/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 193 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
(7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(8) CdtProblem
(9) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CdtProblem
(11) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CdtProblem
(13) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(14) CdtProblem
(15) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CdtProblem
(17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 45 ms]
(18) CdtProblem
(19) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) 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:

   h(c(x, y), c(s(z), z), t(w)) -> h(z, c(y, x), t(t(c(x, c(y, t(w))))))
   h(x, c(y, z), t(w)) -> h(c(s(y), x), z, t(c(t(w), w)))
   h(c(s(x), c(s(0), y)), z, t(x)) -> h(y, c(s(0), c(x, z)), t(t(c(x, s(x)))))
   t(t(x)) -> t(c(t(x), x))
   t(x) -> x
   t(x) -> c(0, c(0, c(0, c(0, c(0, 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(c(x_1, x_2)) -> c(encArg(x_1), encArg(x_2))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_h(x_1, x_2, x_3)) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_t(x_1)) -> t(encArg(x_1))
   encode_h(x_1, x_2, x_3) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_c(x_1, x_2) -> c(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_t(x_1) -> t(encArg(x_1))
   encode_0 -> 0


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

(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:

   h(c(x, y), c(s(z), z), t(w)) -> h(z, c(y, x), t(t(c(x, c(y, t(w))))))
   h(x, c(y, z), t(w)) -> h(c(s(y), x), z, t(c(t(w), w)))
   h(c(s(x), c(s(0), y)), z, t(x)) -> h(y, c(s(0), c(x, z)), t(t(c(x, s(x)))))
   t(t(x)) -> t(c(t(x), x))
   t(x) -> x
   t(x) -> c(0, c(0, c(0, c(0, c(0, x)))))

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

   encArg(c(x_1, x_2)) -> c(encArg(x_1), encArg(x_2))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_h(x_1, x_2, x_3)) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_t(x_1)) -> t(encArg(x_1))
   encode_h(x_1, x_2, x_3) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_c(x_1, x_2) -> c(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_t(x_1) -> t(encArg(x_1))
   encode_0 -> 0

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:

   h(c(x, y), c(s(z), z), t(w)) -> h(z, c(y, x), t(t(c(x, c(y, t(w))))))
   h(x, c(y, z), t(w)) -> h(c(s(y), x), z, t(c(t(w), w)))
   h(c(s(x), c(s(0), y)), z, t(x)) -> h(y, c(s(0), c(x, z)), t(t(c(x, s(x)))))
   t(t(x)) -> t(c(t(x), x))
   t(x) -> x
   t(x) -> c(0, c(0, c(0, c(0, c(0, x)))))

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

   encArg(c(x_1, x_2)) -> c(encArg(x_1), encArg(x_2))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_h(x_1, x_2, x_3)) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_t(x_1)) -> t(encArg(x_1))
   encode_h(x_1, x_2, x_3) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_c(x_1, x_2) -> c(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_t(x_1) -> t(encArg(x_1))
   encode_0 -> 0

