WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 64 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms]
    (12) CdtProblem
    (13) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) BOUNDS(1, 1)
(15) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxTRS
    (17) SlicingProof [LOWER BOUND(ID), 0 ms]
    (18) CpxTRS
    (19) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) typed CpxTrs
    (21) OrderProof [LOWER BOUND(ID), 0 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 273 ms]
    (24) BEST
        (25) proven lower bound
            (26) LowerBoundPropagationProof [FINISHED, 0 ms]
            (27) BOUNDS(n^1, INF)
        (28) typed CpxTrs


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

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


The TRS R consists of the following rules:

   a__eq(0, 0) -> true
   a__eq(s(X), s(Y)) -> a__eq(X, Y)
   a__eq(X, Y) -> false
   a__inf(X) -> cons(X, inf(s(X)))
   a__take(0, X) -> nil
   a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
   a__length(nil) -> 0
   a__length(cons(X, L)) -> s(length(L))
   mark(eq(X1, X2)) -> a__eq(X1, X2)
   mark(inf(X)) -> a__inf(mark(X))
   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
   mark(length(X)) -> a__length(mark(X))
   mark(0) -> 0
   mark(true) -> true
   mark(s(X)) -> s(X)
   mark(false) -> false
   mark(cons(X1, X2)) -> cons(X1, X2)
   mark(nil) -> nil
   a__eq(X1, X2) -> eq(X1, X2)
   a__inf(X) -> inf(X)
   a__take(X1, X2) -> take(X1, X2)
   a__length(X) -> length(X)

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

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__eq(0, 0) -> true
   a__eq(s(z0), s(z1)) -> a__eq(z0, z1)
   a__eq(z0, z1) -> false
   a__eq(z0, z1) -> eq(z0, z1)
   a__inf(z0) -> cons(z0, inf(s(z0)))
   a__inf(z0) -> inf(z0)
   a__take(0, z0) -> nil
   a__take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
   a__take(z0, z1) -> take(z0, z1)
   a__length(nil) -> 0
   a__length(cons(z0, z1)) -> s(length(z1))
   a__length(z0) -> length(z0)
   mark(eq(z0, z1)) -> a__eq(z0, z1)
   mark(inf(z0)) -> a__inf(mark(z0))
   mark(take(z0, z1)) -> a__take(mark(z0), mark(z1))
   mark(length(z0)) -> a__length(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(s(z0)) -> s(z0)
   mark(false) -> false
   mark(cons(z0, z1)) -> cons(z0, z1)
   mark(nil) -> nil
Tuples:
   A__EQ(0, 0) -> c
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   A__EQ(z0, z1) -> c2
   A__EQ(z0, z1) -> c3
   A__INF(z0) -> c4
   A__INF(z0) -> c5
   A__TAKE(0, z0) -> c6
   A__TAKE(s(z0), cons(z1, z2)) -> c7
   A__TAKE(z0, z1) -> c8
   A__LENGTH(nil) -> c9
   A__LENGTH(cons(z0, z1)) -> c10
   A__LENGTH(z0) -> c11
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(A__INF(mark(z0)), MARK(z0))
   MARK(take(z0, z1)) -> c14(A__TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
   MARK(length(z0)) -> c15(A__LENGTH(mark(z0)), MARK(z0))
   MARK(0) -> c16
   MARK(true) -> c17
   MARK(s(z0)) -> c18
   MARK(false) -> c19
   MARK(cons(z0, z1)) -> c20
   MARK(nil) -> c21
S tuples:
   A__EQ(0, 0) -> c
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   A__EQ(z0, z1) -> c2
   A__EQ(z0, z1) -> c3
   A__INF(z0) -> c4
   A__INF(z0) -> c5
   A__TAKE(0, z0) -> c6
   A__TAKE(s(z0), cons(z1, z2)) -> c7
   A__TAKE(z0, z1) -> c8
   A__LENGTH(nil) -> c9
   A__LENGTH(cons(z0, z1)) -> c10
   A__LENGTH(z0) -> c11
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(A__INF(mark(z0)), MARK(z0))
   MARK(take(z0, z1)) -> c14(A__TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
   MARK(length(z0)) -> c15(A__LENGTH(mark(z0)), MARK(z0))
   MARK(0) -> c16
   MARK(true) -> c17
   MARK(s(z0)) -> c18
   MARK(false) -> c19
   MARK(cons(z0, z1)) -> c20
   MARK(nil) -> c21
K tuples:none
Defined Rule Symbols:   a__eq_2, a__inf_1, a__take_2, a__length_1, mark_1

