:null_activate:null_from:null_cons:null_nil -> c4
activate :: n__cons:n__from:n__nil:null_activate:null_from:null_cons:null_nil -> n__cons:n__from:n__nil:null_activate:null_from:null_cons:null_nil
c4 :: c3:null_LENGTH -> c4
n__from :: s -> n__cons:n__from:n__nil:null_activate:null_from:null_cons:null_nil
from :: s -> n__cons:n__from:n__nil:null_activate:null_from:null_cons:null_nil
n__nil :: n__cons:n__from:n__nil:null_activate:null_from:null_cons:null_nil
nil :: n__cons:n__from:n__nil:null_activate:null_from:null_cons:null_nil
cons :: s -> n__cons:n__from:n__nil:null_activate:null_from:null_cons:null_nil -> n__cons:n__from:n__nil:null_activate:null_from:null_cons:null_nil
s :: s -> s
null_activate :: n__cons:n__from:n__nil:null_activate:null_from:null_cons:null_nil
null_from :: n__cons:n__from:n__nil:null_activate:null_from:null_cons:null_nil
null_cons :: n__cons:n__from:n__nil:null_activate:null_from:null_cons:null_nil
null_nil :: n__cons:n__from:n__nil:null_activate:null_from:null_cons:null_nil
null_LENGTH :: c3:null_LENGTH
const :: s
const1 :: c4

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

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

   n__nil => 0
   null_activate => 0
   null_from => 0
   null_cons => 0
   null_nil => 0
   null_LENGTH => 0
   const => 0
   const1 => 0

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

(92)
Obligation:
Complexity RNTS consisting of the following rules:

   LENGTH(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   LENGTH(z) -{ 1 }-> 1 + LENGTH1(activate(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   LENGTH1(z) -{ 1 }-> 1 + LENGTH(activate(z0)) :|: z = z0, z0 >= 0
   activate(z) -{ 0 }-> z0 :|: z = z0, z0 >= 0
   activate(z) -{ 0 }-> nil :|: z = 0
   activate(z) -{ 0 }-> from(z0) :|: z = 1 + z0, z0 >= 0
   activate(z) -{ 0 }-> cons(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   cons(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   cons(z, z') -{ 0 }-> 1 + z0 + z1 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   from(z) -{ 0 }-> cons(z0, 1 + (1 + z0)) :|: z = z0, z0 >= 0
   from(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   from(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0
   nil -{ 0 }-> 0 :|: 

Only complete derivations are relevant for the runtime complexity.

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

(93) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LENGTH(n__cons(z0, z1)) -> c3(LENGTH1(activate(z1))) by 
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1)))
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__from(z0)) -> from(z0)
   activate(n__nil) -> nil
   activate(n__cons(z0, z1)) -> cons(z0, z1)
   activate(z0) -> z0
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   cons(z0, z1) -> n__cons(z0, z1)
   nil -> n__nil
Tuples:
   LENGTH1(z0) -> c4(LENGTH(activate(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1)))
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
S tuples:
   LENGTH1(z0) -> c4(LENGTH(activate(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1)))
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
K tuples:none
Defined Rule Symbols:   activate_1, from_1, cons_2, nil

Defined Pair Symbols:   LENGTH1_1, LENGTH_1

Compound Symbols:   c4_1, c3_1


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

(95) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LENGTH1(z0) -> c4(LENGTH(activate(z0))) by 
   LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0)))
   LENGTH1(n__nil) -> c4(LENGTH(nil))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__from(z0)) -> from(z0)
   activate(n__nil) -> nil
   activate(n__cons(z0, z1)) -> cons(z0, z1)
   activate(z0) -> z0
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   cons(z0, z1) -> n__cons(z0, z1)
   nil -> n__nil
Tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1)))
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0)))
   LENGTH1(n__nil) -> c4(LENGTH(nil))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))
S tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1)))
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0)))
   LENGTH1(n__nil) -> c4(LENGTH(nil))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))
K tuples:none
Defined Rule Symbols:   activate_1, from_1, cons_2, nil

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(97) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   activate(n__from(z0)) -> from(z0)
   activate(n__nil) -> nil
   activate(n__cons(z0, z1)) -> cons(z0, z1)
   activate(z0) -> z0

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   cons(z0, z1) -> n__cons(z0, z1)
   nil -> n__nil
Tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1)))
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0)))
   LENGTH1(n__nil) -> c4(LENGTH(nil))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))
S tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1)))
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0)))
   LENGTH1(n__nil) -> c4(LENGTH(nil))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))
K tuples:none
Defined Rule Symbols:   from_1, cons_2, nil

