MAYBE
proof of input_S9lWiSQEV3.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 0 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 346 ms]
    (16) BEST
        (17) proven lower bound
            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
            (19) BOUNDS(n^1, INF)
        (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 144 ms]
            (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 29 ms]
            (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 28 ms]
            (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 404 ms]
            (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 46 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) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (36) CdtProblem
    (37) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CpxTRS
        (39) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
        (40) CpxTRS
        (41) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (42) CpxWeightedTrs
            (43) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
            (44) CpxTypedWeightedTrs
            (45) CompletionProof [UPPER BOUND(ID), 0 ms]
            (46) CpxTypedWeightedCompleteTrs
                (47) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
                (48) CpxTypedWeightedCompleteTrs
                (49) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (50) CpxRNTS
                (51) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
                (52) CpxRNTS
                (53) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
                (54) CpxRNTS
                (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (56) CpxRNTS
                (57) IntTrsBoundProof [UPPER BOUND(ID), 129 ms]
                (58) CpxRNTS
                (59) IntTrsBoundProof [UPPER BOUND(ID), 73 ms]
                (60) CpxRNTS
                (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (62) CpxRNTS
                (63) IntTrsBoundProof [UPPER BOUND(ID), 159 ms]
                (64) CpxRNTS
                (65) IntTrsBoundProof [UPPER BOUND(ID), 104 ms]
                (66) CpxRNTS
                (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (68) CpxRNTS
                (69) IntTrsBoundProof [UPPER BOUND(ID), 479 ms]
                (70) CpxRNTS
                (71) IntTrsBoundProof [UPPER BOUND(ID), 174 ms]
                (72) CpxRNTS
            (73) CompletionProof [UPPER BOUND(ID), 0 ms]
            (74) CpxTypedWeightedCompleteTrs
                (75) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (76) CpxRNTS
    (77) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 35 ms]
    (78) CdtProblem
        (79) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (80) CdtProblem
        (81) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (82) CdtProblem
        (83) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (84) CdtProblem
        (85) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (86) CdtProblem
        (87) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (88) CdtProblem
        (89) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (90) CdtProblem
        (91) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (92) CdtProblem
        (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (94) CdtProblem
(95) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(96) CpxWeightedTrs
    (97) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CpxTypedWeightedTrs
    (99) CompletionProof [UPPER BOUND(ID), 0 ms]
    (100) CpxTypedWeightedCompleteTrs
        (101) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (102) CpxTypedWeightedCompleteTrs
        (103) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (104) CpxRNTS
        (105) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (106) CpxRNTS
        (107) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (108) CpxRNTS
        (109) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (110) CpxRNTS
        (111) IntTrsBoundProof [UPPER BOUND(ID), 803 ms]
        (112) CpxRNTS
        (113) IntTrsBoundProof [UPPER BOUND(ID), 27 ms]
        (114) CpxRNTS
    (115) CompletionProof [UPPER BOUND(ID), 0 ms]
    (116) CpxTypedWeightedCompleteTrs
        (117) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (118) 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:

   eq(0, 0) -> true
   eq(s(X), s(Y)) -> eq(X, Y)
   eq(X, Y) -> false
   inf(X) -> cons(X, inf(s(X)))
   take(0, X) -> nil
   take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
   length(nil) -> 0
   length(cons(X, L)) -> s(length(L))

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:

   eq(0', 0') -> true
   eq(s(X), s(Y)) -> eq(X, Y)
   eq(X, Y) -> false
   inf(X) -> cons(X, inf(s(X)))
   take(0', X) -> nil
   take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
   length(nil) -> 0'
   length(cons(X, L)) -> s(length(L))

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:

   eq(0, 0) -> true
   eq(s(X), s(Y)) -> eq(X, Y)
   eq(X, Y) -> false
   inf(X) -> cons(X, inf(s(X)))
   take(0, X) -> nil
   take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
   length(nil) -> 0
   length(cons(X, L)) -> s(length(L))

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:
   eq(0, 0) -> true
   eq(s(z0), s(z1)) -> eq(z0, z1)
   eq(z0, z1) -> false
   inf(z0) -> cons(z0, inf(s(z0)))
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(z1))
Tuples:
   EQ(0, 0) -> c
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   EQ(z0, z1) -> c2
   INF(z0) -> c3(INF(s(z0)))
   TAKE(0, z0) -> c4
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(nil) -> c6
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
S tuples:
   EQ(0, 0) -> c
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   EQ(z0, z1) -> c2
   INF(z0) -> c3(INF(s(z0)))
   TAKE(0, z0) -> c4
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(nil) -> c6
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
K tuples:none
Defined Rule Symbols:   eq_2, inf_1, take_2, length_1

Defined Pair Symbols:   EQ_2, INF_1, TAKE_2, LENGTH_1

Compound Symbols:   c, c1_1, c2, c3_1, c4, c5_1, c6, c7_1


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

(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:

   EQ(0, 0) -> c
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   EQ(z0, z1) -> c2
   INF(z0) -> c3(INF(s(z0)))
   TAKE(0, z0) -> c4
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(nil) -> c6
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))

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

   eq(0, 0) -> true
   eq(s(z0), s(z1)) -> eq(z0, z1)
   eq(z0, z1) -> false
   inf(z0) -> cons(z0, inf(s(z0)))
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(z1))

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:

   EQ(0', 0') -> c
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   EQ(z0, z1) -> c2
   INF(z0) -> c3(INF(s(z0)))
   TAKE(0', z0) -> c4
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(nil) -> c6
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))

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

   eq(0', 0') -> true
   eq(s(z0), s(z1)) -> eq(z0, z1)
   eq(z0, z1) -> false
   inf(z0) -> cons(z0, inf(s(z0)))
   take(0', z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
   length(nil) -> 0'
   length(cons(z0, z1)) -> s(length(z1))

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

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

(12)
Obligation:
Innermost TRS:
Rules:
EQ(0', 0') -> c
EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
EQ(z0, z1) -> c2
INF(z0) -> c3(INF(s(z0)))
TAKE(0', z0) -> c4
TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
LENGTH(nil) -> c6
LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
eq(0', 0') -> true
eq(s(z0), s(z1)) -> eq(z0, z1)
eq(z0, z1) -> false
inf(z0) -> cons(z0, inf(s(z0)))
take(0', z0) -> nil
take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))

Types:
EQ :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c:c1:c2
INF :: 0':s -> c3
c3 :: c3 -> c3
TAKE :: 0':s -> cons:nil -> c4:c5
c4 :: c4:c5
cons :: 0':s -> cons:nil -> cons:nil
c5 :: c4:c5 -> c4:c5
LENGTH :: cons:nil -> c6:c7
nil :: cons:nil
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
eq :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
inf :: 0':s -> cons:nil
take :: 0':s -> cons:nil -> cons:nil
length :: cons:nil -> 0':s
hole_c:c1:c21_8 :: c:c1:c2
hole_0':s2_8 :: 0':s
hole_c33_8 :: c3
hole_c4:c54_8 :: c4:c5
hole_cons:nil5_8 :: cons:nil
hole_c6:c76_8 :: c6:c7
hole_true:false7_8 :: true:false
gen_c:c1:c28_8 :: Nat -> c:c1:c2
gen_0':s9_8 :: Nat -> 0':s
gen_c310_8 :: Nat -> c3
gen_c4:c511_8 :: Nat -> c4:c5
gen_cons:nil12_8 :: Nat -> cons:nil
gen_c6:c713_8 :: Nat -> c6:c7