Defined Pair Symbols:   A__EQ_2, A__INF_1, A__TAKE_2, A__LENGTH_1, MARK_1

Compound Symbols:   c, c1_1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12_1, c13_2, c14_3, c15_2, c16, c17, c18, c19, c20, c21


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 17 trailing nodes:
   MARK(nil) -> c21
   A__LENGTH(cons(z0, z1)) -> c10
   A__LENGTH(z0) -> c11
   A__EQ(z0, z1) -> c3
   A__TAKE(s(z0), cons(z1, z2)) -> c7
   A__TAKE(0, z0) -> c6
   A__INF(z0) -> c5
   MARK(true) -> c17
   MARK(s(z0)) -> c18
   A__TAKE(z0, z1) -> c8
   A__INF(z0) -> c4
   MARK(false) -> c19
   A__EQ(0, 0) -> c
   MARK(0) -> c16
   MARK(cons(z0, z1)) -> c20
   A__LENGTH(nil) -> c9
   A__EQ(z0, z1) -> c2

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__eq(0, 0) -> true
   a__eq(s(z0), s(z1)) -> a__eq(z0, z1)
   a__eq(z0, z1) -> false
   a__eq(z0, z1) -> eq(z0, z1)
   a__inf(z0) -> cons(z0, inf(s(z0)))
   a__inf(z0) -> inf(z0)
   a__take(0, z0) -> nil
   a__take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
   a__take(z0, z1) -> take(z0, z1)
   a__length(nil) -> 0
   a__length(cons(z0, z1)) -> s(length(z1))
   a__length(z0) -> length(z0)
   mark(eq(z0, z1)) -> a__eq(z0, z1)
   mark(inf(z0)) -> a__inf(mark(z0))
   mark(take(z0, z1)) -> a__take(mark(z0), mark(z1))
   mark(length(z0)) -> a__length(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(s(z0)) -> s(z0)
   mark(false) -> false
   mark(cons(z0, z1)) -> cons(z0, z1)
   mark(nil) -> nil
Tuples:
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(A__INF(mark(z0)), MARK(z0))
   MARK(take(z0, z1)) -> c14(A__TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
   MARK(length(z0)) -> c15(A__LENGTH(mark(z0)), MARK(z0))
S tuples:
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(A__INF(mark(z0)), MARK(z0))
   MARK(take(z0, z1)) -> c14(A__TAKE(mark(z0), mark(z1)), MARK(z0), MARK(z1))
   MARK(length(z0)) -> c15(A__LENGTH(mark(z0)), MARK(z0))
K tuples:none
Defined Rule Symbols:   a__eq_2, a__inf_1, a__take_2, a__length_1, mark_1

Defined Pair Symbols:   A__EQ_2, MARK_1

Compound Symbols:   c1_1, c12_1, c13_2, c14_3, c15_2


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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__eq(0, 0) -> true
   a__eq(s(z0), s(z1)) -> a__eq(z0, z1)
   a__eq(z0, z1) -> false
   a__eq(z0, z1) -> eq(z0, z1)
   a__inf(z0) -> cons(z0, inf(s(z0)))
   a__inf(z0) -> inf(z0)
   a__take(0, z0) -> nil
   a__take(s(z0), cons(z1, z2)) -> cons(z1, take(z0, z2))
   a__take(z0, z1) -> take(z0, z1)
   a__length(nil) -> 0
   a__length(cons(z0, z1)) -> s(length(z1))
   a__length(z0) -> length(z0)
   mark(eq(z0, z1)) -> a__eq(z0, z1)
   mark(inf(z0)) -> a__inf(mark(z0))
   mark(take(z0, z1)) -> a__take(mark(z0), mark(z1))
   mark(length(z0)) -> a__length(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(s(z0)) -> s(z0)
   mark(false) -> false
   mark(cons(z0, z1)) -> cons(z0, z1)
   mark(nil) -> nil
Tuples:
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(MARK(z0))
   MARK(take(z0, z1)) -> c14(MARK(z0), MARK(z1))
   MARK(length(z0)) -> c15(MARK(z0))
S tuples:
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(MARK(z0))
   MARK(take(z0, z1)) -> c14(MARK(z0), MARK(z1))
   MARK(length(z0)) -> c15(MARK(z0))
K tuples:none
Defined Rule Symbols:   a__eq_2, a__inf_1, a__take_2, a__length_1, mark_1

