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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 227 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
(7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(8) CdtProblem
(9) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CdtProblem
(11) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(12) CdtProblem
(13) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CdtProblem
(15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 104 ms]
(16) CdtProblem
(17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 42 ms]
(18) CdtProblem
(19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 41 ms]
(20) CdtProblem
(21) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 10 ms]
(22) CdtProblem
(23) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(24) CdtProblem
(25) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(26) CdtProblem
(27) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 20 ms]
(28) CdtProblem
(29) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(30) BOUNDS(1, 1)


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

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


The TRS R consists of the following rules:

   2nd(cons1(X, cons(Y, Z))) -> Y
   2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
   from(X) -> cons(X, n__from(s(X)))
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

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

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

   encArg(cons1(x_1, x_2)) -> cons1(encArg(x_1), encArg(x_2))
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(n__from(x_1)) -> n__from(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_2nd(x_1)) -> 2nd(encArg(x_1))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_2nd(x_1) -> 2nd(encArg(x_1))
   encode_cons1(x_1, x_2) -> cons1(encArg(x_1), encArg(x_2))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_n__from(x_1) -> n__from(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))


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

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


The TRS R consists of the following rules:

   2nd(cons1(X, cons(Y, Z))) -> Y
   2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
   from(X) -> cons(X, n__from(s(X)))
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

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

   encArg(cons1(x_1, x_2)) -> cons1(encArg(x_1), encArg(x_2))
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(n__from(x_1)) -> n__from(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_2nd(x_1)) -> 2nd(encArg(x_1))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_2nd(x_1) -> 2nd(encArg(x_1))
   encode_cons1(x_1, x_2) -> cons1(encArg(x_1), encArg(x_2))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_n__from(x_1) -> n__from(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))

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

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

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


The TRS R consists of the following rules:

   2nd(cons1(X, cons(Y, Z))) -> Y
   2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
   from(X) -> cons(X, n__from(s(X)))
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

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

   encArg(cons1(x_1, x_2)) -> cons1(encArg(x_1), encArg(x_2))
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(n__from(x_1)) -> n__from(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_2nd(x_1)) -> 2nd(encArg(x_1))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_2nd(x_1) -> 2nd(encArg(x_1))
   encode_cons1(x_1, x_2) -> cons1(encArg(x_1), encArg(x_2))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_n__from(x_1) -> n__from(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons1(z0, z1)) -> cons1(encArg(z0), encArg(z1))
   encArg(cons(z0, z1)) -> cons(encArg(z0), encArg(z1))
   encArg(n__from(z0)) -> n__from(encArg(z0))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_2nd(z0)) -> 2nd(encArg(z0))
   encArg(cons_from(z0)) -> from(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   encode_2nd(z0) -> 2nd(encArg(z0))
   encode_cons1(z0, z1) -> cons1(encArg(z0), encArg(z1))
   encode_cons(z0, z1) -> cons(encArg(z0), encArg(z1))
   encode_activate(z0) -> activate(encArg(z0))
   encode_from(z0) -> from(encArg(z0))
   encode_n__from(z0) -> n__from(encArg(z0))
   encode_s(z0) -> s(encArg(z0))
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, activate(z1)))
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(cons1(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(n__from(z0)) -> c2(ENCARG(z0))
   ENCARG(s(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_2nd(z0)) -> c4(2ND(encArg(z0)), ENCARG(z0))
   ENCARG(cons_from(z0)) -> c5(FROM(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c6(ACTIVATE(encArg(z0)), ENCARG(z0))
   ENCODE_2ND(z0) -> c7(2ND(encArg(z0)), ENCARG(z0))
   ENCODE_CONS1(z0, z1) -> c8(ENCARG(z0), ENCARG(z1))
   ENCODE_CONS(z0, z1) -> c9(ENCARG(z0), ENCARG(z1))
   ENCODE_ACTIVATE(z0) -> c10(ACTIVATE(encArg(z0)), ENCARG(z0))
   ENCODE_FROM(z0) -> c11(FROM(encArg(z0)), ENCARG(z0))
   ENCODE_N__FROM(z0) -> c12(ENCARG(z0))
   ENCODE_S(z0) -> c13(ENCARG(z0))
   2ND(cons1(z0, cons(z1, z2))) -> c14
   2ND(cons(z0, z1)) -> c15(2ND(cons1(z0, activate(z1))), ACTIVATE(z1))
   FROM(z0) -> c16
   FROM(z0) -> c17
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
   ACTIVATE(z0) -> c19
S tuples:
   2ND(cons1(z0, cons(z1, z2))) -> c14
   2ND(cons(z0, z1)) -> c15(2ND(cons1(z0, activate(z1))), ACTIVATE(z1))
   FROM(z0) -> c16
   FROM(z0) -> c17
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
   ACTIVATE(z0) -> c19
K tuples:none
Defined Rule Symbols:   2nd_1, from_1, activate_1, encArg_1, encode_2nd_1, encode_cons1_2, encode_cons_2, encode_activate_1, encode_from_1, encode_n__from_1, encode_s_1

Defined Pair Symbols:   ENCARG_1, ENCODE_2ND_1, ENCODE_CONS1_2, ENCODE_CONS_2, ENCODE_ACTIVATE_1, ENCODE_FROM_1, ENCODE_N__FROM_1, ENCODE_S_1, 2ND_1, FROM_1, ACTIVATE_1