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

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
EQ, INF, TAKE, LENGTH, eq, inf, take, length
----------------------------------------

(14)
Obligation:
Innermost TRS:
Rules:
EQ(0', 0') -> c
EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
EQ(z0, z1) -> c2
INF(z0) -> c3(INF(s(z0)))
TAKE(0', z0) -> c4
TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
LENGTH(nil) -> c6
LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
eq(0', 0') -> true
eq(s(z0), s(z1)) -> eq(z0, z1)
eq(z0, z1) -> false
inf(z0) -> cons(z0, inf(s(z0)))
take(0', z0) -> nil
take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))

Types:
EQ :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c:c1:c2
INF :: 0':s -> c3
c3 :: c3 -> c3
TAKE :: 0':s -> cons:nil -> c4:c5
c4 :: c4:c5
cons :: 0':s -> cons:nil -> cons:nil
c5 :: c4:c5 -> c4:c5
LENGTH :: cons:nil -> c6:c7
nil :: cons:nil
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
eq :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
inf :: 0':s -> cons:nil
take :: 0':s -> cons:nil -> cons:nil
length :: cons:nil -> 0':s
hole_c:c1:c21_8 :: c:c1:c2
hole_0':s2_8 :: 0':s
hole_c33_8 :: c3
hole_c4:c54_8 :: c4:c5
hole_cons:nil5_8 :: cons:nil
hole_c6:c76_8 :: c6:c7
hole_true:false7_8 :: true:false
gen_c:c1:c28_8 :: Nat -> c:c1:c2
gen_0':s9_8 :: Nat -> 0':s
gen_c310_8 :: Nat -> c3
gen_c4:c511_8 :: Nat -> c4:c5
gen_cons:nil12_8 :: Nat -> cons:nil
gen_c6:c713_8 :: Nat -> c6:c7


Generator Equations:
gen_c:c1:c28_8(0) <=> c
gen_c:c1:c28_8(+(x, 1)) <=> c1(gen_c:c1:c28_8(x))
gen_0':s9_8(0) <=> 0'
gen_0':s9_8(+(x, 1)) <=> s(gen_0':s9_8(x))
gen_c310_8(0) <=> hole_c33_8
gen_c310_8(+(x, 1)) <=> c3(gen_c310_8(x))
gen_c4:c511_8(0) <=> c4
gen_c4:c511_8(+(x, 1)) <=> c5(gen_c4:c511_8(x))
gen_cons:nil12_8(0) <=> nil
gen_cons:nil12_8(+(x, 1)) <=> cons(0', gen_cons:nil12_8(x))
gen_c6:c713_8(0) <=> c6
gen_c6:c713_8(+(x, 1)) <=> c7(gen_c6:c713_8(x))


The following defined symbols remain to be analysed:
EQ, INF, TAKE, LENGTH, eq, inf, take, length
----------------------------------------

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
EQ(gen_0':s9_8(n15_8), gen_0':s9_8(n15_8)) -> gen_c:c1:c28_8(n15_8), rt in Omega(1 + n15_8)

Induction Base:
EQ(gen_0':s9_8(0), gen_0':s9_8(0)) ->_R^Omega(1)
c

Induction Step:
EQ(gen_0':s9_8(+(n15_8, 1)), gen_0':s9_8(+(n15_8, 1))) ->_R^Omega(1)
c1(EQ(gen_0':s9_8(n15_8), gen_0':s9_8(n15_8))) ->_IH
c1(gen_c:c1:c28_8(c16_8))

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

(16)
Complex Obligation (BEST)

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

(17)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
EQ(0', 0') -> c
EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
EQ(z0, z1) -> c2
INF(z0) -> c3(INF(s(z0)))
TAKE(0', z0) -> c4
TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
LENGTH(nil) -> c6
LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
eq(0', 0') -> true
eq(s(z0), s(z1)) -> eq(z0, z1)
eq(z0, z1) -> false
inf(z0) -> cons(z0, inf(s(z0)))
take(0', z0) -> nil
take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))

Types:
EQ :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c:c1:c2
INF :: 0':s -> c3
c3 :: c3 -> c3
TAKE :: 0':s -> cons:nil -> c4:c5
c4 :: c4:c5
cons :: 0':s -> cons:nil -> cons:nil
c5 :: c4:c5 -> c4:c5
LENGTH :: cons:nil -> c6:c7
nil :: cons:nil
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
eq :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
inf :: 0':s -> cons:nil
take :: 0':s -> cons:nil -> cons:nil
length :: cons:nil -> 0':s
hole_c:c1:c21_8 :: c:c1:c2
hole_0':s2_8 :: 0':s
hole_c33_8 :: c3
hole_c4:c54_8 :: c4:c5
hole_cons:nil5_8 :: cons:nil
hole_c6:c76_8 :: c6:c7
hole_true:false7_8 :: true:false
gen_c:c1:c28_8 :: Nat -> c:c1:c2
gen_0':s9_8 :: Nat -> 0':s
gen_c310_8 :: Nat -> c3
gen_c4:c511_8 :: Nat -> c4:c5
gen_cons:nil12_8 :: Nat -> cons:nil
gen_c6:c713_8 :: Nat -> c6:c7


Generator Equations:
gen_c:c1:c28_8(0) <=> c
gen_c:c1:c28_8(+(x, 1)) <=> c1(gen_c:c1:c28_8(x))
gen_0':s9_8(0) <=> 0'
gen_0':s9_8(+(x, 1)) <=> s(gen_0':s9_8(x))
gen_c310_8(0) <=> hole_c33_8
gen_c310_8(+(x, 1)) <=> c3(gen_c310_8(x))
gen_c4:c511_8(0) <=> c4
gen_c4:c511_8(+(x, 1)) <=> c5(gen_c4:c511_8(x))
gen_cons:nil12_8(0) <=> nil
gen_cons:nil12_8(+(x, 1)) <=> cons(0', gen_cons:nil12_8(x))
gen_c6:c713_8(0) <=> c6
gen_c6:c713_8(+(x, 1)) <=> c7(gen_c6:c713_8(x))


The following defined symbols remain to be analysed:
EQ, INF, TAKE, LENGTH, eq, inf, take, length
----------------------------------------

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

(19)
BOUNDS(n^1, INF)

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

(20)
Obligation:
Innermost TRS:
Rules:
EQ(0', 0') -> c
EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
EQ(z0, z1) -> c2
INF(z0) -> c3(INF(s(z0)))
TAKE(0', z0) -> c4
TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
LENGTH(nil) -> c6
LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
eq(0', 0') -> true
eq(s(z0), s(z1)) -> eq(z0, z1)
eq(z0, z1) -> false
inf(z0) -> cons(z0, inf(s(z0)))
take(0', z0) -> nil
take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))

