/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 60 ms]
(4) CpxRelTRS
(5) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
    (7) CpxTrsToCdtProof [UPPER BOUND(ID), 13 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), 13 ms]
    (16) CdtProblem
    (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 179 ms]
    (18) CdtProblem
    (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 184 ms]
    (20) CdtProblem
    (21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 153 ms]
    (22) CdtProblem
    (23) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 116 ms]
    (24) CdtProblem
    (25) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 107 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), 464 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), 698 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(f(X)) -> a__c(f(g(f(X))))
   a__c(X) -> d(X)
   a__h(X) -> a__c(d(X))
   mark(f(X)) -> a__f(mark(X))
   mark(c(X)) -> a__c(X)
   mark(h(X)) -> a__h(mark(X))
   mark(g(X)) -> g(X)
   mark(d(X)) -> d(X)
   a__f(X) -> f(X)
   a__c(X) -> c(X)
   a__h(X) -> h(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(g(x_1)) -> g(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_a__c(x_1)) -> a__c(encArg(x_1))
   encArg(cons_a__h(x_1)) -> a__h(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__c(x_1) -> a__c(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_a__h(x_1) -> a__h(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_h(x_1) -> h(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(X)) -> a__c(f(g(f(X))))
   a__c(X) -> d(X)
   a__h(X) -> a__c(d(X))
   mark(f(X)) -> a__f(mark(X))
   mark(c(X)) -> a__c(X)
   mark(h(X)) -> a__h(mark(X))
   mark(g(X)) -> g(X)
   mark(d(X)) -> d(X)
   a__f(X) -> f(X)
   a__c(X) -> c(X)
   a__h(X) -> h(X)

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

   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_a__c(x_1)) -> a__c(encArg(x_1))
   encArg(cons_a__h(x_1)) -> a__h(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__c(x_1) -> a__c(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_a__h(x_1) -> a__h(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_h(x_1) -> h(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(X)) -> a__c(f(g(f(X))))
   a__c(X) -> d(X)
   a__h(X) -> a__c(d(X))
   mark(f(X)) -> a__f(mark(X))
   mark(c(X)) -> a__c(X)
   mark(h(X)) -> a__h(mark(X))
   mark(g(X)) -> g(X)
   mark(d(X)) -> d(X)
   a__f(X) -> f(X)
   a__c(X) -> c(X)
   a__h(X) -> h(X)

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

   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_a__c(x_1)) -> a__c(encArg(x_1))
   encArg(cons_a__h(x_1)) -> a__h(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__c(x_1) -> a__c(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_a__h(x_1) -> a__h(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_h(x_1) -> h(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(X)) -> a__c(f(g(f(X))))
   a__c(X) -> d(X)
   a__h(X) -> a__c(d(X))
   mark(f(X)) -> a__f(mark(X))
   mark(c(X)) -> a__c(X)
   mark(h(X)) -> a__h(mark(X))
   mark(g(X)) -> g(X)
   mark(d(X)) -> d(X)
   a__f(X) -> f(X)
   a__c(X) -> c(X)
   a__h(X) -> h(X)

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

   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_a__c(x_1)) -> a__c(encArg(x_1))
   encArg(cons_a__h(x_1)) -> a__h(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__c(x_1) -> a__c(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_a__h(x_1) -> a__h(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_h(x_1) -> h(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(g(z0)) -> g(encArg(z0))
   encArg(d(z0)) -> d(encArg(z0))
   encArg(c(z0)) -> c(encArg(z0))
   encArg(h(z0)) -> h(encArg(z0))
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_a__c(z0)) -> a__c(encArg(z0))
   encArg(cons_a__h(z0)) -> a__h(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__c(z0) -> a__c(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_d(z0) -> d(encArg(z0))
   encode_a__h(z0) -> a__h(encArg(z0))
   encode_mark(z0) -> mark(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   encode_h(z0) -> h(encArg(z0))
   a__f(f(z0)) -> a__c(f(g(f(z0))))
   a__f(z0) -> f(z0)
   a__c(z0) -> d(z0)
   a__c(z0) -> c(z0)
   a__h(z0) -> a__c(d(z0))
   a__h(z0) -> h(z0)
   mark(f(z0)) -> a__f(mark(z0))
   mark(c(z0)) -> a__c(z0)
   mark(h(z0)) -> a__h(mark(z0))
   mark(g(z0)) -> g(z0)
   mark(d(z0)) -> d(z0)
Tuples:
   ENCARG(f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(d(z0)) -> c3(ENCARG(z0))
   ENCARG(c(z0)) -> c4(ENCARG(z0))
   ENCARG(h(z0)) -> c5(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c6(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__c(z0)) -> c7(A__C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__h(z0)) -> c8(A__H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c9(MARK(encArg(z0)), ENCARG(z0))
   ENCODE_A__F(z0) -> c10(A__F(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0) -> c11(ENCARG(z0))
   ENCODE_A__C(z0) -> c12(A__C(encArg(z0)), ENCARG(z0))
   ENCODE_G(z0) -> c13(ENCARG(z0))
   ENCODE_D(z0) -> c14(ENCARG(z0))
   ENCODE_A__H(z0) -> c15(A__H(encArg(z0)), ENCARG(z0))
   ENCODE_MARK(z0) -> c16(MARK(encArg(z0)), ENCARG(z0))
   ENCODE_C(z0) -> c17(ENCARG(z0))
   ENCODE_H(z0) -> c18(ENCARG(z0))
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__C(z0) -> c21
   A__C(z0) -> c22
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
S tuples:
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__C(z0) -> c21
   A__C(z0) -> c22
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
K tuples:none
Defined Rule Symbols:   a__f_1, a__c_1, a__h_1, mark_1, encArg_1, encode_a__f_1, encode_f_1, encode_a__c_1, encode_g_1, encode_d_1, encode_a__h_1, encode_mark_1, encode_c_1, encode_h_1

Defined Pair Symbols:   ENCARG_1, ENCODE_A__F_1, ENCODE_F_1, ENCODE_A__C_1, ENCODE_G_1, ENCODE_D_1, ENCODE_A__H_1, ENCODE_MARK_1, ENCODE_C_1, ENCODE_H_1, A__F_1, A__C_1, A__H_1, MARK_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_1, c5_1, c6_2, c7_2, c8_2, c9_2, c10_2, c11_1, c12_2, c13_1, c14_1, c15_2, c16_2, c17_1, c18_1, c19_1, c20, c21, c22, c23_1, c24, c25_2, c26_1, c27_2, c28, c29