Defined Pair Symbols:   A__EQ_2, MARK_1

Compound Symbols:   c1_1, c12_1, c13_1, c14_2, c15_1


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

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

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(MARK(z0))
   MARK(take(z0, z1)) -> c14(MARK(z0), MARK(z1))
   MARK(length(z0)) -> c15(MARK(z0))
S tuples:
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(MARK(z0))
   MARK(take(z0, z1)) -> c14(MARK(z0), MARK(z1))
   MARK(length(z0)) -> c15(MARK(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   A__EQ_2, MARK_1

Compound Symbols:   c1_1, c12_1, c13_1, c14_2, c15_1


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

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(MARK(z0))
   MARK(length(z0)) -> c15(MARK(z0))
We considered the (Usable) Rules:none
And the Tuples:
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(MARK(z0))
   MARK(take(z0, z1)) -> c14(MARK(z0), MARK(z1))
   MARK(length(z0)) -> c15(MARK(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A__EQ(x_1, x_2)) = [1] + x_1 + x_2
   POL(MARK(x_1)) = [1] + x_1
   POL(c1(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c15(x_1)) = x_1
   POL(eq(x_1, x_2)) = [1] + x_1 + x_2
   POL(inf(x_1)) = [1] + x_1
   POL(length(x_1)) = [1] + x_1
   POL(s(x_1)) = [1] + x_1
   POL(take(x_1, x_2)) = [1] + x_1 + x_2

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(MARK(z0))
   MARK(take(z0, z1)) -> c14(MARK(z0), MARK(z1))
   MARK(length(z0)) -> c15(MARK(z0))
S tuples:
   MARK(take(z0, z1)) -> c14(MARK(z0), MARK(z1))
K tuples:
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(MARK(z0))
   MARK(length(z0)) -> c15(MARK(z0))
Defined Rule Symbols:none

Defined Pair Symbols:   A__EQ_2, MARK_1

Compound Symbols:   c1_1, c12_1, c13_1, c14_2, c15_1


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

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MARK(take(z0, z1)) -> c14(MARK(z0), MARK(z1))
We considered the (Usable) Rules:none
And the Tuples:
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(MARK(z0))
   MARK(take(z0, z1)) -> c14(MARK(z0), MARK(z1))
   MARK(length(z0)) -> c15(MARK(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A__EQ(x_1, x_2)) = [2]x_1 + [2]x_2
   POL(MARK(x_1)) = [3] + [3]x_1
   POL(c1(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c15(x_1)) = x_1
   POL(eq(x_1, x_2)) = [3] + x_1 + x_2
   POL(inf(x_1)) = x_1
   POL(length(x_1)) = x_1
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = [3] + x_1 + x_2

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(MARK(z0))
   MARK(take(z0, z1)) -> c14(MARK(z0), MARK(z1))
   MARK(length(z0)) -> c15(MARK(z0))
S tuples:none
K tuples:
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(MARK(z0))
   MARK(length(z0)) -> c15(MARK(z0))
   MARK(take(z0, z1)) -> c14(MARK(z0), MARK(z1))
Defined Rule Symbols:none

Defined Pair Symbols:   A__EQ_2, MARK_1

Compound Symbols:   c1_1, c12_1, c13_1, c14_2, c15_1


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

(13) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
----------------------------------------

(14)
BOUNDS(1, 1)

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

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

(16)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__eq(0', 0') -> true
   a__eq(s(X), s(Y)) -> a__eq(X, Y)
   a__eq(X, Y) -> false
   a__inf(X) -> cons(X, inf(s(X)))
   a__take(0', X) -> nil
   a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
   a__length(nil) -> 0'
   a__length(cons(X, L)) -> s(length(L))
   mark(eq(X1, X2)) -> a__eq(X1, X2)
   mark(inf(X)) -> a__inf(mark(X))
   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
   mark(length(X)) -> a__length(mark(X))
   mark(0') -> 0'
   mark(true) -> true
   mark(s(X)) -> s(X)
   mark(false) -> false
   mark(cons(X1, X2)) -> cons(X1, X2)
   mark(nil) -> nil
   a__eq(X1, X2) -> eq(X1, X2)
   a__inf(X) -> inf(X)
   a__take(X1, X2) -> take(X1, X2)
   a__length(X) -> length(X)

