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


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


WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 236 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)), 71 ms]
    (18) CdtProblem
    (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 173 ms]
    (20) CdtProblem
    (21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 97 ms]
    (22) CdtProblem
    (23) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 101 ms]
    (24) CdtProblem
    (25) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 108 ms]
    (26) CdtProblem
    (27) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) BOUNDS(1, 1)
(29) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(30) CpxRelTRS
    (31) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) typed CpxTrs
    (33) OrderProof [LOWER BOUND(ID), 0 ms]
    (34) typed CpxTrs
    (35) RewriteLemmaProof [LOWER BOUND(ID), 7000 ms]
    (36) BEST
        (37) proven lower bound
            (38) LowerBoundPropagationProof [FINISHED, 0 ms]
            (39) BOUNDS(n^1, INF)
        (40) typed CpxTrs
            (41) RewriteLemmaProof [LOWER BOUND(ID), 69 ms]
            (42) 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(a, X, X) -> a__f(X, a__b, b)
   a__b -> a
   mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
   mark(b) -> a__b
   mark(a) -> a
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__b -> b

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(a) -> a
   encArg(b) -> b
   encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__b) -> a__b
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a -> a
   encode_a__b -> a__b
   encode_b -> b
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))


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

(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(a, X, X) -> a__f(X, a__b, b)
   a__b -> a
   mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
   mark(b) -> a__b
   mark(a) -> a
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__b -> b

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__b) -> a__b
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a -> a
   encode_a__b -> a__b
   encode_b -> b
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))

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(a, X, X) -> a__f(X, a__b, b)
   a__b -> a
   mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
   mark(b) -> a__b
   mark(a) -> a
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__b -> b

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__b) -> a__b
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a -> a
   encode_a__b -> a__b
   encode_b -> b
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))

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(a, X, X) -> a__f(X, a__b, b)
   a__b -> a
   mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
   mark(b) -> a__b
   mark(a) -> a
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__b -> b

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__b) -> a__b
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a -> a
   encode_a__b -> a__b
   encode_b -> b
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))

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

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(b) -> b
   encArg(f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_a__f(z0, z1, z2)) -> a__f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_a__b) -> a__b
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   encode_a__f(z0, z1, z2) -> a__f(encArg(z0), encArg(z1), encArg(z2))
   encode_a -> a
   encode_a__b -> a__b
   encode_b -> b
   encode_mark(z0) -> mark(encArg(z0))
   encode_f(z0, z1, z2) -> f(encArg(z0), encArg(z1), encArg(z2))
   a__f(a, z0, z0) -> a__f(z0, a__b, b)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__b -> a
   a__b -> b
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(b) -> a__b
   mark(a) -> a
Tuples:
   ENCARG(a) -> c
   ENCARG(b) -> c1
   ENCARG(f(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__f(z0, z1, z2)) -> c3(A__F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__b) -> c4(A__B)
   ENCARG(cons_mark(z0)) -> c5(MARK(encArg(z0)), ENCARG(z0))
   ENCODE_A__F(z0, z1, z2) -> c6(A__F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCODE_A -> c7
   ENCODE_A__B -> c8(A__B)
   ENCODE_B -> c9
   ENCODE_MARK(z0) -> c10(MARK(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0, z1, z2) -> c11(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c13
   A__B -> c14
   A__B -> c15
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
S tuples:
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c13
   A__B -> c14
   A__B -> c15
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
K tuples:none
Defined Rule Symbols:   a__f_3, a__b, mark_1, encArg_1, encode_a__f_3, encode_a, encode_a__b, encode_b, encode_mark_1, encode_f_3

Defined Pair Symbols:   ENCARG_1, ENCODE_A__F_3, ENCODE_A, ENCODE_A__B, ENCODE_B, ENCODE_MARK_1, ENCODE_F_3, A__F_3, A__B, MARK_1

Compound Symbols:   c, c1, c2_3, c3_4, c4_1, c5_2, c6_4, c7, c8_1, c9, c10_2, c11_3, c12_2, c13, c14, c15, c16_2, c17_1, c18


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

(9) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_F(z0, z1, z2) -> c11(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCODE_A__B -> c8(A__B)
Removed 4 trailing nodes:
   ENCODE_A -> c7
   ENCARG(a) -> c
   ENCODE_B -> c9
   ENCARG(b) -> c1