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

(9) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 5 leading nodes:
   ENCODE_F(z0) -> c11(ENCARG(z0))
   ENCODE_G(z0) -> c13(ENCARG(z0))
   ENCODE_D(z0) -> c14(ENCARG(z0))
   ENCODE_C(z0) -> c17(ENCARG(z0))
   ENCODE_H(z0) -> c18(ENCARG(z0))

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(f(z0)) -> f(encArg(z0))
   encArg(g(z0)) -> g(encArg(z0))
   encArg(d(z0)) -> d(encArg(z0))
   encArg(c(z0)) -> c(encArg(z0))
   encArg(h(z0)) -> h(encArg(z0))
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_a__c(z0)) -> a__c(encArg(z0))
   encArg(cons_a__h(z0)) -> a__h(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__c(z0) -> a__c(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_d(z0) -> d(encArg(z0))
   encode_a__h(z0) -> a__h(encArg(z0))
   encode_mark(z0) -> mark(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   encode_h(z0) -> h(encArg(z0))
   a__f(f(z0)) -> a__c(f(g(f(z0))))
   a__f(z0) -> f(z0)
   a__c(z0) -> d(z0)
   a__c(z0) -> c(z0)
   a__h(z0) -> a__c(d(z0))
   a__h(z0) -> h(z0)
   mark(f(z0)) -> a__f(mark(z0))
   mark(c(z0)) -> a__c(z0)
   mark(h(z0)) -> a__h(mark(z0))
   mark(g(z0)) -> g(z0)
   mark(d(z0)) -> d(z0)
Tuples:
   ENCARG(f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(d(z0)) -> c3(ENCARG(z0))
   ENCARG(c(z0)) -> c4(ENCARG(z0))
   ENCARG(h(z0)) -> c5(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c6(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__c(z0)) -> c7(A__C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__h(z0)) -> c8(A__H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c9(MARK(encArg(z0)), ENCARG(z0))
   ENCODE_A__F(z0) -> c10(A__F(encArg(z0)), ENCARG(z0))
   ENCODE_A__C(z0) -> c12(A__C(encArg(z0)), ENCARG(z0))
   ENCODE_A__H(z0) -> c15(A__H(encArg(z0)), ENCARG(z0))
   ENCODE_MARK(z0) -> c16(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__C(z0) -> c21
   A__C(z0) -> c22
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
S tuples:
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__C(z0) -> c21
   A__C(z0) -> c22
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
K tuples:none
Defined Rule Symbols:   a__f_1, a__c_1, a__h_1, mark_1, encArg_1, encode_a__f_1, encode_f_1, encode_a__c_1, encode_g_1, encode_d_1, encode_a__h_1, encode_mark_1, encode_c_1, encode_h_1

Defined Pair Symbols:   ENCARG_1, ENCODE_A__F_1, ENCODE_A__C_1, ENCODE_A__H_1, ENCODE_MARK_1, A__F_1, A__C_1, A__H_1, MARK_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_1, c5_1, c6_2, c7_2, c8_2, c9_2, c10_2, c12_2, c15_2, c16_2, c19_1, c20, c21, c22, c23_1, c24, c25_2, c26_1, c27_2, c28, c29


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

(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(g(z0)) -> g(encArg(z0))
   encArg(d(z0)) -> d(encArg(z0))
   encArg(c(z0)) -> c(encArg(z0))
   encArg(h(z0)) -> h(encArg(z0))
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_a__c(z0)) -> a__c(encArg(z0))
   encArg(cons_a__h(z0)) -> a__h(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__c(z0) -> a__c(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_d(z0) -> d(encArg(z0))
   encode_a__h(z0) -> a__h(encArg(z0))
   encode_mark(z0) -> mark(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   encode_h(z0) -> h(encArg(z0))
   a__f(f(z0)) -> a__c(f(g(f(z0))))
   a__f(z0) -> f(z0)
   a__c(z0) -> d(z0)
   a__c(z0) -> c(z0)
   a__h(z0) -> a__c(d(z0))
   a__h(z0) -> h(z0)
   mark(f(z0)) -> a__f(mark(z0))
   mark(c(z0)) -> a__c(z0)
   mark(h(z0)) -> a__h(mark(z0))
   mark(g(z0)) -> g(z0)
   mark(d(z0)) -> d(z0)
Tuples:
   ENCARG(f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(d(z0)) -> c3(ENCARG(z0))
   ENCARG(c(z0)) -> c4(ENCARG(z0))
   ENCARG(h(z0)) -> c5(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c6(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__c(z0)) -> c7(A__C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__h(z0)) -> c8(A__H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c9(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__C(z0) -> c21
   A__C(z0) -> c22
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
   ENCODE_A__F(z0) -> c11(A__F(encArg(z0)))
   ENCODE_A__F(z0) -> c11(ENCARG(z0))
   ENCODE_A__C(z0) -> c11(A__C(encArg(z0)))
   ENCODE_A__C(z0) -> c11(ENCARG(z0))
   ENCODE_A__H(z0) -> c11(A__H(encArg(z0)))
   ENCODE_A__H(z0) -> c11(ENCARG(z0))
   ENCODE_MARK(z0) -> c11(MARK(encArg(z0)))
   ENCODE_MARK(z0) -> c11(ENCARG(z0))
S tuples:
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__C(z0) -> c21
   A__C(z0) -> c22
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
K tuples:none
Defined Rule Symbols:   a__f_1, a__c_1, a__h_1, mark_1, encArg_1, encode_a__f_1, encode_f_1, encode_a__c_1, encode_g_1, encode_d_1, encode_a__h_1, encode_mark_1, encode_c_1, encode_h_1

Defined Pair Symbols:   ENCARG_1, A__F_1, A__C_1, A__H_1, MARK_1, ENCODE_A__F_1, ENCODE_A__C_1, ENCODE_A__H_1, ENCODE_MARK_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_1, c5_1, c6_2, c7_2, c8_2, c9_2, c19_1, c20, c21, c22, c23_1, c24, c25_2, c26_1, c27_2, c28, c29, c11_1