Compound Symbols:   c_2, c1_2, c2_1, c3_1, c4_2, c5_2, c6_2, c7_2, c8_2, c9_2, c10_2, c11_2, c12_1, c13_1, c14, c15_2, c16, c17, c18_1, c19


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 4 leading nodes:
   ENCODE_CONS1(z0, z1) -> c8(ENCARG(z0), ENCARG(z1))
   ENCODE_CONS(z0, z1) -> c9(ENCARG(z0), ENCARG(z1))
   ENCODE_N__FROM(z0) -> c12(ENCARG(z0))
   ENCODE_S(z0) -> c13(ENCARG(z0))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons1(z0, z1)) -> cons1(encArg(z0), encArg(z1))
   encArg(cons(z0, z1)) -> cons(encArg(z0), encArg(z1))
   encArg(n__from(z0)) -> n__from(encArg(z0))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_2nd(z0)) -> 2nd(encArg(z0))
   encArg(cons_from(z0)) -> from(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   encode_2nd(z0) -> 2nd(encArg(z0))
   encode_cons1(z0, z1) -> cons1(encArg(z0), encArg(z1))
   encode_cons(z0, z1) -> cons(encArg(z0), encArg(z1))
   encode_activate(z0) -> activate(encArg(z0))
   encode_from(z0) -> from(encArg(z0))
   encode_n__from(z0) -> n__from(encArg(z0))
   encode_s(z0) -> s(encArg(z0))
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, activate(z1)))
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(cons1(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(n__from(z0)) -> c2(ENCARG(z0))
   ENCARG(s(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_2nd(z0)) -> c4(2ND(encArg(z0)), ENCARG(z0))
   ENCARG(cons_from(z0)) -> c5(FROM(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c6(ACTIVATE(encArg(z0)), ENCARG(z0))
   ENCODE_2ND(z0) -> c7(2ND(encArg(z0)), ENCARG(z0))
   ENCODE_ACTIVATE(z0) -> c10(ACTIVATE(encArg(z0)), ENCARG(z0))
   ENCODE_FROM(z0) -> c11(FROM(encArg(z0)), ENCARG(z0))
   2ND(cons1(z0, cons(z1, z2))) -> c14
   2ND(cons(z0, z1)) -> c15(2ND(cons1(z0, activate(z1))), ACTIVATE(z1))
   FROM(z0) -> c16
   FROM(z0) -> c17
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
   ACTIVATE(z0) -> c19
S tuples:
   2ND(cons1(z0, cons(z1, z2))) -> c14
   2ND(cons(z0, z1)) -> c15(2ND(cons1(z0, activate(z1))), ACTIVATE(z1))
   FROM(z0) -> c16
   FROM(z0) -> c17
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
   ACTIVATE(z0) -> c19
K tuples:none
Defined Rule Symbols:   2nd_1, from_1, activate_1, encArg_1, encode_2nd_1, encode_cons1_2, encode_cons_2, encode_activate_1, encode_from_1, encode_n__from_1, encode_s_1

Defined Pair Symbols:   ENCARG_1, ENCODE_2ND_1, ENCODE_ACTIVATE_1, ENCODE_FROM_1, 2ND_1, FROM_1, ACTIVATE_1

Compound Symbols:   c_2, c1_2, c2_1, c3_1, c4_2, c5_2, c6_2, c7_2, c10_2, c11_2, c14, c15_2, c16, c17, c18_1, c19


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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons1(z0, z1)) -> cons1(encArg(z0), encArg(z1))
   encArg(cons(z0, z1)) -> cons(encArg(z0), encArg(z1))
   encArg(n__from(z0)) -> n__from(encArg(z0))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_2nd(z0)) -> 2nd(encArg(z0))
   encArg(cons_from(z0)) -> from(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   encode_2nd(z0) -> 2nd(encArg(z0))
   encode_cons1(z0, z1) -> cons1(encArg(z0), encArg(z1))
   encode_cons(z0, z1) -> cons(encArg(z0), encArg(z1))
   encode_activate(z0) -> activate(encArg(z0))
   encode_from(z0) -> from(encArg(z0))
   encode_n__from(z0) -> n__from(encArg(z0))
   encode_s(z0) -> s(encArg(z0))
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, activate(z1)))
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(cons1(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(n__from(z0)) -> c2(ENCARG(z0))
   ENCARG(s(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_2nd(z0)) -> c4(2ND(encArg(z0)), ENCARG(z0))
   ENCARG(cons_from(z0)) -> c5(FROM(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c6(ACTIVATE(encArg(z0)), ENCARG(z0))
   2ND(cons1(z0, cons(z1, z2))) -> c14
   FROM(z0) -> c16
   FROM(z0) -> c17
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
   ACTIVATE(z0) -> c19
   ENCODE_2ND(z0) -> c8(2ND(encArg(z0)))
   ENCODE_2ND(z0) -> c8(ENCARG(z0))
   ENCODE_ACTIVATE(z0) -> c8(ACTIVATE(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c8(ENCARG(z0))
   ENCODE_FROM(z0) -> c8(FROM(encArg(z0)))
   ENCODE_FROM(z0) -> c8(ENCARG(z0))
   2ND(cons(z0, z1)) -> c8(2ND(cons1(z0, activate(z1))))
   2ND(cons(z0, z1)) -> c8(ACTIVATE(z1))
S tuples:
   2ND(cons1(z0, cons(z1, z2))) -> c14
   FROM(z0) -> c16
   FROM(z0) -> c17
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
   ACTIVATE(z0) -> c19
   2ND(cons(z0, z1)) -> c8(2ND(cons1(z0, activate(z1))))
   2ND(cons(z0, z1)) -> c8(ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:   2nd_1, from_1, activate_1, encArg_1, encode_2nd_1, encode_cons1_2, encode_cons_2, encode_activate_1, encode_from_1, encode_n__from_1, encode_s_1