Types:
EQ :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c:c1:c2
INF :: 0':s -> c3
c3 :: c3 -> c3
TAKE :: 0':s -> cons:nil -> c4:c5
c4 :: c4:c5
cons :: 0':s -> cons:nil -> cons:nil
c5 :: c4:c5 -> c4:c5
LENGTH :: cons:nil -> c6:c7
nil :: cons:nil
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
eq :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
inf :: 0':s -> cons:nil
take :: 0':s -> cons:nil -> cons:nil
length :: cons:nil -> 0':s
hole_c:c1:c21_8 :: c:c1:c2
hole_0':s2_8 :: 0':s
hole_c33_8 :: c3
hole_c4:c54_8 :: c4:c5
hole_cons:nil5_8 :: cons:nil
hole_c6:c76_8 :: c6:c7
hole_true:false7_8 :: true:false
gen_c:c1:c28_8 :: Nat -> c:c1:c2
gen_0':s9_8 :: Nat -> 0':s
gen_c310_8 :: Nat -> c3
gen_c4:c511_8 :: Nat -> c4:c5
gen_cons:nil12_8 :: Nat -> cons:nil
gen_c6:c713_8 :: Nat -> c6:c7


Lemmas:
EQ(gen_0':s9_8(n15_8), gen_0':s9_8(n15_8)) -> gen_c:c1:c28_8(n15_8), rt in Omega(1 + n15_8)


Generator Equations:
gen_c:c1:c28_8(0) <=> c
gen_c:c1:c28_8(+(x, 1)) <=> c1(gen_c:c1:c28_8(x))
gen_0':s9_8(0) <=> 0'
gen_0':s9_8(+(x, 1)) <=> s(gen_0':s9_8(x))
gen_c310_8(0) <=> hole_c33_8
gen_c310_8(+(x, 1)) <=> c3(gen_c310_8(x))
gen_c4:c511_8(0) <=> c4
gen_c4:c511_8(+(x, 1)) <=> c5(gen_c4:c511_8(x))
gen_cons:nil12_8(0) <=> nil
gen_cons:nil12_8(+(x, 1)) <=> cons(0', gen_cons:nil12_8(x))
gen_c6:c713_8(0) <=> c6
gen_c6:c713_8(+(x, 1)) <=> c7(gen_c6:c713_8(x))


The following defined symbols remain to be analysed:
INF, TAKE, LENGTH, eq, inf, take, length
----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
TAKE(gen_0':s9_8(n577_8), gen_cons:nil12_8(n577_8)) -> gen_c4:c511_8(n577_8), rt in Omega(1 + n577_8)

Induction Base:
TAKE(gen_0':s9_8(0), gen_cons:nil12_8(0)) ->_R^Omega(1)
c4

Induction Step:
TAKE(gen_0':s9_8(+(n577_8, 1)), gen_cons:nil12_8(+(n577_8, 1))) ->_R^Omega(1)
c5(TAKE(gen_0':s9_8(n577_8), gen_cons:nil12_8(n577_8))) ->_IH
c5(gen_c4:c511_8(c578_8))

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

(22)
Obligation:
Innermost TRS:
Rules:
EQ(0', 0') -> c
EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
EQ(z0, z1) -> c2
INF(z0) -> c3(INF(s(z0)))
TAKE(0', z0) -> c4
TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
LENGTH(nil) -> c6
LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
eq(0', 0') -> true
eq(s(z0), s(z1)) -> eq(z0, z1)
eq(z0, z1) -> false
inf(z0) -> cons(z0, inf(s(z0)))
take(0', z0) -> nil
take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))

Types:
EQ :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c:c1:c2
INF :: 0':s -> c3
c3 :: c3 -> c3
TAKE :: 0':s -> cons:nil -> c4:c5
c4 :: c4:c5
cons :: 0':s -> cons:nil -> cons:nil
c5 :: c4:c5 -> c4:c5
LENGTH :: cons:nil -> c6:c7
nil :: cons:nil
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
eq :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
inf :: 0':s -> cons:nil
take :: 0':s -> cons:nil -> cons:nil
length :: cons:nil -> 0':s
hole_c:c1:c21_8 :: c:c1:c2
hole_0':s2_8 :: 0':s
hole_c33_8 :: c3
hole_c4:c54_8 :: c4:c5
hole_cons:nil5_8 :: cons:nil
hole_c6:c76_8 :: c6:c7
hole_true:false7_8 :: true:false
gen_c:c1:c28_8 :: Nat -> c:c1:c2
gen_0':s9_8 :: Nat -> 0':s
gen_c310_8 :: Nat -> c3
gen_c4:c511_8 :: Nat -> c4:c5
gen_cons:nil12_8 :: Nat -> cons:nil
gen_c6:c713_8 :: Nat -> c6:c7


Lemmas:
EQ(gen_0':s9_8(n15_8), gen_0':s9_8(n15_8)) -> gen_c:c1:c28_8(n15_8), rt in Omega(1 + n15_8)
TAKE(gen_0':s9_8(n577_8), gen_cons:nil12_8(n577_8)) -> gen_c4:c511_8(n577_8), rt in Omega(1 + n577_8)


Generator Equations:
gen_c:c1:c28_8(0) <=> c
gen_c:c1:c28_8(+(x, 1)) <=> c1(gen_c:c1:c28_8(x))
gen_0':s9_8(0) <=> 0'
gen_0':s9_8(+(x, 1)) <=> s(gen_0':s9_8(x))
gen_c310_8(0) <=> hole_c33_8
gen_c310_8(+(x, 1)) <=> c3(gen_c310_8(x))
gen_c4:c511_8(0) <=> c4
gen_c4:c511_8(+(x, 1)) <=> c5(gen_c4:c511_8(x))
gen_cons:nil12_8(0) <=> nil
gen_cons:nil12_8(+(x, 1)) <=> cons(0', gen_cons:nil12_8(x))
gen_c6:c713_8(0) <=> c6
gen_c6:c713_8(+(x, 1)) <=> c7(gen_c6:c713_8(x))


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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LENGTH(gen_cons:nil12_8(n1072_8)) -> gen_c6:c713_8(n1072_8), rt in Omega(1 + n1072_8)

Induction Base:
LENGTH(gen_cons:nil12_8(0)) ->_R^Omega(1)
c6

Induction Step:
LENGTH(gen_cons:nil12_8(+(n1072_8, 1))) ->_R^Omega(1)
c7(LENGTH(gen_cons:nil12_8(n1072_8))) ->_IH
c7(gen_c6:c713_8(c1073_8))

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

(24)
Obligation:
Innermost TRS:
Rules:
EQ(0', 0') -> c
EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
EQ(z0, z1) -> c2
INF(z0) -> c3(INF(s(z0)))
TAKE(0', z0) -> c4
TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
LENGTH(nil) -> c6
LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
eq(0', 0') -> true
eq(s(z0), s(z1)) -> eq(z0, z1)
eq(z0, z1) -> false
inf(z0) -> cons(z0, inf(s(z0)))
take(0', z0) -> nil
take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))