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(b) -> b
   encArg(f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_a__f(z0, z1, z2)) -> a__f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_a__b) -> a__b
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   encode_a__f(z0, z1, z2) -> a__f(encArg(z0), encArg(z1), encArg(z2))
   encode_a -> a
   encode_a__b -> a__b
   encode_b -> b
   encode_mark(z0) -> mark(encArg(z0))
   encode_f(z0, z1, z2) -> f(encArg(z0), encArg(z1), encArg(z2))
   a__f(a, z0, z0) -> a__f(z0, a__b, b)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__b -> a
   a__b -> b
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(b) -> a__b
   mark(a) -> a
Tuples:
   ENCARG(f(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__f(z0, z1, z2)) -> c3(A__F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__b) -> c4(A__B)
   ENCARG(cons_mark(z0)) -> c5(MARK(encArg(z0)), ENCARG(z0))
   ENCODE_A__F(z0, z1, z2) -> c6(A__F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCODE_MARK(z0) -> c10(MARK(encArg(z0)), ENCARG(z0))
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c13
   A__B -> c14
   A__B -> c15
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
S tuples:
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c13
   A__B -> c14
   A__B -> c15
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
K tuples:none
Defined Rule Symbols:   a__f_3, a__b, mark_1, encArg_1, encode_a__f_3, encode_a, encode_a__b, encode_b, encode_mark_1, encode_f_3

Defined Pair Symbols:   ENCARG_1, ENCODE_A__F_3, ENCODE_MARK_1, A__F_3, A__B, MARK_1

Compound Symbols:   c2_3, c3_4, c4_1, c5_2, c6_4, c10_2, c12_2, c13, c14, c15, c16_2, c17_1, c18


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

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(b) -> b
   encArg(f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_a__f(z0, z1, z2)) -> a__f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_a__b) -> a__b
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   encode_a__f(z0, z1, z2) -> a__f(encArg(z0), encArg(z1), encArg(z2))
   encode_a -> a
   encode_a__b -> a__b
   encode_b -> b
   encode_mark(z0) -> mark(encArg(z0))
   encode_f(z0, z1, z2) -> f(encArg(z0), encArg(z1), encArg(z2))
   a__f(a, z0, z0) -> a__f(z0, a__b, b)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__b -> a
   a__b -> b
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(b) -> a__b
   mark(a) -> a
Tuples:
   ENCARG(f(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__f(z0, z1, z2)) -> c3(A__F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__b) -> c4(A__B)
   ENCARG(cons_mark(z0)) -> c5(MARK(encArg(z0)), ENCARG(z0))
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c13
   A__B -> c14
   A__B -> c15
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
   ENCODE_A__F(z0, z1, z2) -> c(A__F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_A__F(z0, z1, z2) -> c(ENCARG(z0))
   ENCODE_A__F(z0, z1, z2) -> c(ENCARG(z1))
   ENCODE_A__F(z0, z1, z2) -> c(ENCARG(z2))
   ENCODE_MARK(z0) -> c(MARK(encArg(z0)))
   ENCODE_MARK(z0) -> c(ENCARG(z0))
S tuples:
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c13
   A__B -> c14
   A__B -> c15
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
K tuples:none
Defined Rule Symbols:   a__f_3, a__b, mark_1, encArg_1, encode_a__f_3, encode_a, encode_a__b, encode_b, encode_mark_1, encode_f_3

Defined Pair Symbols:   ENCARG_1, A__F_3, A__B, MARK_1, ENCODE_A__F_3, ENCODE_MARK_1

Compound Symbols:   c2_3, c3_4, c4_1, c5_2, c12_2, c13, c14, c15, c16_2, c17_1, c18, c_1