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

(13) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 4 leading nodes:
   ENCODE_A__F(z0) -> c11(ENCARG(z0))
   ENCODE_A__C(z0) -> c11(ENCARG(z0))
   ENCODE_A__H(z0) -> c11(ENCARG(z0))
   ENCODE_MARK(z0) -> c11(ENCARG(z0))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(f(z0)) -> f(encArg(z0))
   encArg(g(z0)) -> g(encArg(z0))
   encArg(d(z0)) -> d(encArg(z0))
   encArg(c(z0)) -> c(encArg(z0))
   encArg(h(z0)) -> h(encArg(z0))
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_a__c(z0)) -> a__c(encArg(z0))
   encArg(cons_a__h(z0)) -> a__h(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__c(z0) -> a__c(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_d(z0) -> d(encArg(z0))
   encode_a__h(z0) -> a__h(encArg(z0))
   encode_mark(z0) -> mark(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   encode_h(z0) -> h(encArg(z0))
   a__f(f(z0)) -> a__c(f(g(f(z0))))
   a__f(z0) -> f(z0)
   a__c(z0) -> d(z0)
   a__c(z0) -> c(z0)
   a__h(z0) -> a__c(d(z0))
   a__h(z0) -> h(z0)
   mark(f(z0)) -> a__f(mark(z0))
   mark(c(z0)) -> a__c(z0)
   mark(h(z0)) -> a__h(mark(z0))
   mark(g(z0)) -> g(z0)
   mark(d(z0)) -> d(z0)
Tuples:
   ENCARG(f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(d(z0)) -> c3(ENCARG(z0))
   ENCARG(c(z0)) -> c4(ENCARG(z0))
   ENCARG(h(z0)) -> c5(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c6(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__c(z0)) -> c7(A__C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__h(z0)) -> c8(A__H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c9(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__C(z0) -> c21
   A__C(z0) -> c22
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
   ENCODE_A__F(z0) -> c11(A__F(encArg(z0)))
   ENCODE_A__C(z0) -> c11(A__C(encArg(z0)))
   ENCODE_A__H(z0) -> c11(A__H(encArg(z0)))
   ENCODE_MARK(z0) -> c11(MARK(encArg(z0)))
S tuples:
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__C(z0) -> c21
   A__C(z0) -> c22
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
K tuples:none
Defined Rule Symbols:   a__f_1, a__c_1, a__h_1, mark_1, encArg_1, encode_a__f_1, encode_f_1, encode_a__c_1, encode_g_1, encode_d_1, encode_a__h_1, encode_mark_1, encode_c_1, encode_h_1

Defined Pair Symbols:   ENCARG_1, A__F_1, A__C_1, A__H_1, MARK_1, ENCODE_A__F_1, ENCODE_A__C_1, ENCODE_A__H_1, ENCODE_MARK_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_1, c5_1, c6_2, c7_2, c8_2, c9_2, c19_1, c20, c21, c22, c23_1, c24, c25_2, c26_1, c27_2, c28, c29, c11_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__c(z0) -> a__c(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_d(z0) -> d(encArg(z0))
   encode_a__h(z0) -> a__h(encArg(z0))
   encode_mark(z0) -> mark(encArg(z0))
   encode_c(z0) -> c(encArg(z0))
   encode_h(z0) -> h(encArg(z0))

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(f(z0)) -> f(encArg(z0))
   encArg(g(z0)) -> g(encArg(z0))
   encArg(d(z0)) -> d(encArg(z0))
   encArg(c(z0)) -> c(encArg(z0))
   encArg(h(z0)) -> h(encArg(z0))
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_a__c(z0)) -> a__c(encArg(z0))
   encArg(cons_a__h(z0)) -> a__h(encArg(z0))
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__f(f(z0)) -> a__c(f(g(f(z0))))
   a__f(z0) -> f(z0)
   a__c(z0) -> d(z0)
   a__c(z0) -> c(z0)
   a__h(z0) -> a__c(d(z0))
   a__h(z0) -> h(z0)
   mark(f(z0)) -> a__f(mark(z0))
   mark(c(z0)) -> a__c(z0)
   mark(h(z0)) -> a__h(mark(z0))
   mark(g(z0)) -> g(z0)
   mark(d(z0)) -> d(z0)
Tuples:
   ENCARG(f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(d(z0)) -> c3(ENCARG(z0))
   ENCARG(c(z0)) -> c4(ENCARG(z0))
   ENCARG(h(z0)) -> c5(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c6(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__c(z0)) -> c7(A__C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__h(z0)) -> c8(A__H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c9(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__C(z0) -> c21
   A__C(z0) -> c22
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
   ENCODE_A__F(z0) -> c11(A__F(encArg(z0)))
   ENCODE_A__C(z0) -> c11(A__C(encArg(z0)))
   ENCODE_A__H(z0) -> c11(A__H(encArg(z0)))
   ENCODE_MARK(z0) -> c11(MARK(encArg(z0)))
S tuples:
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__C(z0) -> c21
   A__C(z0) -> c22
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
K tuples:none
Defined Rule Symbols:   encArg_1, a__f_1, a__c_1, a__h_1, mark_1

Defined Pair Symbols:   ENCARG_1, A__F_1, A__C_1, A__H_1, MARK_1, ENCODE_A__F_1, ENCODE_A__C_1, ENCODE_A__H_1, ENCODE_MARK_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_1, c5_1, c6_2, c7_2, c8_2, c9_2, c19_1, c20, c21, c22, c23_1, c24, c25_2, c26_1, c27_2, c28, c29, c11_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(c(z0)) -> c26(A__C(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(d(z0)) -> c3(ENCARG(z0))
   ENCARG(c(z0)) -> c4(ENCARG(z0))
   ENCARG(h(z0)) -> c5(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c6(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__c(z0)) -> c7(A__C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__h(z0)) -> c8(A__H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c9(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__C(z0) -> c21
   A__C(z0) -> c22
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
   ENCODE_A__F(z0) -> c11(A__F(encArg(z0)))
   ENCODE_A__C(z0) -> c11(A__C(encArg(z0)))
   ENCODE_A__H(z0) -> c11(A__H(encArg(z0)))
   ENCODE_MARK(z0) -> c11(MARK(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A__C(x_1)) = 0
   POL(A__F(x_1)) = 0
   POL(A__H(x_1)) = 0
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_A__C(x_1)) = 0
   POL(ENCODE_A__F(x_1)) = 0
   POL(ENCODE_A__H(x_1)) = 0
   POL(ENCODE_MARK(x_1)) = [1]
   POL(MARK(x_1)) = [1]
   POL(a__c(x_1)) = [1] + x_1
   POL(a__f(x_1)) = [1]
   POL(a__h(x_1)) = [1] + x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20) = 0
   POL(c21) = 0
   POL(c22) = 0
   POL(c23(x_1)) = x_1
   POL(c24) = 0
   POL(c25(x_1, x_2)) = x_1 + x_2
   POL(c26(x_1)) = x_1
   POL(c27(x_1, x_2)) = x_1 + x_2
   POL(c28) = 0
   POL(c29) = 0
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(cons_a__c(x_1)) = [1] + x_1
   POL(cons_a__f(x_1)) = [1] + x_1
   POL(cons_a__h(x_1)) = [1] + x_1
   POL(cons_mark(x_1)) = [1] + x_1
   POL(d(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(f(x_1)) = x_1
   POL(g(x_1)) = [1] + x_1
   POL(h(x_1)) = x_1
   POL(mark(x_1)) = x_1