Defined Pair Symbols:   ENCARG_1, 2ND_1, FROM_1, ACTIVATE_1, ENCODE_2ND_1, ENCODE_ACTIVATE_1, ENCODE_FROM_1

Compound Symbols:   c_2, c1_2, c2_1, c3_1, c4_2, c5_2, c6_2, c14, c16, c17, c18_1, c19, c8_1


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

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 3 leading nodes:
   ENCODE_2ND(z0) -> c8(ENCARG(z0))
   ENCODE_ACTIVATE(z0) -> c8(ENCARG(z0))
   ENCODE_FROM(z0) -> c8(ENCARG(z0))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons1(z0, z1)) -> cons1(encArg(z0), encArg(z1))
   encArg(cons(z0, z1)) -> cons(encArg(z0), encArg(z1))
   encArg(n__from(z0)) -> n__from(encArg(z0))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_2nd(z0)) -> 2nd(encArg(z0))
   encArg(cons_from(z0)) -> from(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   encode_2nd(z0) -> 2nd(encArg(z0))
   encode_cons1(z0, z1) -> cons1(encArg(z0), encArg(z1))
   encode_cons(z0, z1) -> cons(encArg(z0), encArg(z1))
   encode_activate(z0) -> activate(encArg(z0))
   encode_from(z0) -> from(encArg(z0))
   encode_n__from(z0) -> n__from(encArg(z0))
   encode_s(z0) -> s(encArg(z0))
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, activate(z1)))
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(cons1(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(n__from(z0)) -> c2(ENCARG(z0))
   ENCARG(s(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_2nd(z0)) -> c4(2ND(encArg(z0)), ENCARG(z0))
   ENCARG(cons_from(z0)) -> c5(FROM(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c6(ACTIVATE(encArg(z0)), ENCARG(z0))
   2ND(cons1(z0, cons(z1, z2))) -> c14
   FROM(z0) -> c16
   FROM(z0) -> c17
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
   ACTIVATE(z0) -> c19
   ENCODE_2ND(z0) -> c8(2ND(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c8(ACTIVATE(encArg(z0)))
   ENCODE_FROM(z0) -> c8(FROM(encArg(z0)))
   2ND(cons(z0, z1)) -> c8(2ND(cons1(z0, activate(z1))))
   2ND(cons(z0, z1)) -> c8(ACTIVATE(z1))
S tuples:
   2ND(cons1(z0, cons(z1, z2))) -> c14
   FROM(z0) -> c16
   FROM(z0) -> c17
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
   ACTIVATE(z0) -> c19
   2ND(cons(z0, z1)) -> c8(2ND(cons1(z0, activate(z1))))
   2ND(cons(z0, z1)) -> c8(ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:   2nd_1, from_1, activate_1, encArg_1, encode_2nd_1, encode_cons1_2, encode_cons_2, encode_activate_1, encode_from_1, encode_n__from_1, encode_s_1

Defined Pair Symbols:   ENCARG_1, 2ND_1, FROM_1, ACTIVATE_1, ENCODE_2ND_1, ENCODE_ACTIVATE_1, ENCODE_FROM_1

Compound Symbols:   c_2, c1_2, c2_1, c3_1, c4_2, c5_2, c6_2, c14, c16, c17, c18_1, c19, c8_1


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

(13) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_2nd(z0) -> 2nd(encArg(z0))
   encode_cons1(z0, z1) -> cons1(encArg(z0), encArg(z1))
   encode_cons(z0, z1) -> cons(encArg(z0), encArg(z1))
   encode_activate(z0) -> activate(encArg(z0))
   encode_from(z0) -> from(encArg(z0))
   encode_n__from(z0) -> n__from(encArg(z0))
   encode_s(z0) -> s(encArg(z0))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons1(z0, z1)) -> cons1(encArg(z0), encArg(z1))
   encArg(cons(z0, z1)) -> cons(encArg(z0), encArg(z1))
   encArg(n__from(z0)) -> n__from(encArg(z0))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_2nd(z0)) -> 2nd(encArg(z0))
   encArg(cons_from(z0)) -> from(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, activate(z1)))
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(cons1(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(n__from(z0)) -> c2(ENCARG(z0))
   ENCARG(s(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_2nd(z0)) -> c4(2ND(encArg(z0)), ENCARG(z0))
   ENCARG(cons_from(z0)) -> c5(FROM(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c6(ACTIVATE(encArg(z0)), ENCARG(z0))
   2ND(cons1(z0, cons(z1, z2))) -> c14
   FROM(z0) -> c16
   FROM(z0) -> c17
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
   ACTIVATE(z0) -> c19
   ENCODE_2ND(z0) -> c8(2ND(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c8(ACTIVATE(encArg(z0)))
   ENCODE_FROM(z0) -> c8(FROM(encArg(z0)))
   2ND(cons(z0, z1)) -> c8(2ND(cons1(z0, activate(z1))))
   2ND(cons(z0, z1)) -> c8(ACTIVATE(z1))
S tuples:
   2ND(cons1(z0, cons(z1, z2))) -> c14
   FROM(z0) -> c16
   FROM(z0) -> c17
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
   ACTIVATE(z0) -> c19
   2ND(cons(z0, z1)) -> c8(2ND(cons1(z0, activate(z1))))
   2ND(cons(z0, z1)) -> c8(ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:   encArg_1, 2nd_1, from_1, activate_1