Types:
EQ :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c:c1:c2
INF :: 0':s -> c3
c3 :: c3 -> c3
TAKE :: 0':s -> cons:nil -> c4:c5
c4 :: c4:c5
cons :: 0':s -> cons:nil -> cons:nil
c5 :: c4:c5 -> c4:c5
LENGTH :: cons:nil -> c6:c7
nil :: cons:nil
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
eq :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
inf :: 0':s -> cons:nil
take :: 0':s -> cons:nil -> cons:nil
length :: cons:nil -> 0':s
hole_c:c1:c21_8 :: c:c1:c2
hole_0':s2_8 :: 0':s
hole_c33_8 :: c3
hole_c4:c54_8 :: c4:c5
hole_cons:nil5_8 :: cons:nil
hole_c6:c76_8 :: c6:c7
hole_true:false7_8 :: true:false
gen_c:c1:c28_8 :: Nat -> c:c1:c2
gen_0':s9_8 :: Nat -> 0':s
gen_c310_8 :: Nat -> c3
gen_c4:c511_8 :: Nat -> c4:c5
gen_cons:nil12_8 :: Nat -> cons:nil
gen_c6:c713_8 :: Nat -> c6:c7


Lemmas:
EQ(gen_0':s9_8(n15_8), gen_0':s9_8(n15_8)) -> gen_c:c1:c28_8(n15_8), rt in Omega(1 + n15_8)
TAKE(gen_0':s9_8(n577_8), gen_cons:nil12_8(n577_8)) -> gen_c4:c511_8(n577_8), rt in Omega(1 + n577_8)
LENGTH(gen_cons:nil12_8(n1072_8)) -> gen_c6:c713_8(n1072_8), rt in Omega(1 + n1072_8)


Generator Equations:
gen_c:c1:c28_8(0) <=> c
gen_c:c1:c28_8(+(x, 1)) <=> c1(gen_c:c1:c28_8(x))
gen_0':s9_8(0) <=> 0'
gen_0':s9_8(+(x, 1)) <=> s(gen_0':s9_8(x))
gen_c310_8(0) <=> hole_c33_8
gen_c310_8(+(x, 1)) <=> c3(gen_c310_8(x))
gen_c4:c511_8(0) <=> c4
gen_c4:c511_8(+(x, 1)) <=> c5(gen_c4:c511_8(x))
gen_cons:nil12_8(0) <=> nil
gen_cons:nil12_8(+(x, 1)) <=> cons(0', gen_cons:nil12_8(x))
gen_c6:c713_8(0) <=> c6
gen_c6:c713_8(+(x, 1)) <=> c7(gen_c6:c713_8(x))


The following defined symbols remain to be analysed:
eq, inf, take, length
----------------------------------------

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
eq(gen_0':s9_8(n1529_8), gen_0':s9_8(n1529_8)) -> true, rt in Omega(0)

Induction Base:
eq(gen_0':s9_8(0), gen_0':s9_8(0)) ->_R^Omega(0)
true

Induction Step:
eq(gen_0':s9_8(+(n1529_8, 1)), gen_0':s9_8(+(n1529_8, 1))) ->_R^Omega(0)
eq(gen_0':s9_8(n1529_8), gen_0':s9_8(n1529_8)) ->_IH
true

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

(26)
Obligation:
Innermost TRS:
Rules:
EQ(0', 0') -> c
EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
EQ(z0, z1) -> c2
INF(z0) -> c3(INF(s(z0)))
TAKE(0', z0) -> c4
TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
LENGTH(nil) -> c6
LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
eq(0', 0') -> true
eq(s(z0), s(z1)) -> eq(z0, z1)
eq(z0, z1) -> false
inf(z0) -> cons(z0, inf(s(z0)))
take(0', z0) -> nil
take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))

Types:
EQ :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c:c1:c2
INF :: 0':s -> c3
c3 :: c3 -> c3
TAKE :: 0':s -> cons:nil -> c4:c5
c4 :: c4:c5
cons :: 0':s -> cons:nil -> cons:nil
c5 :: c4:c5 -> c4:c5
LENGTH :: cons:nil -> c6:c7
nil :: cons:nil
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
eq :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
inf :: 0':s -> cons:nil
take :: 0':s -> cons:nil -> cons:nil
length :: cons:nil -> 0':s
hole_c:c1:c21_8 :: c:c1:c2
hole_0':s2_8 :: 0':s
hole_c33_8 :: c3
hole_c4:c54_8 :: c4:c5
hole_cons:nil5_8 :: cons:nil
hole_c6:c76_8 :: c6:c7
hole_true:false7_8 :: true:false
gen_c:c1:c28_8 :: Nat -> c:c1:c2
gen_0':s9_8 :: Nat -> 0':s
gen_c310_8 :: Nat -> c3
gen_c4:c511_8 :: Nat -> c4:c5
gen_cons:nil12_8 :: Nat -> cons:nil
gen_c6:c713_8 :: Nat -> c6:c7


Lemmas:
EQ(gen_0':s9_8(n15_8), gen_0':s9_8(n15_8)) -> gen_c:c1:c28_8(n15_8), rt in Omega(1 + n15_8)
TAKE(gen_0':s9_8(n577_8), gen_cons:nil12_8(n577_8)) -> gen_c4:c511_8(n577_8), rt in Omega(1 + n577_8)
LENGTH(gen_cons:nil12_8(n1072_8)) -> gen_c6:c713_8(n1072_8), rt in Omega(1 + n1072_8)
eq(gen_0':s9_8(n1529_8), gen_0':s9_8(n1529_8)) -> true, rt in Omega(0)


Generator Equations:
gen_c:c1:c28_8(0) <=> c
gen_c:c1:c28_8(+(x, 1)) <=> c1(gen_c:c1:c28_8(x))
gen_0':s9_8(0) <=> 0'
gen_0':s9_8(+(x, 1)) <=> s(gen_0':s9_8(x))
gen_c310_8(0) <=> hole_c33_8
gen_c310_8(+(x, 1)) <=> c3(gen_c310_8(x))
gen_c4:c511_8(0) <=> c4
gen_c4:c511_8(+(x, 1)) <=> c5(gen_c4:c511_8(x))
gen_cons:nil12_8(0) <=> nil
gen_cons:nil12_8(+(x, 1)) <=> cons(0', gen_cons:nil12_8(x))
gen_c6:c713_8(0) <=> c6
gen_c6:c713_8(+(x, 1)) <=> c7(gen_c6:c713_8(x))


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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
take(gen_0':s9_8(n1993_8), gen_cons:nil12_8(n1993_8)) -> gen_cons:nil12_8(n1993_8), rt in Omega(0)

Induction Base:
take(gen_0':s9_8(0), gen_cons:nil12_8(0)) ->_R^Omega(0)
nil

Induction Step:
take(gen_0':s9_8(+(n1993_8, 1)), gen_cons:nil12_8(+(n1993_8, 1))) ->_R^Omega(0)
cons(0', take(gen_0':s9_8(n1993_8), gen_cons:nil12_8(n1993_8))) ->_IH
cons(0', gen_cons:nil12_8(c1994_8))