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(f(z0)) -> f(encArg(z0))
   encArg(g(z0)) -> g(encArg(z0))
   encArg(d(z0)) -> d(encArg(z0))
   encArg(c(z0)) -> c(encArg(z0))
   encArg(h(z0)) -> h(encArg(z0))
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_a__c(z0)) -> a__c(encArg(z0))
   encArg(cons_a__h(z0)) -> a__h(encArg(z0))
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__f(f(z0)) -> a__c(f(g(f(z0))))
   a__f(z0) -> f(z0)
   a__c(z0) -> d(z0)
   a__c(z0) -> c(z0)
   a__h(z0) -> a__c(d(z0))
   a__h(z0) -> h(z0)
   mark(f(z0)) -> a__f(mark(z0))
   mark(c(z0)) -> a__c(z0)
   mark(h(z0)) -> a__h(mark(z0))
   mark(g(z0)) -> g(z0)
   mark(d(z0)) -> d(z0)
Tuples:
   ENCARG(f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(d(z0)) -> c3(ENCARG(z0))
   ENCARG(c(z0)) -> c4(ENCARG(z0))
   ENCARG(h(z0)) -> c5(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c6(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__c(z0)) -> c7(A__C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__h(z0)) -> c8(A__H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c9(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__C(z0) -> c21
   A__C(z0) -> c22
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
   ENCODE_A__F(z0) -> c11(A__F(encArg(z0)))
   ENCODE_A__C(z0) -> c11(A__C(encArg(z0)))
   ENCODE_A__H(z0) -> c11(A__H(encArg(z0)))
   ENCODE_MARK(z0) -> c11(MARK(encArg(z0)))
S tuples:
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__C(z0) -> c21
   A__C(z0) -> c22
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
K tuples:
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
Defined Rule Symbols:   encArg_1, a__f_1, a__c_1, a__h_1, mark_1

Defined Pair Symbols:   ENCARG_1, A__F_1, A__C_1, A__H_1, MARK_1, ENCODE_A__F_1, ENCODE_A__C_1, ENCODE_A__H_1, ENCODE_MARK_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_1, c5_1, c6_2, c7_2, c8_2, c9_2, c19_1, c20, c21, c22, c23_1, c24, c25_2, c26_1, c27_2, c28, c29, c11_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(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
We considered the (Usable) Rules:
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_a__h(z0)) -> a__h(encArg(z0))
   mark(d(z0)) -> d(z0)
   encArg(h(z0)) -> h(encArg(z0))
   encArg(d(z0)) -> d(encArg(z0))
   mark(h(z0)) -> a__h(mark(z0))
   mark(c(z0)) -> a__c(z0)
   encArg(g(z0)) -> g(encArg(z0))
   mark(g(z0)) -> g(z0)
   encArg(cons_a__c(z0)) -> a__c(encArg(z0))
   encArg(c(z0)) -> c(encArg(z0))
   a__f(f(z0)) -> a__c(f(g(f(z0))))
   a__c(z0) -> c(z0)
   a__c(z0) -> d(z0)
   a__f(z0) -> f(z0)
   a__h(z0) -> a__c(d(z0))
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   mark(f(z0)) -> a__f(mark(z0))
   a__h(z0) -> h(z0)
   encArg(f(z0)) -> f(encArg(z0))
And the Tuples:
   ENCARG(f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(d(z0)) -> c3(ENCARG(z0))
   ENCARG(c(z0)) -> c4(ENCARG(z0))
   ENCARG(h(z0)) -> c5(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c6(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__c(z0)) -> c7(A__C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__h(z0)) -> c8(A__H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c9(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__C(z0) -> c21
   A__C(z0) -> c22
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
   ENCODE_A__F(z0) -> c11(A__F(encArg(z0)))
   ENCODE_A__C(z0) -> c11(A__C(encArg(z0)))
   ENCODE_A__H(z0) -> c11(A__H(encArg(z0)))
   ENCODE_MARK(z0) -> c11(MARK(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A__C(x_1)) = 0
   POL(A__F(x_1)) = [2]
   POL(A__H(x_1)) = 0
   POL(ENCARG(x_1)) = [2]x_1^2
   POL(ENCODE_A__C(x_1)) = [2] + [2]x_1^2
   POL(ENCODE_A__F(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(ENCODE_A__H(x_1)) = [1] + x_1 + [2]x_1^2
   POL(ENCODE_MARK(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(MARK(x_1)) = x_1
   POL(a__c(x_1)) = x_1
   POL(a__f(x_1)) = [2] + x_1
   POL(a__h(x_1)) = x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20) = 0
   POL(c21) = 0
   POL(c22) = 0
   POL(c23(x_1)) = x_1
   POL(c24) = 0
   POL(c25(x_1, x_2)) = x_1 + x_2
   POL(c26(x_1)) = x_1
   POL(c27(x_1, x_2)) = x_1 + x_2
   POL(c28) = 0
   POL(c29) = 0
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(cons_a__c(x_1)) = x_1
   POL(cons_a__f(x_1)) = [2] + x_1
   POL(cons_a__h(x_1)) = x_1
   POL(cons_mark(x_1)) = [2] + x_1
   POL(d(x_1)) = x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1)) = [2] + x_1
   POL(g(x_1)) = x_1
   POL(h(x_1)) = x_1
   POL(mark(x_1)) = x_1

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(f(z0)) -> f(encArg(z0))
   encArg(g(z0)) -> g(encArg(z0))
   encArg(d(z0)) -> d(encArg(z0))
   encArg(c(z0)) -> c(encArg(z0))
   encArg(h(z0)) -> h(encArg(z0))
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_a__c(z0)) -> a__c(encArg(z0))
   encArg(cons_a__h(z0)) -> a__h(encArg(z0))
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__f(f(z0)) -> a__c(f(g(f(z0))))
   a__f(z0) -> f(z0)
   a__c(z0) -> d(z0)
   a__c(z0) -> c(z0)
   a__h(z0) -> a__c(d(z0))
   a__h(z0) -> h(z0)
   mark(f(z0)) -> a__f(mark(z0))
   mark(c(z0)) -> a__c(z0)
   mark(h(z0)) -> a__h(mark(z0))
   mark(g(z0)) -> g(z0)
   mark(d(z0)) -> d(z0)
Tuples:
   ENCARG(f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(d(z0)) -> c3(ENCARG(z0))
   ENCARG(c(z0)) -> c4(ENCARG(z0))
   ENCARG(h(z0)) -> c5(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c6(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__c(z0)) -> c7(A__C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__h(z0)) -> c8(A__H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c9(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__C(z0) -> c21
   A__C(z0) -> c22
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
   ENCODE_A__F(z0) -> c11(A__F(encArg(z0)))
   ENCODE_A__C(z0) -> c11(A__C(encArg(z0)))
   ENCODE_A__H(z0) -> c11(A__H(encArg(z0)))
   ENCODE_MARK(z0) -> c11(MARK(encArg(z0)))
S tuples:
   A__C(z0) -> c21
   A__C(z0) -> c22
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
K tuples:
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
Defined Rule Symbols:   encArg_1, a__f_1, a__c_1, a__h_1, mark_1