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0, z1)) -> c(encArg(z0), encArg(z1))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_h(z0, z1, z2)) -> h(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_t(z0)) -> t(encArg(z0))
   encode_h(z0, z1, z2) -> h(encArg(z0), encArg(z1), encArg(z2))
   encode_c(z0, z1) -> c(encArg(z0), encArg(z1))
   encode_s(z0) -> s(encArg(z0))
   encode_t(z0) -> t(encArg(z0))
   encode_0 -> 0
   h(c(z0, z1), c(s(z2), z2), t(z3)) -> h(z2, c(z1, z0), t(t(c(z0, c(z1, t(z3))))))
   h(z0, c(z1, z2), t(z3)) -> h(c(s(z1), z0), z2, t(c(t(z3), z3)))
   h(c(s(z0), c(s(0), z1)), z2, t(z0)) -> h(z1, c(s(0), c(z0, z2)), t(t(c(z0, s(z0)))))
   t(t(z0)) -> t(c(t(z0), z0))
   t(z0) -> z0
   t(z0) -> c(0, c(0, c(0, c(0, c(0, z0)))))
Tuples:
   ENCARG(c(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(s(z0)) -> c2(ENCARG(z0))
   ENCARG(0) -> c3
   ENCARG(cons_h(z0, z1, z2)) -> c4(H(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_t(z0)) -> c5(T(encArg(z0)), ENCARG(z0))
   ENCODE_H(z0, z1, z2) -> c6(H(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCODE_C(z0, z1) -> c7(ENCARG(z0), ENCARG(z1))
   ENCODE_S(z0) -> c8(ENCARG(z0))
   ENCODE_T(z0) -> c9(T(encArg(z0)), ENCARG(z0))
   ENCODE_0 -> c10
   H(c(z0, z1), c(s(z2), z2), t(z3)) -> c11(H(z2, c(z1, z0), t(t(c(z0, c(z1, t(z3)))))), T(t(c(z0, c(z1, t(z3))))), T(c(z0, c(z1, t(z3)))), T(z3))
   H(z0, c(z1, z2), t(z3)) -> c12(H(c(s(z1), z0), z2, t(c(t(z3), z3))), T(c(t(z3), z3)), T(z3))
   H(c(s(z0), c(s(0), z1)), z2, t(z0)) -> c13(H(z1, c(s(0), c(z0, z2)), t(t(c(z0, s(z0))))), T(t(c(z0, s(z0)))), T(c(z0, s(z0))))
   T(t(z0)) -> c14(T(c(t(z0), z0)), T(z0))
   T(z0) -> c15
   T(z0) -> c16
S tuples:
   H(c(z0, z1), c(s(z2), z2), t(z3)) -> c11(H(z2, c(z1, z0), t(t(c(z0, c(z1, t(z3)))))), T(t(c(z0, c(z1, t(z3))))), T(c(z0, c(z1, t(z3)))), T(z3))
   H(z0, c(z1, z2), t(z3)) -> c12(H(c(s(z1), z0), z2, t(c(t(z3), z3))), T(c(t(z3), z3)), T(z3))
   H(c(s(z0), c(s(0), z1)), z2, t(z0)) -> c13(H(z1, c(s(0), c(z0, z2)), t(t(c(z0, s(z0))))), T(t(c(z0, s(z0)))), T(c(z0, s(z0))))
   T(t(z0)) -> c14(T(c(t(z0), z0)), T(z0))
   T(z0) -> c15
   T(z0) -> c16
K tuples:none
Defined Rule Symbols:   h_3, t_1, encArg_1, encode_h_3, encode_c_2, encode_s_1, encode_t_1, encode_0

Defined Pair Symbols:   ENCARG_1, ENCODE_H_3, ENCODE_C_2, ENCODE_S_1, ENCODE_T_1, ENCODE_0, H_3, T_1

Compound Symbols:   c1_2, c2_1, c3, c4_4, c5_2, c6_4, c7_2, c8_1, c9_2, c10, c11_4, c12_3, c13_3, c14_2, c15, c16


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 6 leading nodes:
   ENCODE_C(z0, z1) -> c7(ENCARG(z0), ENCARG(z1))
   ENCODE_S(z0) -> c8(ENCARG(z0))
   H(c(z0, z1), c(s(z2), z2), t(z3)) -> c11(H(z2, c(z1, z0), t(t(c(z0, c(z1, t(z3)))))), T(t(c(z0, c(z1, t(z3))))), T(c(z0, c(z1, t(z3)))), T(z3))
   H(z0, c(z1, z2), t(z3)) -> c12(H(c(s(z1), z0), z2, t(c(t(z3), z3))), T(c(t(z3), z3)), T(z3))
   H(c(s(z0), c(s(0), z1)), z2, t(z0)) -> c13(H(z1, c(s(0), c(z0, z2)), t(t(c(z0, s(z0))))), T(t(c(z0, s(z0)))), T(c(z0, s(z0))))
   T(t(z0)) -> c14(T(c(t(z0), z0)), T(z0))
Removed 2 trailing nodes:
   ENCARG(0) -> c3
   ENCODE_0 -> c10

