WORST_CASE(Omega(n^1), O(n^3))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^3).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 74 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
    (7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (8) CdtProblem
    (9) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (12) CdtProblem
    (13) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CdtProblem
    (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 89 ms]
    (16) CdtProblem
    (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 110 ms]
    (18) CdtProblem
    (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 12 ms]
    (20) CdtProblem
    (21) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CdtProblem
    (23) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CdtProblem
    (25) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CdtProblem
    (27) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 62 ms]
    (28) CdtProblem
    (29) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) BOUNDS(1, 1)
(31) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(32) TRS for Loop Detection
    (33) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (34) BEST
        (35) proven lower bound
            (36) LowerBoundPropagationProof [FINISHED, 0 ms]
            (37) BOUNDS(n^1, INF)
        (38) TRS for Loop Detection


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   a__f(f(a)) -> a__f(g(f(a)))
   mark(f(X)) -> a__f(mark(X))
   mark(a) -> a
   mark(g(X)) -> g(X)
   a__f(X) -> f(X)

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

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(a) -> a
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))


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

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   a__f(f(a)) -> a__f(g(f(a)))
   mark(f(X)) -> a__f(mark(X))
   mark(a) -> a
   mark(g(X)) -> g(X)
   a__f(X) -> f(X)

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

   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(a) -> a
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))

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

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   a__f(f(a)) -> a__f(g(f(a)))
   mark(f(X)) -> a__f(mark(X))
   mark(a) -> a
   mark(g(X)) -> g(X)
   a__f(X) -> f(X)

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

   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(a) -> a
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(f(z0)) -> f(encArg(z0))
   encArg(a) -> a
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   encode_a__f(z0) -> a__f(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_a -> a
   encode_g(z0) -> g(encArg(z0))
   encode_mark(z0) -> mark(encArg(z0))
   a__f(f(a)) -> a__f(g(f(a)))
   a__f(z0) -> f(z0)
   mark(f(z0)) -> a__f(mark(z0))
   mark(a) -> a
   mark(g(z0)) -> g(z0)
Tuples:
   ENCARG(f(z0)) -> c(ENCARG(z0))
   ENCARG(a) -> c1
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c3(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c4(MARK(encArg(z0)), ENCARG(z0))
   ENCODE_A__F(z0) -> c5(A__F(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0) -> c6(ENCARG(z0))
   ENCODE_A -> c7
   ENCODE_G(z0) -> c8(ENCARG(z0))
   ENCODE_MARK(z0) -> c9(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(a)) -> c10(A__F(g(f(a))))
   A__F(z0) -> c11
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   MARK(a) -> c13
   MARK(g(z0)) -> c14
S tuples:
   A__F(f(a)) -> c10(A__F(g(f(a))))
   A__F(z0) -> c11
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   MARK(a) -> c13
   MARK(g(z0)) -> c14
K tuples:none
Defined Rule Symbols:   a__f_1, mark_1, encArg_1, encode_a__f_1, encode_f_1, encode_a, encode_g_1, encode_mark_1

Defined Pair Symbols:   ENCARG_1, ENCODE_A__F_1, ENCODE_F_1, ENCODE_A, ENCODE_G_1, ENCODE_MARK_1, A__F_1, MARK_1

Compound Symbols:   c_1, c1, c2_1, c3_2, c4_2, c5_2, c6_1, c7, c8_1, c9_2, c10_1, c11, c12_2, c13, c14


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_F(z0) -> c6(ENCARG(z0))
   ENCODE_G(z0) -> c8(ENCARG(z0))
Removed 2 trailing nodes:
   ENCODE_A -> c7
   ENCARG(a) -> c1