Defined Pair Symbols:   ENCARG_1, A__F_1, A__C_1, A__H_1, MARK_1, ENCODE_A__F_1, ENCODE_A__C_1, ENCODE_A__H_1, ENCODE_MARK_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_1, c5_1, c6_2, c7_2, c8_2, c9_2, c19_1, c20, c21, c22, c23_1, c24, c25_2, c26_1, c27_2, c28, c29, c11_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__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
We considered the (Usable) Rules:
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_a__h(z0)) -> a__h(encArg(z0))
   mark(d(z0)) -> d(z0)
   encArg(h(z0)) -> h(encArg(z0))
   encArg(d(z0)) -> d(encArg(z0))
   mark(h(z0)) -> a__h(mark(z0))
   mark(c(z0)) -> a__c(z0)
   encArg(g(z0)) -> g(encArg(z0))
   mark(g(z0)) -> g(z0)
   encArg(cons_a__c(z0)) -> a__c(encArg(z0))
   encArg(c(z0)) -> c(encArg(z0))
   a__f(f(z0)) -> a__c(f(g(f(z0))))
   a__c(z0) -> c(z0)
   a__c(z0) -> d(z0)
   a__f(z0) -> f(z0)
   a__h(z0) -> a__c(d(z0))
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   mark(f(z0)) -> a__f(mark(z0))
   a__h(z0) -> h(z0)
   encArg(f(z0)) -> f(encArg(z0))
And the Tuples:
   ENCARG(f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(d(z0)) -> c3(ENCARG(z0))
   ENCARG(c(z0)) -> c4(ENCARG(z0))
   ENCARG(h(z0)) -> c5(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c6(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__c(z0)) -> c7(A__C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__h(z0)) -> c8(A__H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c9(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__C(z0) -> c21
   A__C(z0) -> c22
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
   ENCODE_A__F(z0) -> c11(A__F(encArg(z0)))
   ENCODE_A__C(z0) -> c11(A__C(encArg(z0)))
   ENCODE_A__H(z0) -> c11(A__H(encArg(z0)))
   ENCODE_MARK(z0) -> c11(MARK(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A__C(x_1)) = 0
   POL(A__F(x_1)) = 0
   POL(A__H(x_1)) = [1]
   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_A__C(x_1)) = [1] + x_1^2
   POL(ENCODE_A__F(x_1)) = [2] + [2]x_1^2
   POL(ENCODE_A__H(x_1)) = [1] + [2]x_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__c(x_1)) = x_1
   POL(a__f(x_1)) = x_1
   POL(a__h(x_1)) = [1] + x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20) = 0
   POL(c21) = 0
   POL(c22) = 0
   POL(c23(x_1)) = x_1
   POL(c24) = 0
   POL(c25(x_1, x_2)) = x_1 + x_2
   POL(c26(x_1)) = x_1
   POL(c27(x_1, x_2)) = x_1 + x_2
   POL(c28) = 0
   POL(c29) = 0
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(cons_a__c(x_1)) = x_1
   POL(cons_a__f(x_1)) = x_1
   POL(cons_a__h(x_1)) = [2] + x_1
   POL(cons_mark(x_1)) = [1] + x_1
   POL(d(x_1)) = x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1)) = x_1
   POL(g(x_1)) = x_1
   POL(h(x_1)) = [1] + x_1
   POL(mark(x_1)) = x_1

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(f(z0)) -> f(encArg(z0))
   encArg(g(z0)) -> g(encArg(z0))
   encArg(d(z0)) -> d(encArg(z0))
   encArg(c(z0)) -> c(encArg(z0))
   encArg(h(z0)) -> h(encArg(z0))
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_a__c(z0)) -> a__c(encArg(z0))
   encArg(cons_a__h(z0)) -> a__h(encArg(z0))
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__f(f(z0)) -> a__c(f(g(f(z0))))
   a__f(z0) -> f(z0)
   a__c(z0) -> d(z0)
   a__c(z0) -> c(z0)
   a__h(z0) -> a__c(d(z0))
   a__h(z0) -> h(z0)
   mark(f(z0)) -> a__f(mark(z0))
   mark(c(z0)) -> a__c(z0)
   mark(h(z0)) -> a__h(mark(z0))
   mark(g(z0)) -> g(z0)
   mark(d(z0)) -> d(z0)
Tuples:
   ENCARG(f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(d(z0)) -> c3(ENCARG(z0))
   ENCARG(c(z0)) -> c4(ENCARG(z0))
   ENCARG(h(z0)) -> c5(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c6(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__c(z0)) -> c7(A__C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__h(z0)) -> c8(A__H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c9(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__C(z0) -> c21
   A__C(z0) -> c22
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
   ENCODE_A__F(z0) -> c11(A__F(encArg(z0)))
   ENCODE_A__C(z0) -> c11(A__C(encArg(z0)))
   ENCODE_A__H(z0) -> c11(A__H(encArg(z0)))
   ENCODE_MARK(z0) -> c11(MARK(encArg(z0)))
S tuples:
   A__C(z0) -> c21
   A__C(z0) -> c22
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
K tuples:
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
Defined Rule Symbols:   encArg_1, a__f_1, a__c_1, a__h_1, mark_1