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

(17) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
cons/0

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

(18)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__eq(0', 0') -> true
   a__eq(s(X), s(Y)) -> a__eq(X, Y)
   a__eq(X, Y) -> false
   a__inf(X) -> cons(inf(s(X)))
   a__take(0', X) -> nil
   a__take(s(X), cons(L)) -> cons(take(X, L))
   a__length(nil) -> 0'
   a__length(cons(L)) -> s(length(L))
   mark(eq(X1, X2)) -> a__eq(X1, X2)
   mark(inf(X)) -> a__inf(mark(X))
   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
   mark(length(X)) -> a__length(mark(X))
   mark(0') -> 0'
   mark(true) -> true
   mark(s(X)) -> s(X)
   mark(false) -> false
   mark(cons(X2)) -> cons(X2)
   mark(nil) -> nil
   a__eq(X1, X2) -> eq(X1, X2)
   a__inf(X) -> inf(X)
   a__take(X1, X2) -> take(X1, X2)
   a__length(X) -> length(X)

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

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

(20)
Obligation:
Innermost TRS:
Rules:
a__eq(0', 0') -> true
a__eq(s(X), s(Y)) -> a__eq(X, Y)
a__eq(X, Y) -> false
a__inf(X) -> cons(inf(s(X)))
a__take(0', X) -> nil
a__take(s(X), cons(L)) -> cons(take(X, L))
a__length(nil) -> 0'
a__length(cons(L)) -> s(length(L))
mark(eq(X1, X2)) -> a__eq(X1, X2)
mark(inf(X)) -> a__inf(mark(X))
mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
mark(length(X)) -> a__length(mark(X))
mark(0') -> 0'
mark(true) -> true
mark(s(X)) -> s(X)
mark(false) -> false
mark(cons(X2)) -> cons(X2)
mark(nil) -> nil
a__eq(X1, X2) -> eq(X1, X2)
a__inf(X) -> inf(X)
a__take(X1, X2) -> take(X1, X2)
a__length(X) -> length(X)

Types:
a__eq :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
0' :: 0':true:s:false:inf:cons:nil:take:length:eq
true :: 0':true:s:false:inf:cons:nil:take:length:eq
s :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
false :: 0':true:s:false:inf:cons:nil:take:length:eq
a__inf :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
cons :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
inf :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
a__take :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
nil :: 0':true:s:false:inf:cons:nil:take:length:eq
take :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
a__length :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
length :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
mark :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
eq :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
hole_0':true:s:false:inf:cons:nil:take:length:eq1_0 :: 0':true:s:false:inf:cons:nil:take:length:eq
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0 :: Nat -> 0':true:s:false:inf:cons:nil:take:length:eq

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

(21) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
a__eq, mark

They will be analysed ascendingly in the following order:
a__eq < mark

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

(22)
Obligation:
Innermost TRS:
Rules:
a__eq(0', 0') -> true
a__eq(s(X), s(Y)) -> a__eq(X, Y)
a__eq(X, Y) -> false
a__inf(X) -> cons(inf(s(X)))
a__take(0', X) -> nil
a__take(s(X), cons(L)) -> cons(take(X, L))
a__length(nil) -> 0'
a__length(cons(L)) -> s(length(L))
mark(eq(X1, X2)) -> a__eq(X1, X2)
mark(inf(X)) -> a__inf(mark(X))
mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
mark(length(X)) -> a__length(mark(X))
mark(0') -> 0'
mark(true) -> true
mark(s(X)) -> s(X)
mark(false) -> false
mark(cons(X2)) -> cons(X2)
mark(nil) -> nil
a__eq(X1, X2) -> eq(X1, X2)
a__inf(X) -> inf(X)
a__take(X1, X2) -> take(X1, X2)
a__length(X) -> length(X)

Types:
a__eq :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
0' :: 0':true:s:false:inf:cons:nil:take:length:eq
true :: 0':true:s:false:inf:cons:nil:take:length:eq
s :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
false :: 0':true:s:false:inf:cons:nil:take:length:eq
a__inf :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
cons :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
inf :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
a__take :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
nil :: 0':true:s:false:inf:cons:nil:take:length:eq
take :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
a__length :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
length :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
mark :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
eq :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
hole_0':true:s:false:inf:cons:nil:take:length:eq1_0 :: 0':true:s:false:inf:cons:nil:take:length:eq
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0 :: Nat -> 0':true:s:false:inf:cons:nil:take:length:eq


Generator Equations:
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(0) <=> 0'
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(+(x, 1)) <=> s(gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(x))