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

(13) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 4 leading nodes:
   ENCODE_A__F(z0, z1, z2) -> c(ENCARG(z0))
   ENCODE_A__F(z0, z1, z2) -> c(ENCARG(z1))
   ENCODE_A__F(z0, z1, z2) -> c(ENCARG(z2))
   ENCODE_MARK(z0) -> c(ENCARG(z0))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(b) -> b
   encArg(f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_a__f(z0, z1, z2)) -> a__f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_a__b) -> a__b
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   encode_a__f(z0, z1, z2) -> a__f(encArg(z0), encArg(z1), encArg(z2))
   encode_a -> a
   encode_a__b -> a__b
   encode_b -> b
   encode_mark(z0) -> mark(encArg(z0))
   encode_f(z0, z1, z2) -> f(encArg(z0), encArg(z1), encArg(z2))
   a__f(a, z0, z0) -> a__f(z0, a__b, b)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__b -> a
   a__b -> b
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(b) -> a__b
   mark(a) -> a
Tuples:
   ENCARG(f(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__f(z0, z1, z2)) -> c3(A__F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__b) -> c4(A__B)
   ENCARG(cons_mark(z0)) -> c5(MARK(encArg(z0)), ENCARG(z0))
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c13
   A__B -> c14
   A__B -> c15
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
   ENCODE_A__F(z0, z1, z2) -> c(A__F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_MARK(z0) -> c(MARK(encArg(z0)))
S tuples:
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c13
   A__B -> c14
   A__B -> c15
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
K tuples:none
Defined Rule Symbols:   a__f_3, a__b, mark_1, encArg_1, encode_a__f_3, encode_a, encode_a__b, encode_b, encode_mark_1, encode_f_3

Defined Pair Symbols:   ENCARG_1, A__F_3, A__B, MARK_1, ENCODE_A__F_3, ENCODE_MARK_1

Compound Symbols:   c2_3, c3_4, c4_1, c5_2, c12_2, c13, c14, c15, c16_2, c17_1, c18, c_1


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

(15) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_a__f(z0, z1, z2) -> a__f(encArg(z0), encArg(z1), encArg(z2))
   encode_a -> a
   encode_a__b -> a__b
   encode_b -> b
   encode_mark(z0) -> mark(encArg(z0))
   encode_f(z0, z1, z2) -> f(encArg(z0), encArg(z1), encArg(z2))

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(b) -> b
   encArg(f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_a__f(z0, z1, z2)) -> a__f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_a__b) -> a__b
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__f(a, z0, z0) -> a__f(z0, a__b, b)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__b -> a
   a__b -> b
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(b) -> a__b
   mark(a) -> a
Tuples:
   ENCARG(f(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__f(z0, z1, z2)) -> c3(A__F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__b) -> c4(A__B)
   ENCARG(cons_mark(z0)) -> c5(MARK(encArg(z0)), ENCARG(z0))
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c13
   A__B -> c14
   A__B -> c15
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
   ENCODE_A__F(z0, z1, z2) -> c(A__F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_MARK(z0) -> c(MARK(encArg(z0)))
S tuples:
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c13
   A__B -> c14
   A__B -> c15
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
K tuples:none
Defined Rule Symbols:   encArg_1, a__f_3, a__b, mark_1

Defined Pair Symbols:   ENCARG_1, A__F_3, A__B, MARK_1, ENCODE_A__F_3, ENCODE_MARK_1

Compound Symbols:   c2_3, c3_4, c4_1, c5_2, c12_2, c13, c14, c15, c16_2, c17_1, c18, c_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(b) -> c17(A__B)
   MARK(a) -> c18
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(f(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__f(z0, z1, z2)) -> c3(A__F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__b) -> c4(A__B)
   ENCARG(cons_mark(z0)) -> c5(MARK(encArg(z0)), ENCARG(z0))
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c13
   A__B -> c14
   A__B -> c15
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
   ENCODE_A__F(z0, z1, z2) -> c(A__F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_MARK(z0) -> c(MARK(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A__B) = 0
   POL(A__F(x_1, x_2, x_3)) = 0
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_A__F(x_1, x_2, x_3)) = 0
   POL(ENCODE_MARK(x_1)) = [1]
   POL(MARK(x_1)) = [1]
   POL(a) = 0
   POL(a__b) = 0
   POL(a__f(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(b) = 0
   POL(c(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13) = 0
   POL(c14) = 0
   POL(c15) = 0
   POL(c16(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18) = 0
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c4(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(cons_a__b) = [1]
   POL(cons_a__f(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(cons_mark(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(f(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(mark(x_1)) = [1]