Defined Pair Symbols:   ENCARG_1, A__F_1, A__C_1, A__H_1, MARK_1, ENCODE_A__F_1, ENCODE_A__C_1, ENCODE_A__H_1, ENCODE_MARK_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_1, c5_1, c6_2, c7_2, c8_2, c9_2, c19_1, c20, c21, c22, c23_1, c24, c25_2, c26_1, c27_2, c28, c29, c11_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__C(z0) -> c21
   A__C(z0) -> c22
We considered the (Usable) Rules:
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_a__h(z0)) -> a__h(encArg(z0))
   mark(d(z0)) -> d(z0)
   encArg(h(z0)) -> h(encArg(z0))
   encArg(d(z0)) -> d(encArg(z0))
   mark(h(z0)) -> a__h(mark(z0))
   mark(c(z0)) -> a__c(z0)
   encArg(g(z0)) -> g(encArg(z0))
   mark(g(z0)) -> g(z0)
   encArg(cons_a__c(z0)) -> a__c(encArg(z0))
   encArg(c(z0)) -> c(encArg(z0))
   a__f(f(z0)) -> a__c(f(g(f(z0))))
   a__c(z0) -> c(z0)
   a__c(z0) -> d(z0)
   a__f(z0) -> f(z0)
   a__h(z0) -> a__c(d(z0))
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   mark(f(z0)) -> a__f(mark(z0))
   a__h(z0) -> h(z0)
   encArg(f(z0)) -> f(encArg(z0))
And the Tuples:
   ENCARG(f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(d(z0)) -> c3(ENCARG(z0))
   ENCARG(c(z0)) -> c4(ENCARG(z0))
   ENCARG(h(z0)) -> c5(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c6(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__c(z0)) -> c7(A__C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__h(z0)) -> c8(A__H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c9(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__C(z0) -> c21
   A__C(z0) -> c22
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
   ENCODE_A__F(z0) -> c11(A__F(encArg(z0)))
   ENCODE_A__C(z0) -> c11(A__C(encArg(z0)))
   ENCODE_A__H(z0) -> c11(A__H(encArg(z0)))
   ENCODE_MARK(z0) -> c11(MARK(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A__C(x_1)) = [1]
   POL(A__F(x_1)) = [2]
   POL(A__H(x_1)) = [1]
   POL(ENCARG(x_1)) = [2]x_1^2
   POL(ENCODE_A__C(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(ENCODE_A__F(x_1)) = [2] + x_1 + [2]x_1^2
   POL(ENCODE_A__H(x_1)) = [1] + [2]x_1 + x_1^2
   POL(ENCODE_MARK(x_1)) = [1] + [2]x_1 + x_1^2
   POL(MARK(x_1)) = [1] + x_1
   POL(a__c(x_1)) = x_1
   POL(a__f(x_1)) = [2] + x_1
   POL(a__h(x_1)) = [1] + x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20) = 0
   POL(c21) = 0
   POL(c22) = 0
   POL(c23(x_1)) = x_1
   POL(c24) = 0
   POL(c25(x_1, x_2)) = x_1 + x_2
   POL(c26(x_1)) = x_1
   POL(c27(x_1, x_2)) = x_1 + x_2
   POL(c28) = 0
   POL(c29) = 0
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(cons_a__c(x_1)) = [1] + x_1
   POL(cons_a__f(x_1)) = [2] + x_1
   POL(cons_a__h(x_1)) = [2] + x_1
   POL(cons_mark(x_1)) = [2] + x_1
   POL(d(x_1)) = x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1)) = [2] + x_1
   POL(g(x_1)) = x_1
   POL(h(x_1)) = [1] + x_1
   POL(mark(x_1)) = x_1

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(f(z0)) -> f(encArg(z0))
   encArg(g(z0)) -> g(encArg(z0))
   encArg(d(z0)) -> d(encArg(z0))
   encArg(c(z0)) -> c(encArg(z0))
   encArg(h(z0)) -> h(encArg(z0))
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_a__c(z0)) -> a__c(encArg(z0))
   encArg(cons_a__h(z0)) -> a__h(encArg(z0))
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__f(f(z0)) -> a__c(f(g(f(z0))))
   a__f(z0) -> f(z0)
   a__c(z0) -> d(z0)
   a__c(z0) -> c(z0)
   a__h(z0) -> a__c(d(z0))
   a__h(z0) -> h(z0)
   mark(f(z0)) -> a__f(mark(z0))
   mark(c(z0)) -> a__c(z0)
   mark(h(z0)) -> a__h(mark(z0))
   mark(g(z0)) -> g(z0)
   mark(d(z0)) -> d(z0)
Tuples:
   ENCARG(f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(d(z0)) -> c3(ENCARG(z0))
   ENCARG(c(z0)) -> c4(ENCARG(z0))
   ENCARG(h(z0)) -> c5(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c6(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__c(z0)) -> c7(A__C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__h(z0)) -> c8(A__H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c9(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__C(z0) -> c21
   A__C(z0) -> c22
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
   ENCODE_A__F(z0) -> c11(A__F(encArg(z0)))
   ENCODE_A__C(z0) -> c11(A__C(encArg(z0)))
   ENCODE_A__H(z0) -> c11(A__H(encArg(z0)))
   ENCODE_MARK(z0) -> c11(MARK(encArg(z0)))
S tuples:
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
K tuples:
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
   A__C(z0) -> c21
   A__C(z0) -> c22
Defined Rule Symbols:   encArg_1, a__f_1, a__c_1, a__h_1, mark_1

Defined Pair Symbols:   ENCARG_1, A__F_1, A__C_1, A__H_1, MARK_1, ENCODE_A__F_1, ENCODE_A__C_1, ENCODE_A__H_1, ENCODE_MARK_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_1, c5_1, c6_2, c7_2, c8_2, c9_2, c19_1, c20, c21, c22, c23_1, c24, c25_2, c26_1, c27_2, c28, c29, c11_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.
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
We considered the (Usable) Rules:
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_a__h(z0)) -> a__h(encArg(z0))
   mark(d(z0)) -> d(z0)
   encArg(h(z0)) -> h(encArg(z0))
   encArg(d(z0)) -> d(encArg(z0))
   mark(h(z0)) -> a__h(mark(z0))
   mark(c(z0)) -> a__c(z0)
   encArg(g(z0)) -> g(encArg(z0))
   mark(g(z0)) -> g(z0)
   encArg(cons_a__c(z0)) -> a__c(encArg(z0))
   encArg(c(z0)) -> c(encArg(z0))
   a__f(f(z0)) -> a__c(f(g(f(z0))))
   a__c(z0) -> c(z0)
   a__c(z0) -> d(z0)
   a__f(z0) -> f(z0)
   a__h(z0) -> a__c(d(z0))
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   mark(f(z0)) -> a__f(mark(z0))
   a__h(z0) -> h(z0)
   encArg(f(z0)) -> f(encArg(z0))
And the Tuples:
   ENCARG(f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(d(z0)) -> c3(ENCARG(z0))
   ENCARG(c(z0)) -> c4(ENCARG(z0))
   ENCARG(h(z0)) -> c5(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c6(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__c(z0)) -> c7(A__C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__h(z0)) -> c8(A__H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c9(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__C(z0) -> c21
   A__C(z0) -> c22
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
   ENCODE_A__F(z0) -> c11(A__F(encArg(z0)))
   ENCODE_A__C(z0) -> c11(A__C(encArg(z0)))
   ENCODE_A__H(z0) -> c11(A__H(encArg(z0)))
   ENCODE_MARK(z0) -> c11(MARK(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A__C(x_1)) = 0
   POL(A__F(x_1)) = 0
   POL(A__H(x_1)) = 0
   POL(ENCARG(x_1)) = [2]x_1^2
   POL(ENCODE_A__C(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(ENCODE_A__F(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(ENCODE_A__H(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(ENCODE_MARK(x_1)) = [1] + [2]x_1 + [2]x_1^2
   POL(MARK(x_1)) = x_1
   POL(a__c(x_1)) = x_1
   POL(a__f(x_1)) = [1] + x_1
   POL(a__h(x_1)) = x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20) = 0
   POL(c21) = 0
   POL(c22) = 0
   POL(c23(x_1)) = x_1
   POL(c24) = 0
   POL(c25(x_1, x_2)) = x_1 + x_2
   POL(c26(x_1)) = x_1
   POL(c27(x_1, x_2)) = x_1 + x_2
   POL(c28) = 0
   POL(c29) = 0
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(cons_a__c(x_1)) = [1] + x_1
   POL(cons_a__f(x_1)) = [2] + x_1
   POL(cons_a__h(x_1)) = x_1
   POL(cons_mark(x_1)) = [2] + x_1
   POL(d(x_1)) = x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(f(x_1)) = [1] + x_1
   POL(g(x_1)) = x_1
   POL(h(x_1)) = x_1
   POL(mark(x_1)) = x_1