Defined Pair Symbols:   ENCARG_1, 2ND_1, FROM_1, ACTIVATE_1, ENCODE_2ND_1, ENCODE_ACTIVATE_1, ENCODE_FROM_1

Compound Symbols:   c_2, c1_2, c2_1, c3_1, c4_2, c5_2, c6_2, c14, c16, c17, c18_1, c19, c8_1


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

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   2ND(cons1(z0, cons(z1, z2))) -> c14
   FROM(z0) -> c16
   FROM(z0) -> c17
   ACTIVATE(z0) -> c19
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons1(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(n__from(z0)) -> c2(ENCARG(z0))
   ENCARG(s(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_2nd(z0)) -> c4(2ND(encArg(z0)), ENCARG(z0))
   ENCARG(cons_from(z0)) -> c5(FROM(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c6(ACTIVATE(encArg(z0)), ENCARG(z0))
   2ND(cons1(z0, cons(z1, z2))) -> c14
   FROM(z0) -> c16
   FROM(z0) -> c17
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
   ACTIVATE(z0) -> c19
   ENCODE_2ND(z0) -> c8(2ND(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c8(ACTIVATE(encArg(z0)))
   ENCODE_FROM(z0) -> c8(FROM(encArg(z0)))
   2ND(cons(z0, z1)) -> c8(2ND(cons1(z0, activate(z1))))
   2ND(cons(z0, z1)) -> c8(ACTIVATE(z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(2ND(x_1)) = [1]
   POL(2nd(x_1)) = [1] + x_1
   POL(ACTIVATE(x_1)) = [1]
   POL(ENCARG(x_1)) = [1] + x_1
   POL(ENCODE_2ND(x_1)) = [1]
   POL(ENCODE_ACTIVATE(x_1)) = [1]
   POL(ENCODE_FROM(x_1)) = [1]
   POL(FROM(x_1)) = [1]
   POL(activate(x_1)) = [1] + x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c14) = 0
   POL(c16) = 0
   POL(c17) = 0
   POL(c18(x_1)) = x_1
   POL(c19) = 0
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons1(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_2nd(x_1)) = [1] + x_1
   POL(cons_activate(x_1)) = [1] + x_1
   POL(cons_from(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(from(x_1)) = [1] + x_1
   POL(n__from(x_1)) = [1] + x_1
   POL(s(x_1)) = [1] + x_1

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons1(z0, z1)) -> cons1(encArg(z0), encArg(z1))
   encArg(cons(z0, z1)) -> cons(encArg(z0), encArg(z1))
   encArg(n__from(z0)) -> n__from(encArg(z0))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_2nd(z0)) -> 2nd(encArg(z0))
   encArg(cons_from(z0)) -> from(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, activate(z1)))
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(cons1(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(n__from(z0)) -> c2(ENCARG(z0))
   ENCARG(s(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_2nd(z0)) -> c4(2ND(encArg(z0)), ENCARG(z0))
   ENCARG(cons_from(z0)) -> c5(FROM(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c6(ACTIVATE(encArg(z0)), ENCARG(z0))
   2ND(cons1(z0, cons(z1, z2))) -> c14
   FROM(z0) -> c16
   FROM(z0) -> c17
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
   ACTIVATE(z0) -> c19
   ENCODE_2ND(z0) -> c8(2ND(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c8(ACTIVATE(encArg(z0)))
   ENCODE_FROM(z0) -> c8(FROM(encArg(z0)))
   2ND(cons(z0, z1)) -> c8(2ND(cons1(z0, activate(z1))))
   2ND(cons(z0, z1)) -> c8(ACTIVATE(z1))
S tuples:
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
   2ND(cons(z0, z1)) -> c8(2ND(cons1(z0, activate(z1))))
   2ND(cons(z0, z1)) -> c8(ACTIVATE(z1))
K tuples:
   2ND(cons1(z0, cons(z1, z2))) -> c14
   FROM(z0) -> c16
   FROM(z0) -> c17
   ACTIVATE(z0) -> c19
Defined Rule Symbols:   encArg_1, 2nd_1, from_1, activate_1

Defined Pair Symbols:   ENCARG_1, 2ND_1, FROM_1, ACTIVATE_1, ENCODE_2ND_1, ENCODE_ACTIVATE_1, ENCODE_FROM_1