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(b) -> b
   encArg(f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_a__f(z0, z1, z2)) -> a__f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_a__b) -> a__b
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__f(a, z0, z0) -> a__f(z0, a__b, b)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__b -> a
   a__b -> b
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(b) -> a__b
   mark(a) -> a
Tuples:
   ENCARG(f(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__f(z0, z1, z2)) -> c3(A__F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__b) -> c4(A__B)
   ENCARG(cons_mark(z0)) -> c5(MARK(encArg(z0)), ENCARG(z0))
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c13
   A__B -> c14
   A__B -> c15
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
   ENCODE_A__F(z0, z1, z2) -> c(A__F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_MARK(z0) -> c(MARK(encArg(z0)))
S tuples:
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c13
   A__B -> c14
   A__B -> c15
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
K tuples:
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
Defined Rule Symbols:   encArg_1, a__f_3, a__b, mark_1

Defined Pair Symbols:   ENCARG_1, A__F_3, A__B, MARK_1, ENCODE_A__F_3, ENCODE_MARK_1

Compound Symbols:   c2_3, c3_4, c4_1, c5_2, c12_2, c13, c14, c15, c16_2, c17_1, c18, c_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.
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
We considered the (Usable) Rules:
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   encArg(cons_a__b) -> a__b
   mark(a) -> a
   encArg(a) -> a
   a__b -> b
   encArg(cons_a__f(z0, z1, z2)) -> a__f(encArg(z0), encArg(z1), encArg(z2))
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(b) -> a__b
   encArg(f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   a__f(a, z0, z0) -> a__f(z0, a__b, b)
   encArg(b) -> b
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__b -> a
And the Tuples:
   ENCARG(f(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__f(z0, z1, z2)) -> c3(A__F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__b) -> c4(A__B)
   ENCARG(cons_mark(z0)) -> c5(MARK(encArg(z0)), ENCARG(z0))
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c13
   A__B -> c14
   A__B -> c15
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
   ENCODE_A__F(z0, z1, z2) -> c(A__F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_MARK(z0) -> c(MARK(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A__B) = 0
   POL(A__F(x_1, x_2, x_3)) = 0
   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_A__F(x_1, x_2, x_3)) = [2] + x_1 + x_2 + [2]x_3 + [2]x_3^2 + [2]x_2*x_3 + x_1*x_3 + [2]x_1^2 + [2]x_1*x_2 + x_2^2
   POL(ENCODE_MARK(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(MARK(x_1)) = x_1
   POL(a) = 0
   POL(a__b) = 0
   POL(a__f(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(b) = 0
   POL(c(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13) = 0
   POL(c14) = 0
   POL(c15) = 0
   POL(c16(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18) = 0
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c4(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(cons_a__b) = 0
   POL(cons_a__f(x_1, x_2, x_3)) = [2] + x_1 + x_2 + x_3
   POL(cons_mark(x_1)) = [2] + x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(mark(x_1)) = x_1

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(b) -> b
   encArg(f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_a__f(z0, z1, z2)) -> a__f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_a__b) -> a__b
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__f(a, z0, z0) -> a__f(z0, a__b, b)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__b -> a
   a__b -> b
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(b) -> a__b
   mark(a) -> a
Tuples:
   ENCARG(f(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__f(z0, z1, z2)) -> c3(A__F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__b) -> c4(A__B)
   ENCARG(cons_mark(z0)) -> c5(MARK(encArg(z0)), ENCARG(z0))
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c13
   A__B -> c14
   A__B -> c15
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
   ENCODE_A__F(z0, z1, z2) -> c(A__F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_MARK(z0) -> c(MARK(encArg(z0)))
S tuples:
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c13
   A__B -> c14
   A__B -> c15
K tuples:
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
Defined Rule Symbols:   encArg_1, a__f_3, a__b, mark_1

Defined Pair Symbols:   ENCARG_1, A__F_3, A__B, MARK_1, ENCODE_A__F_3, ENCODE_MARK_1

Compound Symbols:   c2_3, c3_4, c4_1, c5_2, c12_2, c13, c14, c15, c16_2, c17_1, c18, c_1