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

(28)
Obligation:
Innermost TRS:
Rules:
EQ(0', 0') -> c
EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
EQ(z0, z1) -> c2
INF(z0) -> c3(INF(s(z0)))
TAKE(0', z0) -> c4
TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
LENGTH(nil) -> c6
LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
eq(0', 0') -> true
eq(s(z0), s(z1)) -> eq(z0, z1)
eq(z0, z1) -> false
inf(z0) -> cons(z0, inf(s(z0)))
take(0', z0) -> nil
take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))

Types:
EQ :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c:c1:c2
INF :: 0':s -> c3
c3 :: c3 -> c3
TAKE :: 0':s -> cons:nil -> c4:c5
c4 :: c4:c5
cons :: 0':s -> cons:nil -> cons:nil
c5 :: c4:c5 -> c4:c5
LENGTH :: cons:nil -> c6:c7
nil :: cons:nil
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
eq :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
inf :: 0':s -> cons:nil
take :: 0':s -> cons:nil -> cons:nil
length :: cons:nil -> 0':s
hole_c:c1:c21_8 :: c:c1:c2
hole_0':s2_8 :: 0':s
hole_c33_8 :: c3
hole_c4:c54_8 :: c4:c5
hole_cons:nil5_8 :: cons:nil
hole_c6:c76_8 :: c6:c7
hole_true:false7_8 :: true:false
gen_c:c1:c28_8 :: Nat -> c:c1:c2
gen_0':s9_8 :: Nat -> 0':s
gen_c310_8 :: Nat -> c3
gen_c4:c511_8 :: Nat -> c4:c5
gen_cons:nil12_8 :: Nat -> cons:nil
gen_c6:c713_8 :: Nat -> c6:c7


Lemmas:
EQ(gen_0':s9_8(n15_8), gen_0':s9_8(n15_8)) -> gen_c:c1:c28_8(n15_8), rt in Omega(1 + n15_8)
TAKE(gen_0':s9_8(n577_8), gen_cons:nil12_8(n577_8)) -> gen_c4:c511_8(n577_8), rt in Omega(1 + n577_8)
LENGTH(gen_cons:nil12_8(n1072_8)) -> gen_c6:c713_8(n1072_8), rt in Omega(1 + n1072_8)
eq(gen_0':s9_8(n1529_8), gen_0':s9_8(n1529_8)) -> true, rt in Omega(0)
take(gen_0':s9_8(n1993_8), gen_cons:nil12_8(n1993_8)) -> gen_cons:nil12_8(n1993_8), rt in Omega(0)


Generator Equations:
gen_c:c1:c28_8(0) <=> c
gen_c:c1:c28_8(+(x, 1)) <=> c1(gen_c:c1:c28_8(x))
gen_0':s9_8(0) <=> 0'
gen_0':s9_8(+(x, 1)) <=> s(gen_0':s9_8(x))
gen_c310_8(0) <=> hole_c33_8
gen_c310_8(+(x, 1)) <=> c3(gen_c310_8(x))
gen_c4:c511_8(0) <=> c4
gen_c4:c511_8(+(x, 1)) <=> c5(gen_c4:c511_8(x))
gen_cons:nil12_8(0) <=> nil
gen_cons:nil12_8(+(x, 1)) <=> cons(0', gen_cons:nil12_8(x))
gen_c6:c713_8(0) <=> c6
gen_c6:c713_8(+(x, 1)) <=> c7(gen_c6:c713_8(x))


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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length(gen_cons:nil12_8(n2544_8)) -> gen_0':s9_8(n2544_8), rt in Omega(0)

Induction Base:
length(gen_cons:nil12_8(0)) ->_R^Omega(0)
0'

Induction Step:
length(gen_cons:nil12_8(+(n2544_8, 1))) ->_R^Omega(0)
s(length(gen_cons:nil12_8(n2544_8))) ->_IH
s(gen_0':s9_8(c2545_8))

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

(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:
   eq(0, 0) -> true
   eq(s(z0), s(z1)) -> eq(z0, z1)
   eq(z0, z1) -> false
   inf(z0) -> cons(z0, inf(s(z0)))
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(z1))
Tuples:
   EQ(0, 0) -> c
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   EQ(z0, z1) -> c2
   INF(z0) -> c3(INF(s(z0)))
   TAKE(0, z0) -> c4
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(nil) -> c6
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
S tuples:
   EQ(0, 0) -> c
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   EQ(z0, z1) -> c2
   INF(z0) -> c3(INF(s(z0)))
   TAKE(0, z0) -> c4
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(nil) -> c6
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
K tuples:none
Defined Rule Symbols:   eq_2, inf_1, take_2, length_1

Defined Pair Symbols:   EQ_2, INF_1, TAKE_2, LENGTH_1

Compound Symbols:   c, c1_1, c2, c3_1, c4, c5_1, c6, c7_1


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

(33) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   TAKE(0, z0) -> c4
   EQ(z0, z1) -> c2
   LENGTH(nil) -> c6
   EQ(0, 0) -> c

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eq(0, 0) -> true
   eq(s(z0), s(z1)) -> eq(z0, z1)
   eq(z0, z1) -> false
   inf(z0) -> cons(z0, inf(s(z0)))
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(z1))
Tuples:
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   INF(z0) -> c3(INF(s(z0)))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
S tuples:
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   INF(z0) -> c3(INF(s(z0)))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
K tuples:none
Defined Rule Symbols:   eq_2, inf_1, take_2, length_1

Defined Pair Symbols:   EQ_2, INF_1, TAKE_2, LENGTH_1

Compound Symbols:   c1_1, c3_1, c5_1, c7_1


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

(35) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   eq(0, 0) -> true
   eq(s(z0), s(z1)) -> eq(z0, z1)
   eq(z0, z1) -> false
   inf(z0) -> cons(z0, inf(s(z0)))
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(z1))

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   INF(z0) -> c3(INF(s(z0)))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
S tuples:
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   INF(z0) -> c3(INF(s(z0)))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   EQ_2, INF_1, TAKE_2, LENGTH_1

Compound Symbols:   c1_1, c3_1, c5_1, c7_1


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

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

(38)
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:

   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   INF(z0) -> c3(INF(s(z0)))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))

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

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

(40)
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:

   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   INF(z0) -> c3(INF(s(z0)))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))

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

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

(42)
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:

   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1)) [1]
   INF(z0) -> c3(INF(s(z0))) [1]
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) [1]
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) [1]

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

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

(44)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1)) [1]
   INF(z0) -> c3(INF(s(z0))) [1]
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) [1]
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) [1]

The TRS has the following type information:
EQ :: s -> s -> c1
s :: s -> s
c1 :: c1 -> c1
INF :: s -> c3
c3 :: c3 -> c3
TAKE :: s -> cons -> c5
cons :: a -> cons -> cons
c5 :: c5 -> c5
LENGTH :: cons -> c7
c7 :: c7 -> c7

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

(45) 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:

   EQ_2
   INF_1
   TAKE_2
   LENGTH_1

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const, const1, const2, const3, const4, const5, const6

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

(46)
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:

   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1)) [1]
   INF(z0) -> c3(INF(s(z0))) [1]
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) [1]
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) [1]