Compound Symbols:   c_2, c1_2, c2_1, c3_1, c4_2, c5_2, c6_2, c14, c16, c17, c18_1, c19, c8_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.
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons1(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(n__from(z0)) -> c2(ENCARG(z0))
   ENCARG(s(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_2nd(z0)) -> c4(2ND(encArg(z0)), ENCARG(z0))
   ENCARG(cons_from(z0)) -> c5(FROM(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c6(ACTIVATE(encArg(z0)), ENCARG(z0))
   2ND(cons1(z0, cons(z1, z2))) -> c14
   FROM(z0) -> c16
   FROM(z0) -> c17
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
   ACTIVATE(z0) -> c19
   ENCODE_2ND(z0) -> c8(2ND(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c8(ACTIVATE(encArg(z0)))
   ENCODE_FROM(z0) -> c8(FROM(encArg(z0)))
   2ND(cons(z0, z1)) -> c8(2ND(cons1(z0, activate(z1))))
   2ND(cons(z0, z1)) -> c8(ACTIVATE(z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(2ND(x_1)) = [1]
   POL(2nd(x_1)) = [1] + x_1
   POL(ACTIVATE(x_1)) = [1]
   POL(ENCARG(x_1)) = [1] + x_1
   POL(ENCODE_2ND(x_1)) = [1]
   POL(ENCODE_ACTIVATE(x_1)) = [1]
   POL(ENCODE_FROM(x_1)) = 0
   POL(FROM(x_1)) = 0
   POL(activate(x_1)) = [1] + x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c14) = 0
   POL(c16) = 0
   POL(c17) = 0
   POL(c18(x_1)) = x_1
   POL(c19) = 0
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons1(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_2nd(x_1)) = [1] + x_1
   POL(cons_activate(x_1)) = [1] + x_1
   POL(cons_from(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(from(x_1)) = [1] + x_1
   POL(n__from(x_1)) = [1] + x_1
   POL(s(x_1)) = [1] + x_1

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons1(z0, z1)) -> cons1(encArg(z0), encArg(z1))
   encArg(cons(z0, z1)) -> cons(encArg(z0), encArg(z1))
   encArg(n__from(z0)) -> n__from(encArg(z0))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_2nd(z0)) -> 2nd(encArg(z0))
   encArg(cons_from(z0)) -> from(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, activate(z1)))
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(cons1(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(n__from(z0)) -> c2(ENCARG(z0))
   ENCARG(s(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_2nd(z0)) -> c4(2ND(encArg(z0)), ENCARG(z0))
   ENCARG(cons_from(z0)) -> c5(FROM(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c6(ACTIVATE(encArg(z0)), ENCARG(z0))
   2ND(cons1(z0, cons(z1, z2))) -> c14
   FROM(z0) -> c16
   FROM(z0) -> c17
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
   ACTIVATE(z0) -> c19
   ENCODE_2ND(z0) -> c8(2ND(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c8(ACTIVATE(encArg(z0)))
   ENCODE_FROM(z0) -> c8(FROM(encArg(z0)))
   2ND(cons(z0, z1)) -> c8(2ND(cons1(z0, activate(z1))))
   2ND(cons(z0, z1)) -> c8(ACTIVATE(z1))
S tuples:
   2ND(cons(z0, z1)) -> c8(2ND(cons1(z0, activate(z1))))
   2ND(cons(z0, z1)) -> c8(ACTIVATE(z1))
K tuples:
   2ND(cons1(z0, cons(z1, z2))) -> c14
   FROM(z0) -> c16
   FROM(z0) -> c17
   ACTIVATE(z0) -> c19
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
Defined Rule Symbols:   encArg_1, 2nd_1, from_1, activate_1

Defined Pair Symbols:   ENCARG_1, 2ND_1, FROM_1, ACTIVATE_1, ENCODE_2ND_1, ENCODE_ACTIVATE_1, ENCODE_FROM_1

Compound Symbols:   c_2, c1_2, c2_1, c3_1, c4_2, c5_2, c6_2, c14, c16, c17, c18_1, c19, c8_1


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

(19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   2ND(cons(z0, z1)) -> c8(ACTIVATE(z1))
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons1(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(n__from(z0)) -> c2(ENCARG(z0))
   ENCARG(s(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_2nd(z0)) -> c4(2ND(encArg(z0)), ENCARG(z0))
   ENCARG(cons_from(z0)) -> c5(FROM(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c6(ACTIVATE(encArg(z0)), ENCARG(z0))
   2ND(cons1(z0, cons(z1, z2))) -> c14
   FROM(z0) -> c16
   FROM(z0) -> c17
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
   ACTIVATE(z0) -> c19
   ENCODE_2ND(z0) -> c8(2ND(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c8(ACTIVATE(encArg(z0)))
   ENCODE_FROM(z0) -> c8(FROM(encArg(z0)))
   2ND(cons(z0, z1)) -> c8(2ND(cons1(z0, activate(z1))))
   2ND(cons(z0, z1)) -> c8(ACTIVATE(z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(2ND(x_1)) = [1]
   POL(2nd(x_1)) = [1] + x_1
   POL(ACTIVATE(x_1)) = 0
   POL(ENCARG(x_1)) = [1] + x_1
   POL(ENCODE_2ND(x_1)) = [1]
   POL(ENCODE_ACTIVATE(x_1)) = 0
   POL(ENCODE_FROM(x_1)) = 0
   POL(FROM(x_1)) = 0
   POL(activate(x_1)) = [1] + x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c14) = 0
   POL(c16) = 0
   POL(c17) = 0
   POL(c18(x_1)) = x_1
   POL(c19) = 0
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons1(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_2nd(x_1)) = [1] + x_1
   POL(cons_activate(x_1)) = [1] + x_1
   POL(cons_from(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(from(x_1)) = [1] + x_1
   POL(n__from(x_1)) = [1] + x_1
   POL(s(x_1)) = [1] + x_1