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(f(z0)) -> f(encArg(z0))
   encArg(a) -> a
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   encode_a__f(z0) -> a__f(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_a -> a
   encode_g(z0) -> g(encArg(z0))
   encode_mark(z0) -> mark(encArg(z0))
   a__f(f(a)) -> a__f(g(f(a)))
   a__f(z0) -> f(z0)
   mark(f(z0)) -> a__f(mark(z0))
   mark(a) -> a
   mark(g(z0)) -> g(z0)
Tuples:
   ENCARG(f(z0)) -> c(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c3(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c4(MARK(encArg(z0)), ENCARG(z0))
   ENCODE_A__F(z0) -> c5(A__F(encArg(z0)), ENCARG(z0))
   ENCODE_MARK(z0) -> c9(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(a)) -> c10(A__F(g(f(a))))
   A__F(z0) -> c11
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   MARK(a) -> c13
   MARK(g(z0)) -> c14
S tuples:
   A__F(f(a)) -> c10(A__F(g(f(a))))
   A__F(z0) -> c11
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   MARK(a) -> c13
   MARK(g(z0)) -> c14
K tuples:none
Defined Rule Symbols:   a__f_1, mark_1, encArg_1, encode_a__f_1, encode_f_1, encode_a, encode_g_1, encode_mark_1

Defined Pair Symbols:   ENCARG_1, ENCODE_A__F_1, ENCODE_MARK_1, A__F_1, MARK_1

Compound Symbols:   c_1, c2_1, c3_2, c4_2, c5_2, c9_2, c10_1, c11, c12_2, c13, c14


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

(9) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(f(z0)) -> f(encArg(z0))
   encArg(a) -> a
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   encode_a__f(z0) -> a__f(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_a -> a
   encode_g(z0) -> g(encArg(z0))
   encode_mark(z0) -> mark(encArg(z0))
   a__f(f(a)) -> a__f(g(f(a)))
   a__f(z0) -> f(z0)
   mark(f(z0)) -> a__f(mark(z0))
   mark(a) -> a
   mark(g(z0)) -> g(z0)
Tuples:
   ENCARG(f(z0)) -> c(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c3(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c4(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(a)) -> c10(A__F(g(f(a))))
   A__F(z0) -> c11
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   MARK(a) -> c13
   MARK(g(z0)) -> c14
   ENCODE_A__F(z0) -> c1(A__F(encArg(z0)))
   ENCODE_A__F(z0) -> c1(ENCARG(z0))
   ENCODE_MARK(z0) -> c1(MARK(encArg(z0)))
   ENCODE_MARK(z0) -> c1(ENCARG(z0))
S tuples:
   A__F(f(a)) -> c10(A__F(g(f(a))))
   A__F(z0) -> c11
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   MARK(a) -> c13
   MARK(g(z0)) -> c14
K tuples:none
Defined Rule Symbols:   a__f_1, mark_1, encArg_1, encode_a__f_1, encode_f_1, encode_a, encode_g_1, encode_mark_1

Defined Pair Symbols:   ENCARG_1, A__F_1, MARK_1, ENCODE_A__F_1, ENCODE_MARK_1

Compound Symbols:   c_1, c2_1, c3_2, c4_2, c10_1, c11, c12_2, c13, c14, c1_1


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

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_A__F(z0) -> c1(ENCARG(z0))
   ENCODE_MARK(z0) -> c1(ENCARG(z0))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(f(z0)) -> f(encArg(z0))
   encArg(a) -> a
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   encode_a__f(z0) -> a__f(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_a -> a
   encode_g(z0) -> g(encArg(z0))
   encode_mark(z0) -> mark(encArg(z0))
   a__f(f(a)) -> a__f(g(f(a)))
   a__f(z0) -> f(z0)
   mark(f(z0)) -> a__f(mark(z0))
   mark(a) -> a
   mark(g(z0)) -> g(z0)
Tuples:
   ENCARG(f(z0)) -> c(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c3(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c4(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(a)) -> c10(A__F(g(f(a))))
   A__F(z0) -> c11
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   MARK(a) -> c13
   MARK(g(z0)) -> c14
   ENCODE_A__F(z0) -> c1(A__F(encArg(z0)))
   ENCODE_MARK(z0) -> c1(MARK(encArg(z0)))
S tuples:
   A__F(f(a)) -> c10(A__F(g(f(a))))
   A__F(z0) -> c11
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   MARK(a) -> c13
   MARK(g(z0)) -> c14
K tuples:none
Defined Rule Symbols:   a__f_1, mark_1, encArg_1, encode_a__f_1, encode_f_1, encode_a, encode_g_1, encode_mark_1