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

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(f(z0)) -> f(encArg(z0))
   encArg(g(z0)) -> g(encArg(z0))
   encArg(d(z0)) -> d(encArg(z0))
   encArg(c(z0)) -> c(encArg(z0))
   encArg(h(z0)) -> h(encArg(z0))
   encArg(cons_a__f(z0)) -> a__f(encArg(z0))
   encArg(cons_a__c(z0)) -> a__c(encArg(z0))
   encArg(cons_a__h(z0)) -> a__h(encArg(z0))
   encArg(cons_mark(z0)) -> mark(encArg(z0))
   a__f(f(z0)) -> a__c(f(g(f(z0))))
   a__f(z0) -> f(z0)
   a__c(z0) -> d(z0)
   a__c(z0) -> c(z0)
   a__h(z0) -> a__c(d(z0))
   a__h(z0) -> h(z0)
   mark(f(z0)) -> a__f(mark(z0))
   mark(c(z0)) -> a__c(z0)
   mark(h(z0)) -> a__h(mark(z0))
   mark(g(z0)) -> g(z0)
   mark(d(z0)) -> d(z0)
Tuples:
   ENCARG(f(z0)) -> c1(ENCARG(z0))
   ENCARG(g(z0)) -> c2(ENCARG(z0))
   ENCARG(d(z0)) -> c3(ENCARG(z0))
   ENCARG(c(z0)) -> c4(ENCARG(z0))
   ENCARG(h(z0)) -> c5(ENCARG(z0))
   ENCARG(cons_a__f(z0)) -> c6(A__F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__c(z0)) -> c7(A__C(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a__h(z0)) -> c8(A__H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_mark(z0)) -> c9(MARK(encArg(z0)), ENCARG(z0))
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__C(z0) -> c21
   A__C(z0) -> c22
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
   ENCODE_A__F(z0) -> c11(A__F(encArg(z0)))
   ENCODE_A__C(z0) -> c11(A__C(encArg(z0)))
   ENCODE_A__H(z0) -> c11(A__H(encArg(z0)))
   ENCODE_MARK(z0) -> c11(MARK(encArg(z0)))
S tuples:none
K tuples:
   MARK(c(z0)) -> c26(A__C(z0))
   MARK(g(z0)) -> c28
   MARK(d(z0)) -> c29
   A__F(f(z0)) -> c19(A__C(f(g(f(z0)))))
   A__F(z0) -> c20
   A__H(z0) -> c23(A__C(d(z0)))
   A__H(z0) -> c24
   MARK(h(z0)) -> c27(A__H(mark(z0)), MARK(z0))
   A__C(z0) -> c21
   A__C(z0) -> c22
   MARK(f(z0)) -> c25(A__F(mark(z0)), MARK(z0))
Defined Rule Symbols:   encArg_1, a__f_1, a__c_1, a__h_1, mark_1

Defined Pair Symbols:   ENCARG_1, A__F_1, A__C_1, A__H_1, MARK_1, ENCODE_A__F_1, ENCODE_A__C_1, ENCODE_A__H_1, ENCODE_MARK_1

Compound Symbols:   c1_1, c2_1, c3_1, c4_1, c5_1, c6_2, c7_2, c8_2, c9_2, c19_1, c20, c21, c22, c23_1, c24, c25_2, c26_1, c27_2, c28, c29, c11_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(f(X)) -> a__c(f(g(f(X))))
   a__c(X) -> d(X)
   a__h(X) -> a__c(d(X))
   mark(f(X)) -> a__f(mark(X))
   mark(c(X)) -> a__c(X)
   mark(h(X)) -> a__h(mark(X))
   mark(g(X)) -> g(X)
   mark(d(X)) -> d(X)
   a__f(X) -> f(X)
   a__c(X) -> c(X)
   a__h(X) -> h(X)

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

   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_a__c(x_1)) -> a__c(encArg(x_1))
   encArg(cons_a__h(x_1)) -> a__h(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__c(x_1) -> a__c(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_a__h(x_1) -> a__h(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))

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

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

(32)
Obligation:
TRS:
Rules:
a__f(f(X)) -> a__c(f(g(f(X))))
a__c(X) -> d(X)
a__h(X) -> a__c(d(X))
mark(f(X)) -> a__f(mark(X))
mark(c(X)) -> a__c(X)
mark(h(X)) -> a__h(mark(X))
mark(g(X)) -> g(X)
mark(d(X)) -> d(X)
a__f(X) -> f(X)
a__c(X) -> c(X)
a__h(X) -> h(X)
encArg(f(x_1)) -> f(encArg(x_1))
encArg(g(x_1)) -> g(encArg(x_1))
encArg(d(x_1)) -> d(encArg(x_1))
encArg(c(x_1)) -> c(encArg(x_1))
encArg(h(x_1)) -> h(encArg(x_1))
encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
encArg(cons_a__c(x_1)) -> a__c(encArg(x_1))
encArg(cons_a__h(x_1)) -> a__h(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__c(x_1) -> a__c(encArg(x_1))
encode_g(x_1) -> g(encArg(x_1))
encode_d(x_1) -> d(encArg(x_1))
encode_a__h(x_1) -> a__h(encArg(x_1))
encode_mark(x_1) -> mark(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))
encode_h(x_1) -> h(encArg(x_1))