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons1(z0, z1)) -> cons1(encArg(z0), encArg(z1))
   encArg(cons(z0, z1)) -> cons(encArg(z0), encArg(z1))
   encArg(n__from(z0)) -> n__from(encArg(z0))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_2nd(z0)) -> 2nd(encArg(z0))
   encArg(cons_from(z0)) -> from(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, activate(z1)))
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(cons1(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(n__from(z0)) -> c2(ENCARG(z0))
   ENCARG(s(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_2nd(z0)) -> c4(2ND(encArg(z0)), ENCARG(z0))
   ENCARG(cons_from(z0)) -> c5(FROM(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c6(ACTIVATE(encArg(z0)), ENCARG(z0))
   2ND(cons1(z0, cons(z1, z2))) -> c14
   FROM(z0) -> c16
   FROM(z0) -> c17
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
   ACTIVATE(z0) -> c19
   ENCODE_2ND(z0) -> c8(2ND(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c8(ACTIVATE(encArg(z0)))
   ENCODE_FROM(z0) -> c8(FROM(encArg(z0)))
   2ND(cons(z0, z1)) -> c8(2ND(cons1(z0, activate(z1))))
   2ND(cons(z0, z1)) -> c8(ACTIVATE(z1))
S tuples:
   2ND(cons(z0, z1)) -> c8(2ND(cons1(z0, activate(z1))))
K tuples:
   2ND(cons1(z0, cons(z1, z2))) -> c14
   FROM(z0) -> c16
   FROM(z0) -> c17
   ACTIVATE(z0) -> c19
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
   2ND(cons(z0, z1)) -> c8(ACTIVATE(z1))
Defined Rule Symbols:   encArg_1, 2nd_1, from_1, activate_1

Defined Pair Symbols:   ENCARG_1, 2ND_1, FROM_1, ACTIVATE_1, ENCODE_2ND_1, ENCODE_ACTIVATE_1, ENCODE_FROM_1

Compound Symbols:   c_2, c1_2, c2_1, c3_1, c4_2, c5_2, c6_2, c14, c16, c17, c18_1, c19, c8_1


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

(21) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_2nd(z0)) -> c4(2ND(encArg(z0)), ENCARG(z0)) by 
   ENCARG(cons_2nd(cons1(z0, z1))) -> c4(2ND(cons1(encArg(z0), encArg(z1))), ENCARG(cons1(z0, z1)))
   ENCARG(cons_2nd(cons(z0, z1))) -> c4(2ND(cons(encArg(z0), encArg(z1))), ENCARG(cons(z0, z1)))
   ENCARG(cons_2nd(n__from(z0))) -> c4(2ND(n__from(encArg(z0))), ENCARG(n__from(z0)))
   ENCARG(cons_2nd(s(z0))) -> c4(2ND(s(encArg(z0))), ENCARG(s(z0)))
   ENCARG(cons_2nd(cons_2nd(z0))) -> c4(2ND(2nd(encArg(z0))), ENCARG(cons_2nd(z0)))
   ENCARG(cons_2nd(cons_from(z0))) -> c4(2ND(from(encArg(z0))), ENCARG(cons_from(z0)))
   ENCARG(cons_2nd(cons_activate(z0))) -> c4(2ND(activate(encArg(z0))), ENCARG(cons_activate(z0)))

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons1(z0, z1)) -> cons1(encArg(z0), encArg(z1))
   encArg(cons(z0, z1)) -> cons(encArg(z0), encArg(z1))
   encArg(n__from(z0)) -> n__from(encArg(z0))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_2nd(z0)) -> 2nd(encArg(z0))
   encArg(cons_from(z0)) -> from(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, activate(z1)))
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(cons1(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(n__from(z0)) -> c2(ENCARG(z0))
   ENCARG(s(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_from(z0)) -> c5(FROM(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c6(ACTIVATE(encArg(z0)), ENCARG(z0))
   2ND(cons1(z0, cons(z1, z2))) -> c14
   FROM(z0) -> c16
   FROM(z0) -> c17
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
   ACTIVATE(z0) -> c19
   ENCODE_2ND(z0) -> c8(2ND(encArg(z0)))
   ENCODE_ACTIVATE(z0) -> c8(ACTIVATE(encArg(z0)))
   ENCODE_FROM(z0) -> c8(FROM(encArg(z0)))
   2ND(cons(z0, z1)) -> c8(2ND(cons1(z0, activate(z1))))
   2ND(cons(z0, z1)) -> c8(ACTIVATE(z1))
   ENCARG(cons_2nd(cons1(z0, z1))) -> c4(2ND(cons1(encArg(z0), encArg(z1))), ENCARG(cons1(z0, z1)))
   ENCARG(cons_2nd(cons(z0, z1))) -> c4(2ND(cons(encArg(z0), encArg(z1))), ENCARG(cons(z0, z1)))
   ENCARG(cons_2nd(n__from(z0))) -> c4(2ND(n__from(encArg(z0))), ENCARG(n__from(z0)))
   ENCARG(cons_2nd(s(z0))) -> c4(2ND(s(encArg(z0))), ENCARG(s(z0)))
   ENCARG(cons_2nd(cons_2nd(z0))) -> c4(2ND(2nd(encArg(z0))), ENCARG(cons_2nd(z0)))
   ENCARG(cons_2nd(cons_from(z0))) -> c4(2ND(from(encArg(z0))), ENCARG(cons_from(z0)))
   ENCARG(cons_2nd(cons_activate(z0))) -> c4(2ND(activate(encArg(z0))), ENCARG(cons_activate(z0)))
S tuples:
   2ND(cons(z0, z1)) -> c8(2ND(cons1(z0, activate(z1))))
K tuples:
   2ND(cons1(z0, cons(z1, z2))) -> c14
   FROM(z0) -> c16
   FROM(z0) -> c17
   ACTIVATE(z0) -> c19
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
   2ND(cons(z0, z1)) -> c8(ACTIVATE(z1))
Defined Rule Symbols:   encArg_1, 2nd_1, from_1, activate_1