Defined Pair Symbols:   ENCARG_1, A__F_1, MARK_1, ENCODE_A__F_1, ENCODE_MARK_1

Compound Symbols:   c_1, c2_1, c3_2, c4_2, c10_1, c11, c12_2, c13, c14, c1_1


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

(13) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_a__f(z0) -> a__f(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_a -> a
   encode_g(z0) -> g(encArg(z0))
   encode_mark(z0) -> mark(encArg(z0))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(f(z0)) -> f(encArg(z0))
   encArg(a) -> a
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__f(f(a)) -> a__f(g(f(a)))
   a__f(z0) -> f(z0)
   mark(f(z0)) -> a__f(mark(z0))
   mark(a) -> a
   mark(g(z0)) -> g(z0)
Tuples:
   ENCARG(f(z0)) -> c(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c3(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c4(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(a)) -> c10(A__F(g(f(a))))
   A__F(z0) -> c11
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   MARK(a) -> c13
   MARK(g(z0)) -> c14
   ENCODE_A__F(z0) -> c1(A__F(encArg(z0)))
   ENCODE_MARK(z0) -> c1(MARK(encArg(z0)))
S tuples:
   A__F(f(a)) -> c10(A__F(g(f(a))))
   A__F(z0) -> c11
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   MARK(a) -> c13
   MARK(g(z0)) -> c14
K tuples:none
Defined Rule Symbols:   encArg_1, a__f_1, mark_1

Defined Pair Symbols:   ENCARG_1, A__F_1, MARK_1, ENCODE_A__F_1, ENCODE_MARK_1

Compound Symbols:   c_1, c2_1, c3_2, c4_2, c10_1, c11, c12_2, c13, c14, c1_1


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

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MARK(a) -> c13
   MARK(g(z0)) -> c14
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(f(z0)) -> c(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c3(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c4(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(a)) -> c10(A__F(g(f(a))))
   A__F(z0) -> c11
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   MARK(a) -> c13
   MARK(g(z0)) -> c14
   ENCODE_A__F(z0) -> c1(A__F(encArg(z0)))
   ENCODE_MARK(z0) -> c1(MARK(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A__F(x_1)) = 0
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_A__F(x_1)) = [1]
   POL(ENCODE_MARK(x_1)) = [1] + x_1
   POL(MARK(x_1)) = [1]
   POL(a) = 0
   POL(a__f(x_1)) = [3]
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11) = 0
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13) = 0
   POL(c14) = 0
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(cons_a__f(x_1)) = x_1
   POL(cons_mark(x_1)) = [1] + x_1
   POL(encArg(x_1)) = 0
   POL(f(x_1)) = x_1
   POL(g(x_1)) = x_1
   POL(mark(x_1)) = 0

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(f(z0)) -> f(encArg(z0))
   encArg(a) -> a
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__f(f(a)) -> a__f(g(f(a)))
   a__f(z0) -> f(z0)
   mark(f(z0)) -> a__f(mark(z0))
   mark(a) -> a
   mark(g(z0)) -> g(z0)
Tuples:
   ENCARG(f(z0)) -> c(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c3(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c4(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(a)) -> c10(A__F(g(f(a))))
   A__F(z0) -> c11
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   MARK(a) -> c13
   MARK(g(z0)) -> c14
   ENCODE_A__F(z0) -> c1(A__F(encArg(z0)))
   ENCODE_MARK(z0) -> c1(MARK(encArg(z0)))
S tuples:
   A__F(f(a)) -> c10(A__F(g(f(a))))
   A__F(z0) -> c11
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
K tuples:
   MARK(a) -> c13
   MARK(g(z0)) -> c14
Defined Rule Symbols:   encArg_1, a__f_1, mark_1