The TRS has the following type information:
EQ :: s -> s -> c1
s :: s -> s
c1 :: c1 -> c1
INF :: s -> c3
c3 :: c3 -> c3
TAKE :: s -> cons -> c5
cons :: a -> cons -> cons
c5 :: c5 -> c5
LENGTH :: cons -> c7
c7 :: c7 -> c7
const :: c1
const1 :: s
const2 :: c3
const3 :: c5
const4 :: cons
const5 :: a
const6 :: c7

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

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

(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:

   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1)) [1]
   INF(z0) -> c3(INF(s(z0))) [1]
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) [1]
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) [1]

The TRS has the following type information:
EQ :: s -> s -> c1
s :: s -> s
c1 :: c1 -> c1
INF :: s -> c3
c3 :: c3 -> c3
TAKE :: s -> cons -> c5
cons :: a -> cons -> cons
c5 :: c5 -> c5
LENGTH :: cons -> c7
c7 :: c7 -> c7
const :: c1
const1 :: s
const2 :: c3
const3 :: c5
const4 :: cons
const5 :: a
const6 :: c7

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

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

   const => 0
   const1 => 0
   const2 => 0
   const3 => 0
   const4 => 0
   const5 => 0
   const6 => 0

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

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

   EQ(z, z') -{ 1 }-> 1 + EQ(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   INF(z) -{ 1 }-> 1 + INF(1 + z0) :|: z = z0, z0 >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   TAKE(z, z') -{ 1 }-> 1 + TAKE(z0, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 1 + z0, z0 >= 0, z2 >= 0


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

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

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

   EQ(z, z') -{ 1 }-> 1 + EQ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   INF(z) -{ 1 }-> 1 + INF(1 + z) :|: z >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   TAKE(z, z') -{ 1 }-> 1 + TAKE(z - 1, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0


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

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

   { EQ }
   { TAKE }
   { INF }
   { LENGTH }

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

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

   EQ(z, z') -{ 1 }-> 1 + EQ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   INF(z) -{ 1 }-> 1 + INF(1 + z) :|: z >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   TAKE(z, z') -{ 1 }-> 1 + TAKE(z - 1, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0

Function symbols to be analyzed: {EQ}, {TAKE}, {INF}, {LENGTH}

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

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

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

   EQ(z, z') -{ 1 }-> 1 + EQ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   INF(z) -{ 1 }-> 1 + INF(1 + z) :|: z >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   TAKE(z, z') -{ 1 }-> 1 + TAKE(z - 1, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0

Function symbols to be analyzed: {EQ}, {TAKE}, {INF}, {LENGTH}

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   EQ(z, z') -{ 1 }-> 1 + EQ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   INF(z) -{ 1 }-> 1 + INF(1 + z) :|: z >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   TAKE(z, z') -{ 1 }-> 1 + TAKE(z - 1, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0

Function symbols to be analyzed: {EQ}, {TAKE}, {INF}, {LENGTH}
Previous analysis results are:
  EQ: runtime: ?, size: O(1) [0]

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

(59) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   EQ(z, z') -{ 1 }-> 1 + EQ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   INF(z) -{ 1 }-> 1 + INF(1 + z) :|: z >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   TAKE(z, z') -{ 1 }-> 1 + TAKE(z - 1, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0

Function symbols to be analyzed: {TAKE}, {INF}, {LENGTH}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]

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

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

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

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   INF(z) -{ 1 }-> 1 + INF(1 + z) :|: z >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   TAKE(z, z') -{ 1 }-> 1 + TAKE(z - 1, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0

Function symbols to be analyzed: {TAKE}, {INF}, {LENGTH}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: TAKE
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:

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   INF(z) -{ 1 }-> 1 + INF(1 + z) :|: z >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   TAKE(z, z') -{ 1 }-> 1 + TAKE(z - 1, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0

Function symbols to be analyzed: {TAKE}, {INF}, {LENGTH}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  TAKE: runtime: ?, size: O(1) [0]

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

(65) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   INF(z) -{ 1 }-> 1 + INF(1 + z) :|: z >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   TAKE(z, z') -{ 1 }-> 1 + TAKE(z - 1, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0

Function symbols to be analyzed: {INF}, {LENGTH}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  TAKE: runtime: O(n^1) [z'], size: O(1) [0]

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

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

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

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   INF(z) -{ 1 }-> 1 + INF(1 + z) :|: z >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   TAKE(z, z') -{ 1 + z2 }-> 1 + s' :|: s' >= 0, s' <= 0, z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0

Function symbols to be analyzed: {INF}, {LENGTH}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  TAKE: runtime: O(n^1) [z'], size: O(1) [0]

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

(69) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: INF
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:

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   INF(z) -{ 1 }-> 1 + INF(1 + z) :|: z >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   TAKE(z, z') -{ 1 + z2 }-> 1 + s' :|: s' >= 0, s' <= 0, z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0

Function symbols to be analyzed: {INF}, {LENGTH}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  TAKE: runtime: O(n^1) [z'], size: O(1) [0]
  INF: runtime: ?, size: O(1) [0]

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

(71) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   INF(z) -{ 1 }-> 1 + INF(1 + z) :|: z >= 0
   LENGTH(z) -{ 1 }-> 1 + LENGTH(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   TAKE(z, z') -{ 1 + z2 }-> 1 + s' :|: s' >= 0, s' <= 0, z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0

Function symbols to be analyzed: {INF}, {LENGTH}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  TAKE: runtime: O(n^1) [z'], size: O(1) [0]
  INF: runtime: INF, size: O(1) [0]

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

(73) 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:

   EQ(v0, v1) -> null_EQ [0]
   TAKE(v0, v1) -> null_TAKE [0]
   LENGTH(v0) -> null_LENGTH [0]

And the following fresh constants:    null_EQ, null_TAKE, null_LENGTH, const, const1, const2, const3

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

(74)
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:

   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1)) [1]
   INF(z0) -> c3(INF(s(z0))) [1]
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) [1]
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) [1]
   EQ(v0, v1) -> null_EQ [0]
   TAKE(v0, v1) -> null_TAKE [0]
   LENGTH(v0) -> null_LENGTH [0]

The TRS has the following type information:
EQ :: s -> s -> c1:null_EQ
s :: s -> s
c1 :: c1:null_EQ -> c1:null_EQ
INF :: s -> c3
c3 :: c3 -> c3
TAKE :: s -> cons -> c5:null_TAKE
cons :: a -> cons -> cons
c5 :: c5:null_TAKE -> c5:null_TAKE
LENGTH :: cons -> c7:null_LENGTH
c7 :: c7:null_LENGTH -> c7:null_LENGTH
null_EQ :: c1:null_EQ
null_TAKE :: c5:null_TAKE
null_LENGTH :: c7:null_LENGTH
const :: s
const1 :: c3
const2 :: cons
const3 :: a

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

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

   null_EQ => 0
   null_TAKE => 0
   null_LENGTH => 0
   const => 0
   const1 => 0
   const2 => 0
   const3 => 0

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

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

   EQ(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   EQ(z, z') -{ 1 }-> 1 + EQ(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   INF(z) -{ 1 }-> 1 + INF(1 + z0) :|: z = z0, z0 >= 0
   LENGTH(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   LENGTH(z) -{ 1 }-> 1 + LENGTH(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   TAKE(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   TAKE(z, z') -{ 1 }-> 1 + TAKE(z0, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 1 + z0, z0 >= 0, z2 >= 0

Only complete derivations are relevant for the runtime complexity.