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(99) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(from(z0))) by 
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0)))))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   cons(z0, z1) -> n__cons(z0, z1)
   nil -> n__nil
Tuples:
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1)))
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0)))
   LENGTH1(n__nil) -> c4(LENGTH(nil))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0)))))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
S tuples:
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1)))
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0)))
   LENGTH1(n__nil) -> c4(LENGTH(nil))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0)))))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
K tuples:none
Defined Rule Symbols:   from_1, cons_2, nil

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(101) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(nil)) by 
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   cons(z0, z1) -> n__cons(z0, z1)
   nil -> n__nil
Tuples:
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1)))
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0)))
   LENGTH1(n__nil) -> c4(LENGTH(nil))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0)))))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
S tuples:
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1)))
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0)))
   LENGTH1(n__nil) -> c4(LENGTH(nil))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0)))))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
K tuples:none
Defined Rule Symbols:   from_1, cons_2, nil

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(103) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(cons(z0, z1))) by 
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   cons(z0, z1) -> n__cons(z0, z1)
   nil -> n__nil
Tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0)))
   LENGTH1(n__nil) -> c4(LENGTH(nil))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0)))))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
S tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0)))
   LENGTH1(n__nil) -> c4(LENGTH(nil))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0)))))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
K tuples:none
Defined Rule Symbols:   from_1, cons_2, nil

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(105) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LENGTH1(n__from(z0)) -> c4(LENGTH(from(z0))) by 
   LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__from(z0)))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   cons(z0, z1) -> n__cons(z0, z1)
   nil -> n__nil
Tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(n__nil) -> c4(LENGTH(nil))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0)))))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__from(z0)))
S tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(n__nil) -> c4(LENGTH(nil))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0)))))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__from(z0)))
K tuples:none
Defined Rule Symbols:   from_1, cons_2, nil

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(107) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__from(z0)))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   cons(z0, z1) -> n__cons(z0, z1)
   nil -> n__nil
Tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(n__nil) -> c4(LENGTH(nil))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0)))))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0)))))
S tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(n__nil) -> c4(LENGTH(nil))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0)))))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0)))))
K tuples:none
Defined Rule Symbols:   from_1, cons_2, nil

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(109) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   nil -> n__nil
   cons(z0, z1) -> n__cons(z0, z1)
Tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(n__nil) -> c4(LENGTH(nil))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0)))))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0)))))
S tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(n__nil) -> c4(LENGTH(nil))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0)))))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0)))))
K tuples:none
Defined Rule Symbols:   nil, cons_2

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(111) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LENGTH1(n__nil) -> c4(LENGTH(nil)) by 
   LENGTH1(n__nil) -> c4(LENGTH(n__nil))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   nil -> n__nil
   cons(z0, z1) -> n__cons(z0, z1)
Tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0)))))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0)))))
   LENGTH1(n__nil) -> c4(LENGTH(n__nil))
S tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0)))))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0)))))
   LENGTH1(n__nil) -> c4(LENGTH(n__nil))
K tuples:none
Defined Rule Symbols:   nil, cons_2

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(113) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LENGTH1(n__nil) -> c4(LENGTH(n__nil))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   nil -> n__nil
   cons(z0, z1) -> n__cons(z0, z1)
Tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0)))))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0)))))
S tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0)))))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0)))))
K tuples:none
Defined Rule Symbols:   nil, cons_2

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(115) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   nil -> n__nil

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   cons(z0, z1) -> n__cons(z0, z1)
Tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0)))))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0)))))
S tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1)))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0)))))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0)))))
K tuples:none
Defined Rule Symbols:   cons_2

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(117) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(cons(z0, z1))) by 
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1)))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   cons(z0, z1) -> n__cons(z0, z1)
Tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0)))))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1)))
S tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0)))))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1)))
K tuples:none
Defined Rule Symbols:   cons_2

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(119) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(cons(z0, n__from(s(z0))))) by 
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   cons(z0, z1) -> n__cons(z0, z1)
Tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
S tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
K tuples:none
Defined Rule Symbols:   cons_2

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(121) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LENGTH1(n__from(z0)) -> c4(LENGTH(cons(z0, n__from(s(z0))))) by 
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   cons(z0, z1) -> n__cons(z0, z1)
Tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
S tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
K tuples:none
Defined Rule Symbols:   cons_2

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(123) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   cons(z0, z1) -> n__cons(z0, z1)