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

(21) 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, z1, z2) -> c13
We considered the (Usable) Rules:
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   encArg(cons_a__b) -> a__b
   mark(a) -> a
   encArg(a) -> a
   a__b -> b
   encArg(cons_a__f(z0, z1, z2)) -> a__f(encArg(z0), encArg(z1), encArg(z2))
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(b) -> a__b
   encArg(f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   a__f(a, z0, z0) -> a__f(z0, a__b, b)
   encArg(b) -> b
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__b -> a
And the Tuples:
   ENCARG(f(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__f(z0, z1, z2)) -> c3(A__F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__b) -> c4(A__B)
   ENCARG(cons_mark(z0)) -> c5(MARK(encArg(z0)), ENCARG(z0))
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c13
   A__B -> c14
   A__B -> c15
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
   ENCODE_A__F(z0, z1, z2) -> c(A__F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_MARK(z0) -> c(MARK(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A__B) = 0
   POL(A__F(x_1, x_2, x_3)) = [1]
   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_A__F(x_1, x_2, x_3)) = [2] + x_1 + x_2 + [2]x_3 + x_3^2 + x_2*x_3 + [2]x_1*x_3 + [2]x_1^2 + [2]x_1*x_2 + x_2^2
   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__b) = 0
   POL(a__f(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(b) = 0
   POL(c(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13) = 0
   POL(c14) = 0
   POL(c15) = 0
   POL(c16(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18) = 0
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c4(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(cons_a__b) = 0
   POL(cons_a__f(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(cons_mark(x_1)) = [1] + x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(mark(x_1)) = x_1

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(b) -> b
   encArg(f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_a__f(z0, z1, z2)) -> a__f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_a__b) -> a__b
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__f(a, z0, z0) -> a__f(z0, a__b, b)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__b -> a
   a__b -> b
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(b) -> a__b
   mark(a) -> a
Tuples:
   ENCARG(f(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__f(z0, z1, z2)) -> c3(A__F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__b) -> c4(A__B)
   ENCARG(cons_mark(z0)) -> c5(MARK(encArg(z0)), ENCARG(z0))
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c13
   A__B -> c14
   A__B -> c15
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
   ENCODE_A__F(z0, z1, z2) -> c(A__F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_MARK(z0) -> c(MARK(encArg(z0)))
S tuples:
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__B -> c14
   A__B -> c15
K tuples:
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   A__F(z0, z1, z2) -> c13
Defined Rule Symbols:   encArg_1, a__f_3, a__b, mark_1

Defined Pair Symbols:   ENCARG_1, A__F_3, A__B, MARK_1, ENCODE_A__F_3, ENCODE_MARK_1

Compound Symbols:   c2_3, c3_4, c4_1, c5_2, c12_2, c13, c14, c15, c16_2, c17_1, c18, c_1


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

(23) 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(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
We considered the (Usable) Rules:
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   encArg(cons_a__b) -> a__b
   mark(a) -> a
   encArg(a) -> a
   a__b -> b
   encArg(cons_a__f(z0, z1, z2)) -> a__f(encArg(z0), encArg(z1), encArg(z2))
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(b) -> a__b
   encArg(f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   a__f(a, z0, z0) -> a__f(z0, a__b, b)
   encArg(b) -> b
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__b -> a
And the Tuples:
   ENCARG(f(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__f(z0, z1, z2)) -> c3(A__F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__b) -> c4(A__B)
   ENCARG(cons_mark(z0)) -> c5(MARK(encArg(z0)), ENCARG(z0))
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c13
   A__B -> c14
   A__B -> c15
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
   ENCODE_A__F(z0, z1, z2) -> c(A__F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_MARK(z0) -> c(MARK(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A__B) = 0
   POL(A__F(x_1, x_2, x_3)) = [2]x_1 + [2]x_3
   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_A__F(x_1, x_2, x_3)) = [2] + [2]x_1 + x_2 + [2]x_3 + [2]x_3^2 + [2]x_2*x_3 + [2]x_1*x_3 + [2]x_1^2 + [2]x_1*x_2 + x_2^2
   POL(ENCODE_MARK(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(MARK(x_1)) = [2]x_1
   POL(a) = [1]
   POL(a__b) = [1]
   POL(a__f(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(b) = 0
   POL(c(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13) = 0
   POL(c14) = 0
   POL(c15) = 0
   POL(c16(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18) = 0
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c4(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(cons_a__b) = [1]
   POL(cons_a__f(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(cons_mark(x_1)) = [1] + x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(mark(x_1)) = [1] + x_1