The following defined symbols remain to be analysed:
a__eq, mark

They will be analysed ascendingly in the following order:
a__eq < mark

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
a__eq(gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(n4_0), gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(n4_0)) -> true, rt in Omega(1 + n4_0)

Induction Base:
a__eq(gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(0), gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(0)) ->_R^Omega(1)
true

Induction Step:
a__eq(gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(+(n4_0, 1)), gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(+(n4_0, 1))) ->_R^Omega(1)
a__eq(gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(n4_0), gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(n4_0)) ->_IH
true

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

(24)
Complex Obligation (BEST)

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

(25)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
a__eq(0', 0') -> true
a__eq(s(X), s(Y)) -> a__eq(X, Y)
a__eq(X, Y) -> false
a__inf(X) -> cons(inf(s(X)))
a__take(0', X) -> nil
a__take(s(X), cons(L)) -> cons(take(X, L))
a__length(nil) -> 0'
a__length(cons(L)) -> s(length(L))
mark(eq(X1, X2)) -> a__eq(X1, X2)
mark(inf(X)) -> a__inf(mark(X))
mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
mark(length(X)) -> a__length(mark(X))
mark(0') -> 0'
mark(true) -> true
mark(s(X)) -> s(X)
mark(false) -> false
mark(cons(X2)) -> cons(X2)
mark(nil) -> nil
a__eq(X1, X2) -> eq(X1, X2)
a__inf(X) -> inf(X)
a__take(X1, X2) -> take(X1, X2)
a__length(X) -> length(X)

Types:
a__eq :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
0' :: 0':true:s:false:inf:cons:nil:take:length:eq
true :: 0':true:s:false:inf:cons:nil:take:length:eq
s :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
false :: 0':true:s:false:inf:cons:nil:take:length:eq
a__inf :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
cons :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
inf :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
a__take :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
nil :: 0':true:s:false:inf:cons:nil:take:length:eq
take :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
a__length :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
length :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
mark :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
eq :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
hole_0':true:s:false:inf:cons:nil:take:length:eq1_0 :: 0':true:s:false:inf:cons:nil:take:length:eq
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0 :: Nat -> 0':true:s:false:inf:cons:nil:take:length:eq


Generator Equations:
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(0) <=> 0'
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(+(x, 1)) <=> s(gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(x))


The following defined symbols remain to be analysed:
a__eq, mark

They will be analysed ascendingly in the following order:
a__eq < mark

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

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

(27)
BOUNDS(n^1, INF)

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

(28)
Obligation:
Innermost TRS:
Rules:
a__eq(0', 0') -> true
a__eq(s(X), s(Y)) -> a__eq(X, Y)
a__eq(X, Y) -> false
a__inf(X) -> cons(inf(s(X)))
a__take(0', X) -> nil
a__take(s(X), cons(L)) -> cons(take(X, L))
a__length(nil) -> 0'
a__length(cons(L)) -> s(length(L))
mark(eq(X1, X2)) -> a__eq(X1, X2)
mark(inf(X)) -> a__inf(mark(X))
mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
mark(length(X)) -> a__length(mark(X))
mark(0') -> 0'
mark(true) -> true
mark(s(X)) -> s(X)
mark(false) -> false
mark(cons(X2)) -> cons(X2)
mark(nil) -> nil
a__eq(X1, X2) -> eq(X1, X2)
a__inf(X) -> inf(X)
a__take(X1, X2) -> take(X1, X2)
a__length(X) -> length(X)

Types:
a__eq :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
0' :: 0':true:s:false:inf:cons:nil:take:length:eq
true :: 0':true:s:false:inf:cons:nil:take:length:eq
s :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
false :: 0':true:s:false:inf:cons:nil:take:length:eq
a__inf :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
cons :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
inf :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
a__take :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
nil :: 0':true:s:false:inf:cons:nil:take:length:eq
take :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
a__length :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
length :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
mark :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
eq :: 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq -> 0':true:s:false:inf:cons:nil:take:length:eq
hole_0':true:s:false:inf:cons:nil:take:length:eq1_0 :: 0':true:s:false:inf:cons:nil:take:length:eq
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0 :: Nat -> 0':true:s:false:inf:cons:nil:take:length:eq


Lemmas:
a__eq(gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(n4_0), gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(n4_0)) -> true, rt in Omega(1 + n4_0)


Generator Equations:
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(0) <=> 0'
gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(+(x, 1)) <=> s(gen_0':true:s:false:inf:cons:nil:take:length:eq2_0(x))


The following defined symbols remain to be analysed:
mark