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

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
S tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH1(z0) -> c4(LENGTH(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(125) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH1(z0) -> c4(LENGTH(z0)) by 
   LENGTH1(n__cons(y0, y1)) -> c4(LENGTH(n__cons(y0, y1)))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__nil)) -> c4(LENGTH(n__cons(y0, n__nil)))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))

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

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__nil)) -> c4(LENGTH(n__cons(y0, n__nil)))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))
S tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__nil)) -> c4(LENGTH(n__cons(y0, n__nil)))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(127) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LENGTH(n__cons(x0, n__nil)) -> c3(LENGTH1(n__nil))

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__nil)) -> c4(LENGTH(n__cons(y0, n__nil)))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))
S tuples:
   LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__nil)) -> c4(LENGTH(n__cons(y0, n__nil)))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(129) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH(n__cons(x0, z0)) -> c3(LENGTH1(z0)) by 
   LENGTH(n__cons(z0, n__cons(y0, y1))) -> c3(LENGTH1(n__cons(y0, y1)))
   LENGTH(n__cons(z0, n__from(y0))) -> c3(LENGTH1(n__from(y0)))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__nil))) -> c3(LENGTH1(n__cons(y0, n__nil)))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))

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

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__nil)) -> c4(LENGTH(n__cons(y0, n__nil)))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__nil))) -> c3(LENGTH1(n__cons(y0, n__nil)))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
S tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__nil)) -> c4(LENGTH(n__cons(y0, n__nil)))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__nil))) -> c3(LENGTH1(n__cons(y0, n__nil)))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(131) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LENGTH1(n__cons(y0, n__nil)) -> c4(LENGTH(n__cons(y0, n__nil)))

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

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__nil))) -> c3(LENGTH1(n__cons(y0, n__nil)))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
S tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__nil))) -> c3(LENGTH1(n__cons(y0, n__nil)))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(133) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1))) by 
   LENGTH1(n__cons(z0, n__from(y1))) -> c4(LENGTH(n__cons(z0, n__from(y1))))
   LENGTH1(n__cons(z0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(z0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))

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

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH1(n__cons(z0, z1)) -> c4(LENGTH(n__cons(z0, z1)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__nil))) -> c3(LENGTH1(n__cons(y0, n__nil)))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
S tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__nil))) -> c3(LENGTH1(n__cons(y0, n__nil)))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(135) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH1(n__cons(y0, y1)) -> c4(LENGTH(n__cons(y0, y1))) by 
   LENGTH1(n__cons(z0, n__from(y1))) -> c4(LENGTH(n__cons(z0, n__from(y1))))
   LENGTH1(n__cons(z0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(z0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))

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

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__nil))) -> c3(LENGTH1(n__cons(y0, n__nil)))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
S tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__nil))) -> c3(LENGTH1(n__cons(y0, n__nil)))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(137) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LENGTH(n__cons(z0, n__cons(y0, n__nil))) -> c3(LENGTH1(n__cons(y0, n__nil)))

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
S tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(139) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1))) by 
   LENGTH(n__cons(z0, n__cons(z1, n__from(y1)))) -> c3(LENGTH1(n__cons(z1, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, y2))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))

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

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__cons(z0, z1))) -> c3(LENGTH1(n__cons(z0, z1)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
S tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(141) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH(n__cons(z0, n__cons(y0, y1))) -> c3(LENGTH1(n__cons(y0, y1))) by 
   LENGTH(n__cons(z0, n__cons(z1, n__from(y1)))) -> c3(LENGTH1(n__cons(z1, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, y2))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))

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

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
S tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(143) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LENGTH1(n__cons(z0, n__cons(y1, n__nil))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__nil))))

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

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
S tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(145) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2)))) by 
   LENGTH1(n__cons(z0, n__cons(z1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))

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

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
S tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(147) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH1(n__cons(z0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(z0, n__cons(y1, y2)))) by 
   LENGTH1(n__cons(z0, n__cons(z1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))

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

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(y0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(y0, n__cons(y1, y2))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
S tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(149) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH1(n__cons(z0, n__cons(y1, y2))) -> c4(LENGTH(n__cons(z0, n__cons(y1, y2)))) by 
   LENGTH1(n__cons(z0, n__cons(z1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))

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

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
S tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(151) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__nil)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__nil))))

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

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
S tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(153) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2)))) by 
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))

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

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))
S tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(155) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, y2)))) by 
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))

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

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(y0, n__cons(y1, y2))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))
S tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(157) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, y2)))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, y2)))) by 
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))

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

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))
S tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(159) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__nil)))))

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

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))
S tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(161) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3))))) by 
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, y4))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))))