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(b) -> b
   encArg(f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_a__f(z0, z1, z2)) -> a__f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_a__b) -> a__b
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__f(a, z0, z0) -> a__f(z0, a__b, b)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__b -> a
   a__b -> b
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(b) -> a__b
   mark(a) -> a
Tuples:
   ENCARG(f(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__f(z0, z1, z2)) -> c3(A__F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__b) -> c4(A__B)
   ENCARG(cons_mark(z0)) -> c5(MARK(encArg(z0)), ENCARG(z0))
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c13
   A__B -> c14
   A__B -> c15
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
   ENCODE_A__F(z0, z1, z2) -> c(A__F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_MARK(z0) -> c(MARK(encArg(z0)))
S tuples:
   A__B -> c14
   A__B -> c15
K tuples:
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   A__F(z0, z1, z2) -> c13
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
Defined Rule Symbols:   encArg_1, a__f_3, a__b, mark_1

Defined Pair Symbols:   ENCARG_1, A__F_3, A__B, MARK_1, ENCODE_A__F_3, ENCODE_MARK_1

Compound Symbols:   c2_3, c3_4, c4_1, c5_2, c12_2, c13, c14, c15, c16_2, c17_1, c18, c_1


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

(25) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   A__B -> c14
   A__B -> c15
We considered the (Usable) Rules:
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   encArg(cons_a__b) -> a__b
   mark(a) -> a
   encArg(a) -> a
   a__b -> b
   encArg(cons_a__f(z0, z1, z2)) -> a__f(encArg(z0), encArg(z1), encArg(z2))
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(b) -> a__b
   encArg(f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   a__f(a, z0, z0) -> a__f(z0, a__b, b)
   encArg(b) -> b
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__b -> a
And the Tuples:
   ENCARG(f(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__f(z0, z1, z2)) -> c3(A__F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__b) -> c4(A__B)
   ENCARG(cons_mark(z0)) -> c5(MARK(encArg(z0)), ENCARG(z0))
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c13
   A__B -> c14
   A__B -> c15
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
   ENCODE_A__F(z0, z1, z2) -> c(A__F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_MARK(z0) -> c(MARK(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A__B) = [1]
   POL(A__F(x_1, x_2, x_3)) = [2]x_1 + [2]x_3
   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_A__F(x_1, x_2, x_3)) = [1] + [2]x_1 + x_2 + [2]x_3 + [2]x_3^2 + [2]x_2*x_3 + [2]x_1*x_3 + [2]x_1^2 + [2]x_1*x_2 + [2]x_2^2
   POL(ENCODE_MARK(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(MARK(x_1)) = [1] + [2]x_1
   POL(a) = [1]
   POL(a__b) = [1]
   POL(a__f(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(b) = 0
   POL(c(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13) = 0
   POL(c14) = 0
   POL(c15) = 0
   POL(c16(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18) = 0
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c4(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(cons_a__b) = [1]
   POL(cons_a__f(x_1, x_2, x_3)) = [2] + x_1 + x_2 + x_3
   POL(cons_mark(x_1)) = [2] + x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(mark(x_1)) = [2] + x_1

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

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(b) -> b
   encArg(f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_a__f(z0, z1, z2)) -> a__f(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_a__b) -> a__b
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__f(a, z0, z0) -> a__f(z0, a__b, b)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__b -> a
   a__b -> b
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(b) -> a__b
   mark(a) -> a
Tuples:
   ENCARG(f(z0, z1, z2)) -> c2(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__f(z0, z1, z2)) -> c3(A__F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_a__b) -> c4(A__B)
   ENCARG(cons_mark(z0)) -> c5(MARK(encArg(z0)), ENCARG(z0))
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c13
   A__B -> c14
   A__B -> c15
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
   ENCODE_A__F(z0, z1, z2) -> c(A__F(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_MARK(z0) -> c(MARK(encArg(z0)))
S tuples:none
K tuples:
   MARK(b) -> c17(A__B)
   MARK(a) -> c18
   MARK(f(z0, z1, z2)) -> c16(A__F(z0, mark(z1), z2), MARK(z1))
   A__F(z0, z1, z2) -> c13
   A__F(a, z0, z0) -> c12(A__F(z0, a__b, b), A__B)
   A__B -> c14
   A__B -> c15
Defined Rule Symbols:   encArg_1, a__f_3, a__b, mark_1

Defined Pair Symbols:   ENCARG_1, A__F_3, A__B, MARK_1, ENCODE_A__F_3, ENCODE_MARK_1