Defined Pair Symbols:   ENCARG_1, 2ND_1, FROM_1, ACTIVATE_1, ENCODE_2ND_1, ENCODE_ACTIVATE_1, ENCODE_FROM_1

Compound Symbols:   c_2, c1_2, c2_1, c3_1, c5_2, c6_2, c14, c16, c17, c18_1, c19, c8_1, c4_2


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

(23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 8 trailing nodes:
   2ND(cons1(z0, cons(z1, z2))) -> c14
   FROM(z0) -> c16
   2ND(cons(z0, z1)) -> c8(ACTIVATE(z1))
   FROM(z0) -> c17
   ACTIVATE(z0) -> c19
   ENCODE_FROM(z0) -> c8(FROM(encArg(z0)))
   ACTIVATE(n__from(z0)) -> c18(FROM(z0))
   ENCODE_ACTIVATE(z0) -> c8(ACTIVATE(encArg(z0)))

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons1(z0, z1)) -> cons1(encArg(z0), encArg(z1))
   encArg(cons(z0, z1)) -> cons(encArg(z0), encArg(z1))
   encArg(n__from(z0)) -> n__from(encArg(z0))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_2nd(z0)) -> 2nd(encArg(z0))
   encArg(cons_from(z0)) -> from(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, activate(z1)))
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(cons1(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(n__from(z0)) -> c2(ENCARG(z0))
   ENCARG(s(z0)) -> c3(ENCARG(z0))
   ENCARG(cons_from(z0)) -> c5(FROM(encArg(z0)), ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c6(ACTIVATE(encArg(z0)), ENCARG(z0))
   ENCODE_2ND(z0) -> c8(2ND(encArg(z0)))
   2ND(cons(z0, z1)) -> c8(2ND(cons1(z0, activate(z1))))
   ENCARG(cons_2nd(cons1(z0, z1))) -> c4(2ND(cons1(encArg(z0), encArg(z1))), ENCARG(cons1(z0, z1)))
   ENCARG(cons_2nd(cons(z0, z1))) -> c4(2ND(cons(encArg(z0), encArg(z1))), ENCARG(cons(z0, z1)))
   ENCARG(cons_2nd(n__from(z0))) -> c4(2ND(n__from(encArg(z0))), ENCARG(n__from(z0)))
   ENCARG(cons_2nd(s(z0))) -> c4(2ND(s(encArg(z0))), ENCARG(s(z0)))
   ENCARG(cons_2nd(cons_2nd(z0))) -> c4(2ND(2nd(encArg(z0))), ENCARG(cons_2nd(z0)))
   ENCARG(cons_2nd(cons_from(z0))) -> c4(2ND(from(encArg(z0))), ENCARG(cons_from(z0)))
   ENCARG(cons_2nd(cons_activate(z0))) -> c4(2ND(activate(encArg(z0))), ENCARG(cons_activate(z0)))
S tuples:
   2ND(cons(z0, z1)) -> c8(2ND(cons1(z0, activate(z1))))
K tuples:none
Defined Rule Symbols:   encArg_1, 2nd_1, from_1, activate_1

Defined Pair Symbols:   ENCARG_1, ENCODE_2ND_1, 2ND_1

Compound Symbols:   c_2, c1_2, c2_1, c3_1, c5_2, c6_2, c8_1, c4_2


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

(25) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing tuple parts
----------------------------------------

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons1(z0, z1)) -> cons1(encArg(z0), encArg(z1))
   encArg(cons(z0, z1)) -> cons(encArg(z0), encArg(z1))
   encArg(n__from(z0)) -> n__from(encArg(z0))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_2nd(z0)) -> 2nd(encArg(z0))
   encArg(cons_from(z0)) -> from(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, activate(z1)))
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(cons1(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(n__from(z0)) -> c2(ENCARG(z0))
   ENCARG(s(z0)) -> c3(ENCARG(z0))
   ENCODE_2ND(z0) -> c8(2ND(encArg(z0)))
   ENCARG(cons_2nd(cons(z0, z1))) -> c4(2ND(cons(encArg(z0), encArg(z1))), ENCARG(cons(z0, z1)))
   ENCARG(cons_2nd(cons_2nd(z0))) -> c4(2ND(2nd(encArg(z0))), ENCARG(cons_2nd(z0)))
   ENCARG(cons_2nd(cons_from(z0))) -> c4(2ND(from(encArg(z0))), ENCARG(cons_from(z0)))
   ENCARG(cons_2nd(cons_activate(z0))) -> c4(2ND(activate(encArg(z0))), ENCARG(cons_activate(z0)))
   ENCARG(cons_from(z0)) -> c5(ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c6(ENCARG(z0))
   2ND(cons(z0, z1)) -> c8
   ENCARG(cons_2nd(cons1(z0, z1))) -> c4(ENCARG(cons1(z0, z1)))
   ENCARG(cons_2nd(n__from(z0))) -> c4(ENCARG(n__from(z0)))
   ENCARG(cons_2nd(s(z0))) -> c4(ENCARG(s(z0)))
S tuples:
   2ND(cons(z0, z1)) -> c8
K tuples:none
Defined Rule Symbols:   encArg_1, 2nd_1, from_1, activate_1