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

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))))
S tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(163) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))
We considered the (Usable) Rules:none
And the Tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))))
The order we found is given by the following interpretation:

Matrix interpretation [MATRO]:

Non-tuple symbols: 
<<<
 M( s_1(x_1) ) =  	[[2], [0]] 	+ 	[[1, 2], [0, 0]] 	* 	x_1
>>>

<<<
 M( n__nil ) =  	[[0], [1]]
>>>

<<<
 M( n__from_1(x_1) ) =  	[[1], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1
>>>

<<<
 M( n__cons_2(x_1, x_2) ) =  	[[1], [1]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1 	+ 	[[0, 2], [0, 1]] 	* 	x_2
>>>

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

<<<
 M( LENGTH1_1(x_1) ) =  	[[1], [4]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1
>>>

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

<<<
 M( c4_1(x_1) ) =  	[[0], [0]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1
>>>



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.
----------------------------------------

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))))
S tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))))
K tuples:
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

(165) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, y4))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(z1, n__from(y1)))) -> c3(LENGTH1(n__cons(z1, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(z1, n__from(y1)))) -> c3(LENGTH1(n__cons(z1, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))

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

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(y1, n__cons(y2, y3)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))))
S tuples:
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__from(z0)))
   LENGTH(n__cons(x0, n__from(z0))) -> c3(LENGTH1(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__from(z0)) -> c4(LENGTH(n__cons(z0, n__from(s(z0)))))
   LENGTH1(n__cons(y0, n__from(y1))) -> c4(LENGTH(n__cons(y0, n__from(y1))))
K tuples:
   LENGTH(n__cons(z0, n__cons(y0, n__from(y1)))) -> c3(LENGTH1(n__cons(y0, n__from(y1))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__from(y2))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__from(y2)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(y1, n__cons(y2, y3))))) -> c3(LENGTH1(n__cons(z1, n__cons(y1, n__cons(y2, y3)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__from(y3))))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__nil))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__nil)))))
   LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4)))))) -> c3(LENGTH1(n__cons(z1, n__cons(z2, n__cons(y2, n__cons(y3, y4))))))
   LENGTH1(n__cons(z0, n__cons(y1, n__from(y2)))) -> c4(LENGTH(n__cons(z0, n__cons(y1, n__from(y2)))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__from(y3))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(y2, n__cons(y3, y4))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__from(y4)))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__nil))))))
   LENGTH1(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))) -> c4(LENGTH(n__cons(z0, n__cons(z1, n__cons(z2, n__cons(y3, n__cons(y4, y5)))))))
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, LENGTH1_1

Compound Symbols:   c3_1, c4_1


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

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

(168)
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:

   from(X) -> cons(X, n__from(s(X))) [1]
   length(n__nil) -> 0 [1]
   length(n__cons(X, Y)) -> s(length1(activate(Y))) [1]
   length1(X) -> length(activate(X)) [1]
   from(X) -> n__from(X) [1]
   nil -> n__nil [1]
   cons(X1, X2) -> n__cons(X1, X2) [1]
   activate(n__from(X)) -> from(X) [1]
   activate(n__nil) -> nil [1]
   activate(n__cons(X1, X2)) -> cons(X1, X2) [1]
   activate(X) -> X [1]

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

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

(170)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   from(X) -> cons(X, n__from(s(X))) [1]
   length(n__nil) -> 0 [1]
   length(n__cons(X, Y)) -> s(length1(activate(Y))) [1]
   length1(X) -> length(activate(X)) [1]
   from(X) -> n__from(X) [1]
   nil -> n__nil [1]
   cons(X1, X2) -> n__cons(X1, X2) [1]
   activate(n__from(X)) -> from(X) [1]
   activate(n__nil) -> nil [1]
   activate(n__cons(X1, X2)) -> cons(X1, X2) [1]
   activate(X) -> X [1]

The TRS has the following type information:
from :: s:0 -> n__from:n__nil:n__cons
cons :: s:0 -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
n__from :: s:0 -> n__from:n__nil:n__cons
s :: s:0 -> s:0
length :: n__from:n__nil:n__cons -> s:0
n__nil :: n__from:n__nil:n__cons
0 :: s:0
n__cons :: s:0 -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
length1 :: n__from:n__nil:n__cons -> s:0
activate :: n__from:n__nil:n__cons -> n__from:n__nil:n__cons
nil :: n__from:n__nil:n__cons

