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


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


WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 47 ms]
(4) CpxRelTRS
(5) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
    (7) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
    (8) CdtProblem
    (9) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (10) CdtProblem
    (11) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CdtProblem
    (13) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (14) CdtProblem
    (15) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CdtProblem
    (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 101 ms]
    (18) CdtProblem
    (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 72 ms]
    (20) CdtProblem
    (21) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 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)), 86 ms]
    (28) CdtProblem
    (29) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) BOUNDS(1, 1)
(31) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(32) CpxRelTRS
    (33) TypeInferenceProof [BOTH BOUNDS(ID, ID), 11 ms]
    (34) typed CpxTrs
    (35) OrderProof [LOWER BOUND(ID), 0 ms]
    (36) typed CpxTrs
    (37) RewriteLemmaProof [LOWER BOUND(ID), 1008 ms]
    (38) BEST
        (39) proven lower bound
            (40) LowerBoundPropagationProof [FINISHED, 0 ms]
            (41) BOUNDS(n^1, INF)
        (42) typed CpxTrs
            (43) RewriteLemmaProof [LOWER BOUND(ID), 34 ms]
            (44) BOUNDS(1, INF)


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

(0)
Obligation:
The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).


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: FULL
----------------------------------------

(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 (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).


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: FULL
----------------------------------------

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

(4)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).


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: FULL
----------------------------------------

(5) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


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

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

(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(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


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

(9) 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

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

(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))
   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


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

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

(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_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


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

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


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

(15) 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))

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

(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))
   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


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

(17) 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)) = 0
   POL(ENCODE_MARK(x_1)) = [1]
   POL(MARK(x_1)) = [1]
   POL(a) = 0
   POL(a__f(x_1)) = [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)) = [1]
   POL(f(x_1)) = x_1
   POL(g(x_1)) = x_1
   POL(mark(x_1)) = [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
   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


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

(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
   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)) = [1]
   POL(ENCARG(x_1)) = [2]x_1^2
   POL(ENCODE_A__F(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(ENCODE_MARK(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(MARK(x_1)) = x_1
   POL(a) = [2]
   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)) = [1] + x_1
   POL(cons_mark(x_1)) = [2] + x_1
   POL(encArg(x_1)) = [2] + [2]x_1
   POL(f(x_1)) = [2] + x_1
   POL(g(x_1)) = x_1
   POL(mark(x_1)) = [2] + 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
   A__F(z0) -> c11
   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


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

(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
   A__F(z0) -> c11
   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, c11, c12_2, c13, c14, c1_1, c4_2


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

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

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

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

(32)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


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: FULL
----------------------------------------

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

(34)
Obligation:
TRS:
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)
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))

Types:
a__f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
a :: a:f:g:cons_a__f:cons_mark
g :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
mark :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encArg :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
cons_a__f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
cons_mark :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_a__f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_a :: a:f:g:cons_a__f:cons_mark
encode_g :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_mark :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
hole_a:f:g:cons_a__f:cons_mark1_0 :: a:f:g:cons_a__f:cons_mark
gen_a:f:g:cons_a__f:cons_mark2_0 :: Nat -> a:f:g:cons_a__f:cons_mark

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

(35) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
a__f, mark, encArg

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

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

(36)
Obligation:
TRS:
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)
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))

Types:
a__f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
a :: a:f:g:cons_a__f:cons_mark
g :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
mark :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encArg :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
cons_a__f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
cons_mark :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_a__f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_a :: a:f:g:cons_a__f:cons_mark
encode_g :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_mark :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
hole_a:f:g:cons_a__f:cons_mark1_0 :: a:f:g:cons_a__f:cons_mark
gen_a:f:g:cons_a__f:cons_mark2_0 :: Nat -> a:f:g:cons_a__f:cons_mark


Generator Equations:
gen_a:f:g:cons_a__f:cons_mark2_0(0) <=> a
gen_a:f:g:cons_a__f:cons_mark2_0(+(x, 1)) <=> f(gen_a:f:g:cons_a__f:cons_mark2_0(x))


The following defined symbols remain to be analysed:
a__f, mark, encArg

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

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

(37) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_a:f:g:cons_a__f:cons_mark2_0(+(1, n29_0))) -> *3_0, rt in Omega(n29_0)

Induction Base:
mark(gen_a:f:g:cons_a__f:cons_mark2_0(+(1, 0)))

Induction Step:
mark(gen_a:f:g:cons_a__f:cons_mark2_0(+(1, +(n29_0, 1)))) ->_R^Omega(1)
a__f(mark(gen_a:f:g:cons_a__f:cons_mark2_0(+(1, n29_0)))) ->_IH
a__f(*3_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(38)
Complex Obligation (BEST)

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

(39)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
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)
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))

Types:
a__f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
a :: a:f:g:cons_a__f:cons_mark
g :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
mark :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encArg :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
cons_a__f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
cons_mark :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_a__f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_a :: a:f:g:cons_a__f:cons_mark
encode_g :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_mark :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
hole_a:f:g:cons_a__f:cons_mark1_0 :: a:f:g:cons_a__f:cons_mark
gen_a:f:g:cons_a__f:cons_mark2_0 :: Nat -> a:f:g:cons_a__f:cons_mark


Generator Equations:
gen_a:f:g:cons_a__f:cons_mark2_0(0) <=> a
gen_a:f:g:cons_a__f:cons_mark2_0(+(x, 1)) <=> f(gen_a:f:g:cons_a__f:cons_mark2_0(x))


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

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

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

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

(41)
BOUNDS(n^1, INF)

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

(42)
Obligation:
TRS:
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)
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))

Types:
a__f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
a :: a:f:g:cons_a__f:cons_mark
g :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
mark :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encArg :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
cons_a__f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
cons_mark :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_a__f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_a :: a:f:g:cons_a__f:cons_mark
encode_g :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_mark :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
hole_a:f:g:cons_a__f:cons_mark1_0 :: a:f:g:cons_a__f:cons_mark
gen_a:f:g:cons_a__f:cons_mark2_0 :: Nat -> a:f:g:cons_a__f:cons_mark


Lemmas:
mark(gen_a:f:g:cons_a__f:cons_mark2_0(+(1, n29_0))) -> *3_0, rt in Omega(n29_0)


Generator Equations:
gen_a:f:g:cons_a__f:cons_mark2_0(0) <=> a
gen_a:f:g:cons_a__f:cons_mark2_0(+(x, 1)) <=> f(gen_a:f:g:cons_a__f:cons_mark2_0(x))


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

(43) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_a:f:g:cons_a__f:cons_mark2_0(n5029_0)) -> gen_a:f:g:cons_a__f:cons_mark2_0(n5029_0), rt in Omega(0)

Induction Base:
encArg(gen_a:f:g:cons_a__f:cons_mark2_0(0)) ->_R^Omega(0)
a

Induction Step:
encArg(gen_a:f:g:cons_a__f:cons_mark2_0(+(n5029_0, 1))) ->_R^Omega(0)
f(encArg(gen_a:f:g:cons_a__f:cons_mark2_0(n5029_0))) ->_IH
f(gen_a:f:g:cons_a__f:cons_mark2_0(c5030_0))

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

(44)
BOUNDS(1, INF)