Defined Pair Symbols:   ENCARG_1, ENCODE_2ND_1, 2ND_1

Compound Symbols:   c_2, c1_2, c2_1, c3_1, c8_1, c4_2, c5_1, c6_1, c8, c4_1


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

(27) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   2ND(cons(z0, z1)) -> c8
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons1(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(n__from(z0)) -> c2(ENCARG(z0))
   ENCARG(s(z0)) -> c3(ENCARG(z0))
   ENCODE_2ND(z0) -> c8(2ND(encArg(z0)))
   ENCARG(cons_2nd(cons(z0, z1))) -> c4(2ND(cons(encArg(z0), encArg(z1))), ENCARG(cons(z0, z1)))
   ENCARG(cons_2nd(cons_2nd(z0))) -> c4(2ND(2nd(encArg(z0))), ENCARG(cons_2nd(z0)))
   ENCARG(cons_2nd(cons_from(z0))) -> c4(2ND(from(encArg(z0))), ENCARG(cons_from(z0)))
   ENCARG(cons_2nd(cons_activate(z0))) -> c4(2ND(activate(encArg(z0))), ENCARG(cons_activate(z0)))
   ENCARG(cons_from(z0)) -> c5(ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c6(ENCARG(z0))
   2ND(cons(z0, z1)) -> c8
   ENCARG(cons_2nd(cons1(z0, z1))) -> c4(ENCARG(cons1(z0, z1)))
   ENCARG(cons_2nd(n__from(z0))) -> c4(ENCARG(n__from(z0)))
   ENCARG(cons_2nd(s(z0))) -> c4(ENCARG(s(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(2ND(x_1)) = [1]
   POL(2nd(x_1)) = [1] + x_1
   POL(ENCARG(x_1)) = [1] + x_1
   POL(ENCODE_2ND(x_1)) = [1]
   POL(activate(x_1)) = [1] + x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c8) = 0
   POL(c8(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons1(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_2nd(x_1)) = [1] + x_1
   POL(cons_activate(x_1)) = [1] + x_1
   POL(cons_from(x_1)) = [1] + x_1
   POL(encArg(x_1)) = x_1
   POL(from(x_1)) = [1] + x_1
   POL(n__from(x_1)) = [1] + x_1
   POL(s(x_1)) = [1] + x_1

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

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons1(z0, z1)) -> cons1(encArg(z0), encArg(z1))
   encArg(cons(z0, z1)) -> cons(encArg(z0), encArg(z1))
   encArg(n__from(z0)) -> n__from(encArg(z0))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(cons_2nd(z0)) -> 2nd(encArg(z0))
   encArg(cons_from(z0)) -> from(encArg(z0))
   encArg(cons_activate(z0)) -> activate(encArg(z0))
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, activate(z1)))
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   ENCARG(cons1(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(n__from(z0)) -> c2(ENCARG(z0))
   ENCARG(s(z0)) -> c3(ENCARG(z0))
   ENCODE_2ND(z0) -> c8(2ND(encArg(z0)))
   ENCARG(cons_2nd(cons(z0, z1))) -> c4(2ND(cons(encArg(z0), encArg(z1))), ENCARG(cons(z0, z1)))
   ENCARG(cons_2nd(cons_2nd(z0))) -> c4(2ND(2nd(encArg(z0))), ENCARG(cons_2nd(z0)))
   ENCARG(cons_2nd(cons_from(z0))) -> c4(2ND(from(encArg(z0))), ENCARG(cons_from(z0)))
   ENCARG(cons_2nd(cons_activate(z0))) -> c4(2ND(activate(encArg(z0))), ENCARG(cons_activate(z0)))
   ENCARG(cons_from(z0)) -> c5(ENCARG(z0))
   ENCARG(cons_activate(z0)) -> c6(ENCARG(z0))
   2ND(cons(z0, z1)) -> c8
   ENCARG(cons_2nd(cons1(z0, z1))) -> c4(ENCARG(cons1(z0, z1)))
   ENCARG(cons_2nd(n__from(z0))) -> c4(ENCARG(n__from(z0)))
   ENCARG(cons_2nd(s(z0))) -> c4(ENCARG(s(z0)))
S tuples:none
K tuples:
   2ND(cons(z0, z1)) -> c8
Defined Rule Symbols:   encArg_1, 2nd_1, from_1, activate_1

Defined Pair Symbols:   ENCARG_1, ENCODE_2ND_1, 2ND_1

Compound Symbols:   c_2, c1_2, c2_1, c3_1, c8_1, c4_2, c5_1, c6_1, c8, c4_1


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

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

(30)
BOUNDS(1, 1)