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

(171) 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:

   length_1
   length1_1

(c) The following functions are completely defined:

   activate_1
   nil
   from_1
   cons_2

Due to the following rules being added:
none

And the following fresh constants: none

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

(172)
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:

   from(X) -> cons(X, n__from(s(X))) [1]
   length(n__nil) -> 0 [1]
   length(n__cons(X, Y)) -> s(length1(activate(Y))) [1]
   length1(X) -> length(activate(X)) [1]
   from(X) -> n__from(X) [1]
   nil -> n__nil [1]
   cons(X1, X2) -> n__cons(X1, X2) [1]
   activate(n__from(X)) -> from(X) [1]
   activate(n__nil) -> nil [1]
   activate(n__cons(X1, X2)) -> cons(X1, X2) [1]
   activate(X) -> X [1]

The TRS has the following type information:
from :: s:0 -> n__from:n__nil:n__cons
cons :: s:0 -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
n__from :: s:0 -> n__from:n__nil:n__cons
s :: s:0 -> s:0
length :: n__from:n__nil:n__cons -> s:0
n__nil :: n__from:n__nil:n__cons
0 :: s:0
n__cons :: s:0 -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
length1 :: n__from:n__nil:n__cons -> s:0
activate :: n__from:n__nil:n__cons -> n__from:n__nil:n__cons
nil :: n__from:n__nil:n__cons

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

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

(174)
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:

   from(X) -> cons(X, n__from(s(X))) [1]
   length(n__nil) -> 0 [1]
   length(n__cons(X, n__from(X'))) -> s(length1(from(X'))) [2]
   length(n__cons(X, n__nil)) -> s(length1(nil)) [2]
   length(n__cons(X, n__cons(X1', X2'))) -> s(length1(cons(X1', X2'))) [2]
   length(n__cons(X, Y)) -> s(length1(Y)) [2]
   length1(n__from(X'')) -> length(from(X'')) [2]
   length1(n__nil) -> length(nil) [2]
   length1(n__cons(X1'', X2'')) -> length(cons(X1'', X2'')) [2]
   length1(X) -> length(X) [2]
   from(X) -> n__from(X) [1]
   nil -> n__nil [1]
   cons(X1, X2) -> n__cons(X1, X2) [1]
   activate(n__from(X)) -> from(X) [1]
   activate(n__nil) -> nil [1]
   activate(n__cons(X1, X2)) -> cons(X1, X2) [1]
   activate(X) -> X [1]

The TRS has the following type information:
from :: s:0 -> n__from:n__nil:n__cons
cons :: s:0 -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
n__from :: s:0 -> n__from:n__nil:n__cons
s :: s:0 -> s:0
length :: n__from:n__nil:n__cons -> s:0
n__nil :: n__from:n__nil:n__cons
0 :: s:0
n__cons :: s:0 -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
length1 :: n__from:n__nil:n__cons -> s:0
activate :: n__from:n__nil:n__cons -> n__from:n__nil:n__cons
nil :: n__from:n__nil:n__cons

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

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

   n__nil => 0
   0 => 0

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

(176)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> nil :|: z = 0
   activate(z) -{ 1 }-> from(X) :|: z = 1 + X, X >= 0
   activate(z) -{ 1 }-> cons(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   cons(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
   from(z) -{ 1 }-> cons(X, 1 + (1 + X)) :|: X >= 0, z = X
   from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   length(z) -{ 2 }-> 1 + length1(nil) :|: z = 1 + X + 0, X >= 0
   length(z) -{ 2 }-> 1 + length1(from(X')) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0
   length(z) -{ 2 }-> 1 + length1(cons(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2')
   length1(z) -{ 2 }-> length(X) :|: X >= 0, z = X
   length1(z) -{ 2 }-> length(nil) :|: z = 0
   length1(z) -{ 2 }-> length(from(X'')) :|: z = 1 + X'', X'' >= 0
   length1(z) -{ 2 }-> length(cons(X1'', X2'')) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0
   nil -{ 1 }-> 0 :|: 


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

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

   nil -{ 1 }-> 0 :|: 
   cons(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
   from(z) -{ 1 }-> cons(X, 1 + (1 + X)) :|: X >= 0, z = X
   from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X

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

(178)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z = 1 + X, X >= 0, X' >= 0, X = X'
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X'
   activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
   cons(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
   from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   from(z) -{ 2 }-> 1 + X1 + X2 :|: X >= 0, z = X, X1 >= 0, X2 >= 0, X = X1, 1 + (1 + X) = X2
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(0) :|: z = 1 + X + 0, X >= 0
   length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2
   length1(z) -{ 2 }-> length(X) :|: X >= 0, z = X
   length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z = 1 + X'', X'' >= 0, X >= 0, X'' = X
   length1(z) -{ 3 }-> length(0) :|: z = 0
   length1(z) -{ 3 }-> length(1 + X) :|: z = 1 + X'', X'' >= 0, X >= 0, X'' = X
   length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2
   nil -{ 1 }-> 0 :|: 