Compound Symbols:   c2_3, c3_4, c4_1, c5_2, c12_2, c13, c14, c15, c16_2, c17_1, c18, c_1


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

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

(28)
BOUNDS(1, 1)

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

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

(30)
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(a, X, X) -> a__f(X, a__b, b)
   a__b -> a
   mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
   mark(b) -> a__b
   mark(a) -> a
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__b -> b

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__b) -> a__b
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a -> a
   encode_a__b -> a__b
   encode_b -> b
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: FULL
----------------------------------------

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

(32)
Obligation:
TRS:
Rules:
a__f(a, X, X) -> a__f(X, a__b, b)
a__b -> a
mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
mark(b) -> a__b
mark(a) -> a
a__f(X1, X2, X3) -> f(X1, X2, X3)
a__b -> b
encArg(a) -> a
encArg(b) -> b
encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__b) -> a__b
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
encode_a -> a
encode_a__b -> a__b
encode_b -> b
encode_mark(x_1) -> mark(encArg(x_1))
encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
a__f :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
a :: a:b:f:cons_a__f:cons_a__b:cons_mark
a__b :: a:b:f:cons_a__f:cons_a__b:cons_mark
b :: a:b:f:cons_a__f:cons_a__b:cons_mark
mark :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
f :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
encArg :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
cons_a__f :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
cons_a__b :: a:b:f:cons_a__f:cons_a__b:cons_mark
cons_mark :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
encode_a__f :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
encode_a :: a:b:f:cons_a__f:cons_a__b:cons_mark
encode_a__b :: a:b:f:cons_a__f:cons_a__b:cons_mark
encode_b :: a:b:f:cons_a__f:cons_a__b:cons_mark
encode_mark :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
encode_f :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
hole_a:b:f:cons_a__f:cons_a__b:cons_mark1_0 :: a:b:f:cons_a__f:cons_a__b:cons_mark
gen_a:b:f:cons_a__f:cons_a__b:cons_mark2_0 :: Nat -> a:b:f:cons_a__f:cons_a__b:cons_mark

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

(33) 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

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

(34)
Obligation:
TRS:
Rules:
a__f(a, X, X) -> a__f(X, a__b, b)
a__b -> a
mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
mark(b) -> a__b
mark(a) -> a
a__f(X1, X2, X3) -> f(X1, X2, X3)
a__b -> b
encArg(a) -> a
encArg(b) -> b
encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__b) -> a__b
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
encode_a -> a
encode_a__b -> a__b
encode_b -> b
encode_mark(x_1) -> mark(encArg(x_1))
encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
a__f :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
a :: a:b:f:cons_a__f:cons_a__b:cons_mark
a__b :: a:b:f:cons_a__f:cons_a__b:cons_mark
b :: a:b:f:cons_a__f:cons_a__b:cons_mark
mark :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
f :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
encArg :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
cons_a__f :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
cons_a__b :: a:b:f:cons_a__f:cons_a__b:cons_mark
cons_mark :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
encode_a__f :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
encode_a :: a:b:f:cons_a__f:cons_a__b:cons_mark
encode_a__b :: a:b:f:cons_a__f:cons_a__b:cons_mark
encode_b :: a:b:f:cons_a__f:cons_a__b:cons_mark
encode_mark :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
encode_f :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
hole_a:b:f:cons_a__f:cons_a__b:cons_mark1_0 :: a:b:f:cons_a__f:cons_a__b:cons_mark
gen_a:b:f:cons_a__f:cons_a__b:cons_mark2_0 :: Nat -> a:b:f:cons_a__f:cons_a__b:cons_mark


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


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

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

(35) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_a:b:f:cons_a__f:cons_a__b:cons_mark2_0(+(1, n62_0))) -> *3_0, rt in Omega(n62_0)

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

Induction Step:
mark(gen_a:b:f:cons_a__f:cons_a__b:cons_mark2_0(+(1, +(n62_0, 1)))) ->_R^Omega(1)
a__f(a, mark(gen_a:b:f:cons_a__f:cons_a__b:cons_mark2_0(+(1, n62_0))), a) ->_IH
a__f(a, *3_0, a)

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

(36)
Complex Obligation (BEST)