Types:
a__f :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
f :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
a__c :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
g :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
d :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
a__h :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
mark :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
c :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
h :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encArg :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
cons_a__f :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
cons_a__c :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
cons_a__h :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
cons_mark :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_a__f :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_f :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_a__c :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_g :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_d :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_a__h :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_mark :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_c :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_h :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
hole_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark1_0 :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
gen_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark2_0 :: Nat -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark

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

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

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

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

(34)
Obligation:
TRS:
Rules:
a__f(f(X)) -> a__c(f(g(f(X))))
a__c(X) -> d(X)
a__h(X) -> a__c(d(X))
mark(f(X)) -> a__f(mark(X))
mark(c(X)) -> a__c(X)
mark(h(X)) -> a__h(mark(X))
mark(g(X)) -> g(X)
mark(d(X)) -> d(X)
a__f(X) -> f(X)
a__c(X) -> c(X)
a__h(X) -> h(X)
encArg(f(x_1)) -> f(encArg(x_1))
encArg(g(x_1)) -> g(encArg(x_1))
encArg(d(x_1)) -> d(encArg(x_1))
encArg(c(x_1)) -> c(encArg(x_1))
encArg(h(x_1)) -> h(encArg(x_1))
encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
encArg(cons_a__c(x_1)) -> a__c(encArg(x_1))
encArg(cons_a__h(x_1)) -> a__h(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__c(x_1) -> a__c(encArg(x_1))
encode_g(x_1) -> g(encArg(x_1))
encode_d(x_1) -> d(encArg(x_1))
encode_a__h(x_1) -> a__h(encArg(x_1))
encode_mark(x_1) -> mark(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))
encode_h(x_1) -> h(encArg(x_1))

Types:
a__f :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
f :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
a__c :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
g :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
d :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
a__h :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
mark :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
c :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
h :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encArg :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
cons_a__f :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
cons_a__c :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
cons_a__h :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
cons_mark :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_a__f :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_f :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_a__c :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_g :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_d :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_a__h :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_mark :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_c :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_h :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
hole_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark1_0 :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
gen_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark2_0 :: Nat -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark


Generator Equations:
gen_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark2_0(0) <=> hole_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark1_0
gen_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark2_0(+(x, 1)) <=> f(gen_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h: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

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

(35) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
mark(gen_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark2_0(+(1, 0)))

Induction Step:
mark(gen_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
a__f(mark(gen_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark2_0(+(1, n4_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).
----------------------------------------

(36)
Complex Obligation (BEST)

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

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

TRS:
Rules:
a__f(f(X)) -> a__c(f(g(f(X))))
a__c(X) -> d(X)
a__h(X) -> a__c(d(X))
mark(f(X)) -> a__f(mark(X))
mark(c(X)) -> a__c(X)
mark(h(X)) -> a__h(mark(X))
mark(g(X)) -> g(X)
mark(d(X)) -> d(X)
a__f(X) -> f(X)
a__c(X) -> c(X)
a__h(X) -> h(X)
encArg(f(x_1)) -> f(encArg(x_1))
encArg(g(x_1)) -> g(encArg(x_1))
encArg(d(x_1)) -> d(encArg(x_1))
encArg(c(x_1)) -> c(encArg(x_1))
encArg(h(x_1)) -> h(encArg(x_1))
encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
encArg(cons_a__c(x_1)) -> a__c(encArg(x_1))
encArg(cons_a__h(x_1)) -> a__h(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__c(x_1) -> a__c(encArg(x_1))
encode_g(x_1) -> g(encArg(x_1))
encode_d(x_1) -> d(encArg(x_1))
encode_a__h(x_1) -> a__h(encArg(x_1))
encode_mark(x_1) -> mark(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))
encode_h(x_1) -> h(encArg(x_1))

Types:
a__f :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
f :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
a__c :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
g :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
d :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
a__h :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
mark :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
c :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
h :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encArg :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
cons_a__f :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
cons_a__c :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
cons_a__h :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
cons_mark :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_a__f :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_f :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_a__c :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_g :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_d :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_a__h :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_mark :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_c :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_h :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
hole_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark1_0 :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
gen_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark2_0 :: Nat -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark


Generator Equations:
gen_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark2_0(0) <=> hole_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark1_0
gen_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark2_0(+(x, 1)) <=> f(gen_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h: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

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

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

(39)
BOUNDS(n^1, INF)

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

(40)
Obligation:
TRS:
Rules:
a__f(f(X)) -> a__c(f(g(f(X))))
a__c(X) -> d(X)
a__h(X) -> a__c(d(X))
mark(f(X)) -> a__f(mark(X))
mark(c(X)) -> a__c(X)
mark(h(X)) -> a__h(mark(X))
mark(g(X)) -> g(X)
mark(d(X)) -> d(X)
a__f(X) -> f(X)
a__c(X) -> c(X)
a__h(X) -> h(X)
encArg(f(x_1)) -> f(encArg(x_1))
encArg(g(x_1)) -> g(encArg(x_1))
encArg(d(x_1)) -> d(encArg(x_1))
encArg(c(x_1)) -> c(encArg(x_1))
encArg(h(x_1)) -> h(encArg(x_1))
encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
encArg(cons_a__c(x_1)) -> a__c(encArg(x_1))
encArg(cons_a__h(x_1)) -> a__h(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__c(x_1) -> a__c(encArg(x_1))
encode_g(x_1) -> g(encArg(x_1))
encode_d(x_1) -> d(encArg(x_1))
encode_a__h(x_1) -> a__h(encArg(x_1))
encode_mark(x_1) -> mark(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))
encode_h(x_1) -> h(encArg(x_1))

Types:
a__f :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
f :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
a__c :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
g :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
d :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
a__h :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
mark :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
c :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
h :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encArg :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
cons_a__f :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
cons_a__c :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
cons_a__h :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
cons_mark :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_a__f :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_f :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_a__c :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_g :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_d :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_a__h :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_mark :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_c :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
encode_h :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
hole_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark1_0 :: f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark
gen_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark2_0 :: Nat -> f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark


Lemmas:
mark(gen_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark2_0(0) <=> hole_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark1_0
gen_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark2_0(+(x, 1)) <=> f(gen_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark2_0(x))


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

(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark2_0(+(1, n3187_0))) -> *3_0, rt in Omega(0)

Induction Base:
encArg(gen_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark2_0(+(1, 0)))

Induction Step:
encArg(gen_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark2_0(+(1, +(n3187_0, 1)))) ->_R^Omega(0)
f(encArg(gen_f:g:d:c:h:cons_a__f:cons_a__c:cons_a__h:cons_mark2_0(+(1, n3187_0)))) ->_IH
f(*3_0)

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

(42)
BOUNDS(1, INF)