Defined Pair Symbols:   ENCARG_1, A__F_1, MARK_1, ENCODE_A__F_1, ENCODE_MARK_1

Compound Symbols:   c_1, c2_1, c3_2, c4_2, c10_1, c11, c12_2, c13, c14, c1_1


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

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
We considered the (Usable) Rules:
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   mark(a) -> a
   a__f(f(a)) -> a__f(g(f(a)))
   a__f(z0) -> f(z0)
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   mark(f(z0)) -> a__f(mark(z0))
   encArg(a) -> a
   encArg(g(z0)) -> g(encArg(z0))
   mark(g(z0)) -> g(z0)
   encArg(f(z0)) -> f(encArg(z0))
And the Tuples:
   ENCARG(f(z0)) -> c(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c3(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c4(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(a)) -> c10(A__F(g(f(a))))
   A__F(z0) -> c11
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   MARK(a) -> c13
   MARK(g(z0)) -> c14
   ENCODE_A__F(z0) -> c1(A__F(encArg(z0)))
   ENCODE_MARK(z0) -> c1(MARK(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A__F(x_1)) = 0
   POL(ENCARG(x_1)) = x_1^3
   POL(ENCODE_A__F(x_1)) = 0
   POL(ENCODE_MARK(x_1)) = [1] + x_1
   POL(MARK(x_1)) = [1] + x_1
   POL(a) = 0
   POL(a__f(x_1)) = [1] + x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11) = 0
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13) = 0
   POL(c14) = 0
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(cons_a__f(x_1)) = [1] + x_1
   POL(cons_mark(x_1)) = [1] + x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1)) = [1] + x_1
   POL(g(x_1)) = x_1
   POL(mark(x_1)) = x_1

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(f(z0)) -> f(encArg(z0))
   encArg(a) -> a
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__f(f(a)) -> a__f(g(f(a)))
   a__f(z0) -> f(z0)
   mark(f(z0)) -> a__f(mark(z0))
   mark(a) -> a
   mark(g(z0)) -> g(z0)
Tuples:
   ENCARG(f(z0)) -> c(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c3(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c4(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(a)) -> c10(A__F(g(f(a))))
   A__F(z0) -> c11
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   MARK(a) -> c13
   MARK(g(z0)) -> c14
   ENCODE_A__F(z0) -> c1(A__F(encArg(z0)))
   ENCODE_MARK(z0) -> c1(MARK(encArg(z0)))
S tuples:
   A__F(f(a)) -> c10(A__F(g(f(a))))
   A__F(z0) -> c11
K tuples:
   MARK(a) -> c13
   MARK(g(z0)) -> c14
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
Defined Rule Symbols:   encArg_1, a__f_1, mark_1

Defined Pair Symbols:   ENCARG_1, A__F_1, MARK_1, ENCODE_A__F_1, ENCODE_MARK_1

Compound Symbols:   c_1, c2_1, c3_2, c4_2, c10_1, c11, c12_2, c13, c14, c1_1


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

(19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   A__F(z0) -> c11
We considered the (Usable) Rules:
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   mark(a) -> a
   a__f(f(a)) -> a__f(g(f(a)))
   a__f(z0) -> f(z0)
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   mark(f(z0)) -> a__f(mark(z0))
   encArg(a) -> a
   encArg(g(z0)) -> g(encArg(z0))
   mark(g(z0)) -> g(z0)
   encArg(f(z0)) -> f(encArg(z0))
And the Tuples:
   ENCARG(f(z0)) -> c(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c3(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c4(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(a)) -> c10(A__F(g(f(a))))
   A__F(z0) -> c11
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   MARK(a) -> c13
   MARK(g(z0)) -> c14
   ENCODE_A__F(z0) -> c1(A__F(encArg(z0)))
   ENCODE_MARK(z0) -> c1(MARK(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A__F(x_1)) = [1]
   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_A__F(x_1)) = [1] + [2]x_1^2
   POL(ENCODE_MARK(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(MARK(x_1)) = [2]x_1
   POL(a) = 0
   POL(a__f(x_1)) = [2] + x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11) = 0
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13) = 0
   POL(c14) = 0
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(cons_a__f(x_1)) = [2] + x_1
   POL(cons_mark(x_1)) = [1] + x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1)) = [2] + x_1
   POL(g(x_1)) = x_1
   POL(mark(x_1)) = x_1