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

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

(180)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
   cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2
   length1(z) -{ 2 }-> length(z) :|: z >= 0
   length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(0) :|: z = 0
   length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2
   nil -{ 1 }-> 0 :|: 


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

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

   { nil }
   { from }
   { cons }
   { activate }
   { length1, length }

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

(182)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
   cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2
   length1(z) -{ 2 }-> length(z) :|: z >= 0
   length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(0) :|: z = 0
   length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2
   nil -{ 1 }-> 0 :|: 

Function symbols to be analyzed: {nil}, {from}, {cons}, {activate}, {length1,length}

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

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

(184)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
   cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2
   length1(z) -{ 2 }-> length(z) :|: z >= 0
   length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(0) :|: z = 0
   length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2
   nil -{ 1 }-> 0 :|: 

Function symbols to be analyzed: {nil}, {from}, {cons}, {activate}, {length1,length}

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

(185) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(186)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
   cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2
   length1(z) -{ 2 }-> length(z) :|: z >= 0
   length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(0) :|: z = 0
   length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2
   nil -{ 1 }-> 0 :|: 

Function symbols to be analyzed: {nil}, {from}, {cons}, {activate}, {length1,length}
Previous analysis results are:
  nil: runtime: ?, size: O(1) [0]

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

(187) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(188)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
   cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2
   length1(z) -{ 2 }-> length(z) :|: z >= 0
   length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(0) :|: z = 0
   length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2
   nil -{ 1 }-> 0 :|: 

Function symbols to be analyzed: {from}, {cons}, {activate}, {length1,length}
Previous analysis results are:
  nil: runtime: O(1) [1], size: O(1) [0]

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

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

(190)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
   cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2
   length1(z) -{ 2 }-> length(z) :|: z >= 0
   length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(0) :|: z = 0
   length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2
   nil -{ 1 }-> 0 :|: 

Function symbols to be analyzed: {from}, {cons}, {activate}, {length1,length}
Previous analysis results are:
  nil: runtime: O(1) [1], size: O(1) [0]

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

(191) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: from
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + 2*z

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

(192)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
   cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2
   length1(z) -{ 2 }-> length(z) :|: z >= 0
   length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(0) :|: z = 0
   length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2
   nil -{ 1 }-> 0 :|: 

Function symbols to be analyzed: {from}, {cons}, {activate}, {length1,length}
Previous analysis results are:
  nil: runtime: O(1) [1], size: O(1) [0]
  from: runtime: ?, size: O(n^1) [3 + 2*z]

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

(193) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(194)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
   cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2
   length1(z) -{ 2 }-> length(z) :|: z >= 0
   length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(0) :|: z = 0
   length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2
   nil -{ 1 }-> 0 :|: 

Function symbols to be analyzed: {cons}, {activate}, {length1,length}
Previous analysis results are:
  nil: runtime: O(1) [1], size: O(1) [0]
  from: runtime: O(1) [2], size: O(n^1) [3 + 2*z]

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

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

(196)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
   cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2
   length1(z) -{ 2 }-> length(z) :|: z >= 0
   length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(0) :|: z = 0
   length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2
   nil -{ 1 }-> 0 :|: 

Function symbols to be analyzed: {cons}, {activate}, {length1,length}
Previous analysis results are:
  nil: runtime: O(1) [1], size: O(1) [0]
  from: runtime: O(1) [2], size: O(n^1) [3 + 2*z]

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

(197) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(198)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
   cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2
   length1(z) -{ 2 }-> length(z) :|: z >= 0
   length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(0) :|: z = 0
   length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2
   nil -{ 1 }-> 0 :|: 

Function symbols to be analyzed: {cons}, {activate}, {length1,length}
Previous analysis results are:
  nil: runtime: O(1) [1], size: O(1) [0]
  from: runtime: O(1) [2], size: O(n^1) [3 + 2*z]
  cons: runtime: ?, size: O(n^1) [1 + z + z']

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

(199) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(200)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
   cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   length(z) -{ 3 }-> 1 + length1(cons(X'', 1 + (1 + X''))) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2
   length1(z) -{ 2 }-> length(z) :|: z >= 0
   length1(z) -{ 3 }-> length(cons(X, 1 + (1 + X))) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(0) :|: z = 0
   length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2
   nil -{ 1 }-> 0 :|: 

