/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (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), 1 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 54 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms]
    (12) CdtProblem
    (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 5 ms]
    (14) CdtProblem
    (15) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) BOUNDS(1, 1)
(17) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(18) TRS for Loop Detection
    (19) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) TRS for Loop Detection


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

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

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

(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(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)) = x_1 + x_2
   POL(MARK(x_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)) = x_1 + x_2
   POL(inf(x_1)) = [3] + x_1
   POL(length(x_1)) = [3] + x_1
   POL(s(x_1)) = [3] + x_1
   POL(take(x_1, x_2)) = 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(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   MARK(take(z0, z1)) -> c14(MARK(z0), MARK(z1))
K tuples:
   A__EQ(s(z0), s(z1)) -> c1(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(eq(z0, z1)) -> c12(A__EQ(z0, 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)) = [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)) = x_1
   POL(length(x_1)) = x_1
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = [1] + 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:
   MARK(take(z0, z1)) -> c14(MARK(z0), MARK(z1))
K tuples:
   A__EQ(s(z0), s(z1)) -> c1(A__EQ(z0, z1))
   MARK(inf(z0)) -> c13(MARK(z0))
   MARK(length(z0)) -> c15(MARK(z0))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, 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) 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)) = [1] + x_1 + x_2
   POL(MARK(x_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)) = x_1
   POL(length(x_1)) = x_1
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = [1] + x_1 + x_2

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

(14)
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(inf(z0)) -> c13(MARK(z0))
   MARK(length(z0)) -> c15(MARK(z0))
   MARK(eq(z0, z1)) -> c12(A__EQ(z0, z1))
   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


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

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

(16)
BOUNDS(1, 1)

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

(17) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(18)
Obligation:
Analyzing the following TRS for decreasing loops:

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
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(19) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

mark(length(X)) ->^+ a__length(mark(X))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [X / length(X)].

The result substitution is [ ].




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(20)
Complex Obligation (BEST)

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(21)
Obligation:
Proved the lower bound n^1 for the following 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
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(22) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(23)
BOUNDS(n^1, INF)

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(24)
Obligation:
Analyzing the following TRS for decreasing loops:

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