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(f(z0)) -> f(encArg(z0))
   encArg(a) -> a
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__f(f(a)) -> a__f(g(f(a)))
   a__f(z0) -> f(z0)
   mark(f(z0)) -> a__f(mark(z0))
   mark(a) -> a
   mark(g(z0)) -> g(z0)
Tuples:
   ENCARG(f(z0)) -> c(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c3(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c4(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(a)) -> c10(A__F(g(f(a))))
   A__F(z0) -> c11
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   MARK(a) -> c13
   MARK(g(z0)) -> c14
   ENCODE_A__F(z0) -> c1(A__F(encArg(z0)))
   ENCODE_MARK(z0) -> c1(MARK(encArg(z0)))
S tuples:
   A__F(f(a)) -> c10(A__F(g(f(a))))
K tuples:
   MARK(a) -> c13
   MARK(g(z0)) -> c14
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   A__F(z0) -> c11
Defined Rule Symbols:   encArg_1, a__f_1, mark_1

Defined Pair Symbols:   ENCARG_1, A__F_1, MARK_1, ENCODE_A__F_1, ENCODE_MARK_1

Compound Symbols:   c_1, c2_1, c3_2, c4_2, c10_1, c11, c12_2, c13, c14, c1_1


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

(21) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_mark(z0)) -> c4(MARK(encArg(z0)), ENCARG(z0)) by 
   ENCARG(cons_mark(f(z0))) -> c4(MARK(f(encArg(z0))), ENCARG(f(z0)))
   ENCARG(cons_mark(a)) -> c4(MARK(a), ENCARG(a))
   ENCARG(cons_mark(g(z0))) -> c4(MARK(g(encArg(z0))), ENCARG(g(z0)))
   ENCARG(cons_mark(cons_a__f(z0))) -> c4(MARK(a__f(encArg(z0))), ENCARG(cons_a__f(z0)))
   ENCARG(cons_mark(cons_mark(z0))) -> c4(MARK(mark(encArg(z0))), ENCARG(cons_mark(z0)))

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(f(z0)) -> f(encArg(z0))
   encArg(a) -> a
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__f(f(a)) -> a__f(g(f(a)))
   a__f(z0) -> f(z0)
   mark(f(z0)) -> a__f(mark(z0))
   mark(a) -> a
   mark(g(z0)) -> g(z0)
Tuples:
   ENCARG(f(z0)) -> c(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c3(A__F(encArg(z0)), ENCARG(z0))
   A__F(f(a)) -> c10(A__F(g(f(a))))
   A__F(z0) -> c11
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   MARK(a) -> c13
   MARK(g(z0)) -> c14
   ENCODE_A__F(z0) -> c1(A__F(encArg(z0)))
   ENCODE_MARK(z0) -> c1(MARK(encArg(z0)))
   ENCARG(cons_mark(f(z0))) -> c4(MARK(f(encArg(z0))), ENCARG(f(z0)))
   ENCARG(cons_mark(a)) -> c4(MARK(a), ENCARG(a))
   ENCARG(cons_mark(g(z0))) -> c4(MARK(g(encArg(z0))), ENCARG(g(z0)))
   ENCARG(cons_mark(cons_a__f(z0))) -> c4(MARK(a__f(encArg(z0))), ENCARG(cons_a__f(z0)))
   ENCARG(cons_mark(cons_mark(z0))) -> c4(MARK(mark(encArg(z0))), ENCARG(cons_mark(z0)))
S tuples:
   A__F(f(a)) -> c10(A__F(g(f(a))))
K tuples:
   MARK(a) -> c13
   MARK(g(z0)) -> c14
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   A__F(z0) -> c11
Defined Rule Symbols:   encArg_1, a__f_1, mark_1

Defined Pair Symbols:   ENCARG_1, A__F_1, MARK_1, ENCODE_A__F_1, ENCODE_MARK_1

Compound Symbols:   c_1, c2_1, c3_2, c10_1, c11, c12_2, c13, c14, c1_1, c4_2


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

(23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   MARK(g(z0)) -> c14
   ENCARG(cons_mark(a)) -> c4(MARK(a), ENCARG(a))
   MARK(a) -> c13
   A__F(z0) -> c11