Function symbols to be analyzed: {activate}, {length1,length}
Previous analysis results are:
  nil: runtime: O(1) [1], size: O(1) [0]
  from: runtime: O(1) [2], size: O(n^1) [3 + 2*z]
  cons: runtime: O(1) [1], size: O(n^1) [1 + z + z']

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

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

(202)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 3 }-> s'' :|: s'' >= 0, s'' <= X' + (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
   cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   length(z) -{ 4 }-> 1 + length1(s) :|: s >= 0, s <= X'' + (1 + (1 + X'')) + 1, z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2
   length1(z) -{ 4 }-> length(s') :|: s' >= 0, s' <= X + (1 + (1 + X)) + 1, z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 2 }-> length(z) :|: z >= 0
   length1(z) -{ 3 }-> length(0) :|: z = 0
   length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2
   nil -{ 1 }-> 0 :|: 

Function symbols to be analyzed: {activate}, {length1,length}
Previous analysis results are:
  nil: runtime: O(1) [1], size: O(1) [0]
  from: runtime: O(1) [2], size: O(n^1) [3 + 2*z]
  cons: runtime: O(1) [1], size: O(n^1) [1 + z + z']

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

(203) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + 2*z

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

(204)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 3 }-> s'' :|: s'' >= 0, s'' <= X' + (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
   cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   length(z) -{ 4 }-> 1 + length1(s) :|: s >= 0, s <= X'' + (1 + (1 + X'')) + 1, z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2
   length1(z) -{ 4 }-> length(s') :|: s' >= 0, s' <= X + (1 + (1 + X)) + 1, z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 2 }-> length(z) :|: z >= 0
   length1(z) -{ 3 }-> length(0) :|: z = 0
   length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2
   nil -{ 1 }-> 0 :|: 

Function symbols to be analyzed: {activate}, {length1,length}
Previous analysis results are:
  nil: runtime: O(1) [1], size: O(1) [0]
  from: runtime: O(1) [2], size: O(n^1) [3 + 2*z]
  cons: runtime: O(1) [1], size: O(n^1) [1 + z + z']
  activate: runtime: ?, size: O(n^1) [1 + 2*z]

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

(205) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 10

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

(206)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 3 }-> s'' :|: s'' >= 0, s'' <= X' + (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
   cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   length(z) -{ 4 }-> 1 + length1(s) :|: s >= 0, s <= X'' + (1 + (1 + X'')) + 1, z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2
   length1(z) -{ 4 }-> length(s') :|: s' >= 0, s' <= X + (1 + (1 + X)) + 1, z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 2 }-> length(z) :|: z >= 0
   length1(z) -{ 3 }-> length(0) :|: z = 0
   length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2
   nil -{ 1 }-> 0 :|: 

Function symbols to be analyzed: {length1,length}
Previous analysis results are:
  nil: runtime: O(1) [1], size: O(1) [0]
  from: runtime: O(1) [2], size: O(n^1) [3 + 2*z]
  cons: runtime: O(1) [1], size: O(n^1) [1 + z + z']
  activate: runtime: O(1) [10], size: O(n^1) [1 + 2*z]

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

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

(208)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 3 }-> s'' :|: s'' >= 0, s'' <= X' + (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
   cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   length(z) -{ 4 }-> 1 + length1(s) :|: s >= 0, s <= X'' + (1 + (1 + X'')) + 1, z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2
   length1(z) -{ 4 }-> length(s') :|: s' >= 0, s' <= X + (1 + (1 + X)) + 1, z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 2 }-> length(z) :|: z >= 0
   length1(z) -{ 3 }-> length(0) :|: z = 0
   length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2
   nil -{ 1 }-> 0 :|: 

Function symbols to be analyzed: {length1,length}
Previous analysis results are:
  nil: runtime: O(1) [1], size: O(1) [0]
  from: runtime: O(1) [2], size: O(n^1) [3 + 2*z]
  cons: runtime: O(1) [1], size: O(n^1) [1 + z + z']
  activate: runtime: O(1) [10], size: O(n^1) [1 + 2*z]

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

(209) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: length1
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