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

(77) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
We considered the (Usable) Rules:none
And the Tuples:
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   INF(z0) -> c3(INF(s(z0)))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(EQ(x_1, x_2)) = x_1
   POL(INF(x_1)) = 0
   POL(LENGTH(x_1)) = x_1
   POL(TAKE(x_1, x_2)) = x_2
   POL(c1(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(s(x_1)) = [1] + x_1

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   INF(z0) -> c3(INF(s(z0)))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
S tuples:
   INF(z0) -> c3(INF(s(z0)))
K tuples:
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
Defined Rule Symbols:none

Defined Pair Symbols:   EQ_2, INF_1, TAKE_2, LENGTH_1

Compound Symbols:   c1_1, c3_1, c5_1, c7_1


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

(79) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace INF(z0) -> c3(INF(s(z0))) by 
   INF(s(x0)) -> c3(INF(s(s(x0))))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
   INF(s(x0)) -> c3(INF(s(s(x0))))
S tuples:
   INF(s(x0)) -> c3(INF(s(s(x0))))
K tuples:
   EQ(s(z0), s(z1)) -> c1(EQ(z0, z1))
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
Defined Rule Symbols:none

Defined Pair Symbols:   EQ_2, TAKE_2, LENGTH_1, INF_1

Compound Symbols:   c1_1, c5_1, c7_1, c3_1


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

(81) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace EQ(s(z0), s(z1)) -> c1(EQ(z0, z1)) by 
   EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1)))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
   INF(s(x0)) -> c3(INF(s(s(x0))))
   EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1)))
S tuples:
   INF(s(x0)) -> c3(INF(s(s(x0))))
K tuples:
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
   EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1)))
Defined Rule Symbols:none

Defined Pair Symbols:   TAKE_2, LENGTH_1, INF_1, EQ_2

Compound Symbols:   c5_1, c7_1, c3_1, c1_1


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

(83) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace INF(s(x0)) -> c3(INF(s(s(x0)))) by 
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
   EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1)))
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
S tuples:
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
K tuples:
   TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2))
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
   EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1)))
Defined Rule Symbols:none

Defined Pair Symbols:   TAKE_2, LENGTH_1, EQ_2, INF_1

Compound Symbols:   c5_1, c7_1, c1_1, c3_1


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

(85) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace TAKE(s(z0), cons(z1, z2)) -> c5(TAKE(z0, z2)) by 
   TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2)))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
   EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1)))
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
   TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2)))
S tuples:
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
K tuples:
   LENGTH(cons(z0, z1)) -> c7(LENGTH(z1))
   EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1)))
   TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2)))
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH_1, EQ_2, INF_1, TAKE_2

Compound Symbols:   c7_1, c1_1, c3_1, c5_1


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

(87) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH(cons(z0, z1)) -> c7(LENGTH(z1)) by 
   LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1)))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1)))
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
   TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2)))
   LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1)))
S tuples:
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
K tuples:
   EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1)))
   TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2)))
   LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1)))
Defined Rule Symbols:none

Defined Pair Symbols:   EQ_2, INF_1, TAKE_2, LENGTH_1

Compound Symbols:   c1_1, c3_1, c5_1, c7_1


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

(89) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace EQ(s(s(y0)), s(s(y1))) -> c1(EQ(s(y0), s(y1))) by 
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c1(EQ(s(s(y0)), s(s(y1))))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
   TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2)))
   LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c1(EQ(s(s(y0)), s(s(y1))))
S tuples:
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
K tuples:
   TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2)))
   LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c1(EQ(s(s(y0)), s(s(y1))))
Defined Rule Symbols:none

Defined Pair Symbols:   INF_1, TAKE_2, LENGTH_1, EQ_2

Compound Symbols:   c3_1, c5_1, c7_1, c1_1


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

(91) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace TAKE(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(TAKE(s(y0), cons(y1, y2))) by 
   TAKE(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c5(TAKE(s(s(y0)), cons(z2, cons(y2, y3))))

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
   LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c1(EQ(s(s(y0)), s(s(y1))))
   TAKE(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c5(TAKE(s(s(y0)), cons(z2, cons(y2, y3))))
S tuples:
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
K tuples:
   LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1)))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c1(EQ(s(s(y0)), s(s(y1))))
   TAKE(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c5(TAKE(s(s(y0)), cons(z2, cons(y2, y3))))
Defined Rule Symbols:none

Defined Pair Symbols:   INF_1, LENGTH_1, EQ_2, TAKE_2

Compound Symbols:   c3_1, c7_1, c1_1, c5_1


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

(93) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LENGTH(cons(z0, cons(y0, y1))) -> c7(LENGTH(cons(y0, y1))) by 
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c7(LENGTH(cons(z1, cons(y1, y2))))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c1(EQ(s(s(y0)), s(s(y1))))
   TAKE(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c5(TAKE(s(s(y0)), cons(z2, cons(y2, y3))))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c7(LENGTH(cons(z1, cons(y1, y2))))
S tuples:
   INF(s(s(x0))) -> c3(INF(s(s(s(x0)))))
K tuples:
   EQ(s(s(s(y0))), s(s(s(y1)))) -> c1(EQ(s(s(y0)), s(s(y1))))
   TAKE(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c5(TAKE(s(s(y0)), cons(z2, cons(y2, y3))))
   LENGTH(cons(z0, cons(z1, cons(y1, y2)))) -> c7(LENGTH(cons(z1, cons(y1, y2))))
Defined Rule Symbols:none

Defined Pair Symbols:   INF_1, EQ_2, TAKE_2, LENGTH_1

Compound Symbols:   c3_1, c1_1, c5_1, c7_1


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

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

(96)
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:

   eq(0, 0) -> true [1]
   eq(s(X), s(Y)) -> eq(X, Y) [1]
   eq(X, Y) -> false [1]
   inf(X) -> cons(X, inf(s(X))) [1]
   take(0, X) -> nil [1]
   take(s(X), cons(Y, L)) -> cons(Y, take(X, L)) [1]
   length(nil) -> 0 [1]
   length(cons(X, L)) -> s(length(L)) [1]

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

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

(98)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   eq(0, 0) -> true [1]
   eq(s(X), s(Y)) -> eq(X, Y) [1]
   eq(X, Y) -> false [1]
   inf(X) -> cons(X, inf(s(X))) [1]
   take(0, X) -> nil [1]
   take(s(X), cons(Y, L)) -> cons(Y, take(X, L)) [1]
   length(nil) -> 0 [1]
   length(cons(X, L)) -> s(length(L)) [1]