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(f(z0)) -> f(encArg(z0))
   encArg(a) -> a
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__f(f(a)) -> a__f(g(f(a)))
   a__f(z0) -> f(z0)
   mark(f(z0)) -> a__f(mark(z0))
   mark(a) -> a
   mark(g(z0)) -> g(z0)
Tuples:
   ENCARG(f(z0)) -> c(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c3(A__F(encArg(z0)), ENCARG(z0))
   A__F(f(a)) -> c10(A__F(g(f(a))))
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   ENCODE_A__F(z0) -> c1(A__F(encArg(z0)))
   ENCODE_MARK(z0) -> c1(MARK(encArg(z0)))
   ENCARG(cons_mark(f(z0))) -> c4(MARK(f(encArg(z0))), ENCARG(f(z0)))
   ENCARG(cons_mark(g(z0))) -> c4(MARK(g(encArg(z0))), ENCARG(g(z0)))
   ENCARG(cons_mark(cons_a__f(z0))) -> c4(MARK(a__f(encArg(z0))), ENCARG(cons_a__f(z0)))
   ENCARG(cons_mark(cons_mark(z0))) -> c4(MARK(mark(encArg(z0))), ENCARG(cons_mark(z0)))
S tuples:
   A__F(f(a)) -> c10(A__F(g(f(a))))
K tuples:
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
Defined Rule Symbols:   encArg_1, a__f_1, mark_1

Defined Pair Symbols:   ENCARG_1, A__F_1, MARK_1, ENCODE_A__F_1, ENCODE_MARK_1

Compound Symbols:   c_1, c2_1, c3_2, c10_1, c12_2, c1_1, c4_2


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

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

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(f(z0)) -> f(encArg(z0))
   encArg(a) -> a
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__f(f(a)) -> a__f(g(f(a)))
   a__f(z0) -> f(z0)
   mark(f(z0)) -> a__f(mark(z0))
   mark(a) -> a
   mark(g(z0)) -> g(z0)
Tuples:
   ENCARG(f(z0)) -> c(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c3(A__F(encArg(z0)), ENCARG(z0))
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   ENCODE_A__F(z0) -> c1(A__F(encArg(z0)))
   ENCODE_MARK(z0) -> c1(MARK(encArg(z0)))
   ENCARG(cons_mark(f(z0))) -> c4(MARK(f(encArg(z0))), ENCARG(f(z0)))
   ENCARG(cons_mark(cons_a__f(z0))) -> c4(MARK(a__f(encArg(z0))), ENCARG(cons_a__f(z0)))
   ENCARG(cons_mark(cons_mark(z0))) -> c4(MARK(mark(encArg(z0))), ENCARG(cons_mark(z0)))
   A__F(f(a)) -> c10
   ENCARG(cons_mark(g(z0))) -> c4(ENCARG(g(z0)))
S tuples:
   A__F(f(a)) -> c10
K tuples:
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
Defined Rule Symbols:   encArg_1, a__f_1, mark_1