(210)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 3 }-> s'' :|: s'' >= 0, s'' <= X' + (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
   cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   length(z) -{ 4 }-> 1 + length1(s) :|: s >= 0, s <= X'' + (1 + (1 + X'')) + 1, z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2
   length1(z) -{ 4 }-> length(s') :|: s' >= 0, s' <= X + (1 + (1 + X)) + 1, z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 2 }-> length(z) :|: z >= 0
   length1(z) -{ 3 }-> length(0) :|: z = 0
   length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2
   nil -{ 1 }-> 0 :|: 

Function symbols to be analyzed: {length1,length}
Previous analysis results are:
  nil: runtime: O(1) [1], size: O(1) [0]
  from: runtime: O(1) [2], size: O(n^1) [3 + 2*z]
  cons: runtime: O(1) [1], size: O(n^1) [1 + z + z']
  activate: runtime: O(1) [10], size: O(n^1) [1 + 2*z]
  length1: runtime: ?, size: INF
  length: runtime: ?, size: INF

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

(211) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: length1
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

(212)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 3 }-> s'' :|: s'' >= 0, s'' <= X' + (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
   cons(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 2 }-> 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 2 }-> 1 + length1(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   length(z) -{ 4 }-> 1 + length1(s) :|: s >= 0, s <= X'' + (1 + (1 + X'')) + 1, z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(0) :|: z - 1 >= 0
   length(z) -{ 3 }-> 1 + length1(1 + X'') :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   length(z) -{ 3 }-> 1 + length1(1 + X1 + X2) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2'), X1 >= 0, X2 >= 0, X1' = X1, X2' = X2
   length1(z) -{ 4 }-> length(s') :|: s' >= 0, s' <= X + (1 + (1 + X)) + 1, z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 2 }-> length(z) :|: z >= 0
   length1(z) -{ 3 }-> length(0) :|: z = 0
   length1(z) -{ 3 }-> length(1 + X) :|: z - 1 >= 0, X >= 0, z - 1 = X
   length1(z) -{ 3 }-> length(1 + X1 + X2) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0, X1 >= 0, X2 >= 0, X1'' = X1, X2'' = X2
   nil -{ 1 }-> 0 :|: 

Function symbols to be analyzed: {length1,length}
Previous analysis results are:
  nil: runtime: O(1) [1], size: O(1) [0]
  from: runtime: O(1) [2], size: O(n^1) [3 + 2*z]
  cons: runtime: O(1) [1], size: O(n^1) [1 + z + z']
  activate: runtime: O(1) [10], size: O(n^1) [1 + 2*z]
  length1: runtime: INF, size: INF
  length: runtime: ?, size: INF

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

(213) 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

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

(214)
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:

   from(X) -> cons(X, n__from(s(X))) [1]
   length(n__nil) -> 0 [1]
   length(n__cons(X, Y)) -> s(length1(activate(Y))) [1]
   length1(X) -> length(activate(X)) [1]
   from(X) -> n__from(X) [1]
   nil -> n__nil [1]
   cons(X1, X2) -> n__cons(X1, X2) [1]
   activate(n__from(X)) -> from(X) [1]
   activate(n__nil) -> nil [1]
   activate(n__cons(X1, X2)) -> cons(X1, X2) [1]
   activate(X) -> X [1]
   length(v0) -> null_length [0]

The TRS has the following type information:
from :: s:0:null_length -> n__from:n__nil:n__cons
cons :: s:0:null_length -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
n__from :: s:0:null_length -> n__from:n__nil:n__cons
s :: s:0:null_length -> s:0:null_length
length :: n__from:n__nil:n__cons -> s:0:null_length
n__nil :: n__from:n__nil:n__cons
0 :: s:0:null_length
n__cons :: s:0:null_length -> n__from:n__nil:n__cons -> n__from:n__nil:n__cons
length1 :: n__from:n__nil:n__cons -> s:0:null_length
activate :: n__from:n__nil:n__cons -> n__from:n__nil:n__cons
nil :: n__from:n__nil:n__cons
null_length :: s:0:null_length

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

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

   n__nil => 0
   0 => 0
   null_length => 0

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

(216)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> nil :|: z = 0
   activate(z) -{ 1 }-> from(X) :|: z = 1 + X, X >= 0
   activate(z) -{ 1 }-> cons(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   cons(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
   from(z) -{ 1 }-> cons(X, 1 + (1 + X)) :|: X >= 0, z = X
   from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   length(z) -{ 1 }-> 1 + length1(activate(Y)) :|: Y >= 0, z = 1 + X + Y, X >= 0
   length1(z) -{ 1 }-> length(activate(X)) :|: X >= 0, z = X
   nil -{ 1 }-> 0 :|: 

Only complete derivations are relevant for the runtime complexity.