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

(37)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
a__f(a, X, X) -> a__f(X, a__b, b)
a__b -> a
mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
mark(b) -> a__b
mark(a) -> a
a__f(X1, X2, X3) -> f(X1, X2, X3)
a__b -> b
encArg(a) -> a
encArg(b) -> b
encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__b) -> a__b
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
encode_a -> a
encode_a__b -> a__b
encode_b -> b
encode_mark(x_1) -> mark(encArg(x_1))
encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
a__f :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
a :: a:b:f:cons_a__f:cons_a__b:cons_mark
a__b :: a:b:f:cons_a__f:cons_a__b:cons_mark
b :: a:b:f:cons_a__f:cons_a__b:cons_mark
mark :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
f :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
encArg :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
cons_a__f :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
cons_a__b :: a:b:f:cons_a__f:cons_a__b:cons_mark
cons_mark :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
encode_a__f :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
encode_a :: a:b:f:cons_a__f:cons_a__b:cons_mark
encode_a__b :: a:b:f:cons_a__f:cons_a__b:cons_mark
encode_b :: a:b:f:cons_a__f:cons_a__b:cons_mark
encode_mark :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
encode_f :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
hole_a:b:f:cons_a__f:cons_a__b:cons_mark1_0 :: a:b:f:cons_a__f:cons_a__b:cons_mark
gen_a:b:f:cons_a__f:cons_a__b:cons_mark2_0 :: Nat -> a:b:f:cons_a__f:cons_a__b:cons_mark


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


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

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

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

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

(39)
BOUNDS(n^1, INF)

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

(40)
Obligation:
TRS:
Rules:
a__f(a, X, X) -> a__f(X, a__b, b)
a__b -> a
mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
mark(b) -> a__b
mark(a) -> a
a__f(X1, X2, X3) -> f(X1, X2, X3)
a__b -> b
encArg(a) -> a
encArg(b) -> b
encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__b) -> a__b
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
encode_a -> a
encode_a__b -> a__b
encode_b -> b
encode_mark(x_1) -> mark(encArg(x_1))
encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
a__f :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
a :: a:b:f:cons_a__f:cons_a__b:cons_mark
a__b :: a:b:f:cons_a__f:cons_a__b:cons_mark
b :: a:b:f:cons_a__f:cons_a__b:cons_mark
mark :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
f :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
encArg :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
cons_a__f :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
cons_a__b :: a:b:f:cons_a__f:cons_a__b:cons_mark
cons_mark :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
encode_a__f :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
encode_a :: a:b:f:cons_a__f:cons_a__b:cons_mark
encode_a__b :: a:b:f:cons_a__f:cons_a__b:cons_mark
encode_b :: a:b:f:cons_a__f:cons_a__b:cons_mark
encode_mark :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
encode_f :: a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark -> a:b:f:cons_a__f:cons_a__b:cons_mark
hole_a:b:f:cons_a__f:cons_a__b:cons_mark1_0 :: a:b:f:cons_a__f:cons_a__b:cons_mark
gen_a:b:f:cons_a__f:cons_a__b:cons_mark2_0 :: Nat -> a:b:f:cons_a__f:cons_a__b:cons_mark


Lemmas:
mark(gen_a:b:f:cons_a__f:cons_a__b:cons_mark2_0(+(1, n62_0))) -> *3_0, rt in Omega(n62_0)


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


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

(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_a:b:f:cons_a__f:cons_a__b:cons_mark2_0(n72596_0)) -> gen_a:b:f:cons_a__f:cons_a__b:cons_mark2_0(n72596_0), rt in Omega(0)

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

Induction Step:
encArg(gen_a:b:f:cons_a__f:cons_a__b:cons_mark2_0(+(n72596_0, 1))) ->_R^Omega(0)
f(encArg(a), encArg(gen_a:b:f:cons_a__f:cons_a__b:cons_mark2_0(n72596_0)), encArg(a)) ->_R^Omega(0)
f(a, encArg(gen_a:b:f:cons_a__f:cons_a__b:cons_mark2_0(n72596_0)), encArg(a)) ->_IH
f(a, gen_a:b:f:cons_a__f:cons_a__b:cons_mark2_0(c72597_0), encArg(a)) ->_R^Omega(0)
f(a, gen_a:b:f:cons_a__f:cons_a__b:cons_mark2_0(n72596_0), a)

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

(42)
BOUNDS(1, INF)