Defined Pair Symbols:   ENCARG_1, MARK_1, ENCODE_A__F_1, ENCODE_MARK_1, A__F_1

Compound Symbols:   c_1, c2_1, c3_2, c12_2, c1_1, c4_2, c10, c4_1


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

(27) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   A__F(f(a)) -> c10
We considered the (Usable) Rules:
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   mark(a) -> a
   a__f(f(a)) -> a__f(g(f(a)))
   a__f(z0) -> f(z0)
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   mark(f(z0)) -> a__f(mark(z0))
   encArg(a) -> a
   encArg(g(z0)) -> g(encArg(z0))
   mark(g(z0)) -> g(z0)
   encArg(f(z0)) -> f(encArg(z0))
And the Tuples:
   ENCARG(f(z0)) -> c(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c3(A__F(encArg(z0)), ENCARG(z0))
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   ENCODE_A__F(z0) -> c1(A__F(encArg(z0)))
   ENCODE_MARK(z0) -> c1(MARK(encArg(z0)))
   ENCARG(cons_mark(f(z0))) -> c4(MARK(f(encArg(z0))), ENCARG(f(z0)))
   ENCARG(cons_mark(cons_a__f(z0))) -> c4(MARK(a__f(encArg(z0))), ENCARG(cons_a__f(z0)))
   ENCARG(cons_mark(cons_mark(z0))) -> c4(MARK(mark(encArg(z0))), ENCARG(cons_mark(z0)))
   A__F(f(a)) -> c10
   ENCARG(cons_mark(g(z0))) -> c4(ENCARG(g(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A__F(x_1)) = [1]
   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_A__F(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(ENCODE_MARK(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(MARK(x_1)) = [2] + x_1
   POL(a) = 0
   POL(a__f(x_1)) = [2] + x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10) = 0
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(cons_a__f(x_1)) = [2] + x_1
   POL(cons_mark(x_1)) = [2] + x_1
   POL(encArg(x_1)) = [2]x_1
   POL(f(x_1)) = [2] + x_1
   POL(g(x_1)) = x_1
   POL(mark(x_1)) = x_1

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

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(f(z0)) -> f(encArg(z0))
   encArg(a) -> a
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__f(f(a)) -> a__f(g(f(a)))
   a__f(z0) -> f(z0)
   mark(f(z0)) -> a__f(mark(z0))
   mark(a) -> a
   mark(g(z0)) -> g(z0)
Tuples:
   ENCARG(f(z0)) -> c(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c3(A__F(encArg(z0)), ENCARG(z0))
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   ENCODE_A__F(z0) -> c1(A__F(encArg(z0)))
   ENCODE_MARK(z0) -> c1(MARK(encArg(z0)))
   ENCARG(cons_mark(f(z0))) -> c4(MARK(f(encArg(z0))), ENCARG(f(z0)))
   ENCARG(cons_mark(cons_a__f(z0))) -> c4(MARK(a__f(encArg(z0))), ENCARG(cons_a__f(z0)))
   ENCARG(cons_mark(cons_mark(z0))) -> c4(MARK(mark(encArg(z0))), ENCARG(cons_mark(z0)))
   A__F(f(a)) -> c10
   ENCARG(cons_mark(g(z0))) -> c4(ENCARG(g(z0)))
S tuples:none
K tuples:
   MARK(f(z0)) -> c12(A__F(mark(z0)), MARK(z0))
   A__F(f(a)) -> c10
Defined Rule Symbols:   encArg_1, a__f_1, mark_1

Defined Pair Symbols:   ENCARG_1, MARK_1, ENCODE_A__F_1, ENCODE_MARK_1, A__F_1

Compound Symbols:   c_1, c2_1, c3_2, c12_2, c1_1, c4_2, c10, c4_1


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

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

(30)
BOUNDS(1, 1)

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

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

(32)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   a__f(f(a)) -> a__f(g(f(a)))
   mark(f(X)) -> a__f(mark(X))
   mark(a) -> a
   mark(g(X)) -> g(X)
   a__f(X) -> f(X)

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

   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(a) -> a
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))

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

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

The rewrite sequence

mark(f(X)) ->^+ a__f(mark(X))

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

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

The result substitution is [ ].




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

(34)
Complex Obligation (BEST)

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

(35)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   a__f(f(a)) -> a__f(g(f(a)))
   mark(f(X)) -> a__f(mark(X))
   mark(a) -> a
   mark(g(X)) -> g(X)
   a__f(X) -> f(X)

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

   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(a) -> a
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))

Rewrite Strategy: INNERMOST
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(36) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(37)
BOUNDS(n^1, INF)

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(38)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   a__f(f(a)) -> a__f(g(f(a)))
   mark(f(X)) -> a__f(mark(X))
   mark(a) -> a
   mark(g(X)) -> g(X)
   a__f(X) -> f(X)

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

   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(a) -> a
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))

Rewrite Strategy: INNERMOST