The TRS has the following type information:
eq :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
inf :: 0:s -> cons:nil
cons :: 0:s -> cons:nil -> cons:nil
take :: 0:s -> cons:nil -> cons:nil
nil :: cons:nil
length :: cons:nil -> 0:s

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

(99) 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:

   eq_2
   inf_1
   take_2
   length_1

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants: none

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

(100)
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:

   eq(0, 0) -> true [1]
   eq(s(X), s(Y)) -> eq(X, Y) [1]
   eq(X, Y) -> false [1]
   inf(X) -> cons(X, inf(s(X))) [1]
   take(0, X) -> nil [1]
   take(s(X), cons(Y, L)) -> cons(Y, take(X, L)) [1]
   length(nil) -> 0 [1]
   length(cons(X, L)) -> s(length(L)) [1]

The TRS has the following type information:
eq :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
inf :: 0:s -> cons:nil
cons :: 0:s -> cons:nil -> cons:nil
take :: 0:s -> cons:nil -> cons:nil
nil :: cons:nil
length :: cons:nil -> 0:s

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

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

(102)
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:

   eq(0, 0) -> true [1]
   eq(s(X), s(Y)) -> eq(X, Y) [1]
   eq(X, Y) -> false [1]
   inf(X) -> cons(X, inf(s(X))) [1]
   take(0, X) -> nil [1]
   take(s(X), cons(Y, L)) -> cons(Y, take(X, L)) [1]
   length(nil) -> 0 [1]
   length(cons(X, L)) -> s(length(L)) [1]

The TRS has the following type information:
eq :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
inf :: 0:s -> cons:nil
cons :: 0:s -> cons:nil -> cons:nil
take :: 0:s -> cons:nil -> cons:nil
nil :: cons:nil
length :: cons:nil -> 0:s

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

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

   0 => 0
   true => 1
   false => 0
   nil => 0

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

(104)
Obligation:
Complexity RNTS consisting of the following rules:

   eq(z, z') -{ 1 }-> eq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, X >= 0, z = X
   inf(z) -{ 1 }-> 1 + X + inf(1 + X) :|: X >= 0, z = X
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 1 }-> 1 + length(L) :|: z = 1 + X + L, X >= 0, L >= 0
   take(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0
   take(z, z') -{ 1 }-> 1 + Y + take(X, L) :|: z = 1 + X, Y >= 0, X >= 0, L >= 0, z' = 1 + Y + L


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

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

(106)
Obligation:
Complexity RNTS consisting of the following rules:

   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   inf(z) -{ 1 }-> 1 + z + inf(1 + z) :|: z >= 0
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 1 }-> 1 + length(L) :|: z = 1 + X + L, X >= 0, L >= 0
   take(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   take(z, z') -{ 1 }-> 1 + Y + take(z - 1, L) :|: Y >= 0, z - 1 >= 0, L >= 0, z' = 1 + Y + L


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

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

   { inf }
   { take }
   { length }
   { eq }

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

(108)
Obligation:
Complexity RNTS consisting of the following rules:

   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   inf(z) -{ 1 }-> 1 + z + inf(1 + z) :|: z >= 0
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 1 }-> 1 + length(L) :|: z = 1 + X + L, X >= 0, L >= 0
   take(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   take(z, z') -{ 1 }-> 1 + Y + take(z - 1, L) :|: Y >= 0, z - 1 >= 0, L >= 0, z' = 1 + Y + L

Function symbols to be analyzed: {inf}, {take}, {length}, {eq}

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

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

(110)
Obligation:
Complexity RNTS consisting of the following rules:

   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   inf(z) -{ 1 }-> 1 + z + inf(1 + z) :|: z >= 0
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 1 }-> 1 + length(L) :|: z = 1 + X + L, X >= 0, L >= 0
   take(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   take(z, z') -{ 1 }-> 1 + Y + take(z - 1, L) :|: Y >= 0, z - 1 >= 0, L >= 0, z' = 1 + Y + L

Function symbols to be analyzed: {inf}, {take}, {length}, {eq}

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

(111) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(112)
Obligation:
Complexity RNTS consisting of the following rules:

   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   inf(z) -{ 1 }-> 1 + z + inf(1 + z) :|: z >= 0
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 1 }-> 1 + length(L) :|: z = 1 + X + L, X >= 0, L >= 0
   take(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   take(z, z') -{ 1 }-> 1 + Y + take(z - 1, L) :|: Y >= 0, z - 1 >= 0, L >= 0, z' = 1 + Y + L

Function symbols to be analyzed: {inf}, {take}, {length}, {eq}
Previous analysis results are:
  inf: runtime: ?, size: O(1) [0]

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

(113) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(114)
Obligation:
Complexity RNTS consisting of the following rules:

   eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   inf(z) -{ 1 }-> 1 + z + inf(1 + z) :|: z >= 0
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 1 }-> 1 + length(L) :|: z = 1 + X + L, X >= 0, L >= 0
   take(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   take(z, z') -{ 1 }-> 1 + Y + take(z - 1, L) :|: Y >= 0, z - 1 >= 0, L >= 0, z' = 1 + Y + L

Function symbols to be analyzed: {inf}, {take}, {length}, {eq}
Previous analysis results are:
  inf: runtime: INF, size: O(1) [0]

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

(115) 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:

   take(v0, v1) -> null_take [0]
   length(v0) -> null_length [0]

And the following fresh constants:    null_take, null_length

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

(116)
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:

   eq(0, 0) -> true [1]
   eq(s(X), s(Y)) -> eq(X, Y) [1]
   eq(X, Y) -> false [1]
   inf(X) -> cons(X, inf(s(X))) [1]
   take(0, X) -> nil [1]
   take(s(X), cons(Y, L)) -> cons(Y, take(X, L)) [1]
   length(nil) -> 0 [1]
   length(cons(X, L)) -> s(length(L)) [1]
   take(v0, v1) -> null_take [0]
   length(v0) -> null_length [0]

The TRS has the following type information:
eq :: 0:s:null_length -> 0:s:null_length -> true:false
0 :: 0:s:null_length
true :: true:false
s :: 0:s:null_length -> 0:s:null_length
false :: true:false
inf :: 0:s:null_length -> cons:nil:null_take
cons :: 0:s:null_length -> cons:nil:null_take -> cons:nil:null_take
take :: 0:s:null_length -> cons:nil:null_take -> cons:nil:null_take
nil :: cons:nil:null_take
length :: cons:nil:null_take -> 0:s:null_length
null_take :: cons:nil:null_take
null_length :: 0:s:null_length

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

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

   0 => 0
   true => 1
   false => 0
   nil => 0
   null_take => 0
   null_length => 0

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

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

   eq(z, z') -{ 1 }-> eq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0
   eq(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, X >= 0, z = X
   inf(z) -{ 1 }-> 1 + X + inf(1 + X) :|: X >= 0, z = X
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   length(z) -{ 1 }-> 1 + length(L) :|: z = 1 + X + L, X >= 0, L >= 0
   take(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0
   take(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   take(z, z') -{ 1 }-> 1 + Y + take(X, L) :|: z = 1 + X, Y >= 0, X >= 0, L >= 0, z' = 1 + Y + L

Only complete derivations are relevant for the runtime complexity.