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0, z1)) -> c(encArg(z0), encArg(z1))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_h(z0, z1, z2)) -> h(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_t(z0)) -> t(encArg(z0))
   encode_h(z0, z1, z2) -> h(encArg(z0), encArg(z1), encArg(z2))
   encode_c(z0, z1) -> c(encArg(z0), encArg(z1))
   encode_s(z0) -> s(encArg(z0))
   encode_t(z0) -> t(encArg(z0))
   encode_0 -> 0
   h(c(z0, z1), c(s(z2), z2), t(z3)) -> h(z2, c(z1, z0), t(t(c(z0, c(z1, t(z3))))))
   h(z0, c(z1, z2), t(z3)) -> h(c(s(z1), z0), z2, t(c(t(z3), z3)))
   h(c(s(z0), c(s(0), z1)), z2, t(z0)) -> h(z1, c(s(0), c(z0, z2)), t(t(c(z0, s(z0)))))
   t(t(z0)) -> t(c(t(z0), z0))
   t(z0) -> z0
   t(z0) -> c(0, c(0, c(0, c(0, c(0, z0)))))
Tuples:
   ENCARG(c(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(s(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_h(z0, z1, z2)) -> c4(H(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCARG(cons_t(z0)) -> c5(T(encArg(z0)), ENCARG(z0))
   ENCODE_H(z0, z1, z2) -> c6(H(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCODE_T(z0) -> c9(T(encArg(z0)), ENCARG(z0))
   T(z0) -> c15
   T(z0) -> c16
S tuples:
   T(z0) -> c15
   T(z0) -> c16
K tuples:none
Defined Rule Symbols:   h_3, t_1, encArg_1, encode_h_3, encode_c_2, encode_s_1, encode_t_1, encode_0

Defined Pair Symbols:   ENCARG_1, ENCODE_H_3, ENCODE_T_1, T_1

Compound Symbols:   c1_2, c2_1, c4_4, c5_2, c6_4, c9_2, c15, c16


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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0, z1)) -> c(encArg(z0), encArg(z1))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_h(z0, z1, z2)) -> h(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_t(z0)) -> t(encArg(z0))
   encode_h(z0, z1, z2) -> h(encArg(z0), encArg(z1), encArg(z2))
   encode_c(z0, z1) -> c(encArg(z0), encArg(z1))
   encode_s(z0) -> s(encArg(z0))
   encode_t(z0) -> t(encArg(z0))
   encode_0 -> 0
   h(c(z0, z1), c(s(z2), z2), t(z3)) -> h(z2, c(z1, z0), t(t(c(z0, c(z1, t(z3))))))
   h(z0, c(z1, z2), t(z3)) -> h(c(s(z1), z0), z2, t(c(t(z3), z3)))
   h(c(s(z0), c(s(0), z1)), z2, t(z0)) -> h(z1, c(s(0), c(z0, z2)), t(t(c(z0, s(z0)))))
   t(t(z0)) -> t(c(t(z0), z0))
   t(z0) -> z0
   t(z0) -> c(0, c(0, c(0, c(0, c(0, z0)))))
Tuples:
   ENCARG(c(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(s(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_t(z0)) -> c5(T(encArg(z0)), ENCARG(z0))
   ENCODE_T(z0) -> c9(T(encArg(z0)), ENCARG(z0))
   T(z0) -> c15
   T(z0) -> c16
   ENCARG(cons_h(z0, z1, z2)) -> c4(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCODE_H(z0, z1, z2) -> c6(ENCARG(z0), ENCARG(z1), ENCARG(z2))
S tuples:
   T(z0) -> c15
   T(z0) -> c16
K tuples:none
Defined Rule Symbols:   h_3, t_1, encArg_1, encode_h_3, encode_c_2, encode_s_1, encode_t_1, encode_0

Defined Pair Symbols:   ENCARG_1, ENCODE_T_1, T_1, ENCODE_H_3

Compound Symbols:   c1_2, c2_1, c5_2, c9_2, c15, c16, c4_3, c6_3


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

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0, z1)) -> c(encArg(z0), encArg(z1))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_h(z0, z1, z2)) -> h(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_t(z0)) -> t(encArg(z0))
   encode_h(z0, z1, z2) -> h(encArg(z0), encArg(z1), encArg(z2))
   encode_c(z0, z1) -> c(encArg(z0), encArg(z1))
   encode_s(z0) -> s(encArg(z0))
   encode_t(z0) -> t(encArg(z0))
   encode_0 -> 0
   h(c(z0, z1), c(s(z2), z2), t(z3)) -> h(z2, c(z1, z0), t(t(c(z0, c(z1, t(z3))))))
   h(z0, c(z1, z2), t(z3)) -> h(c(s(z1), z0), z2, t(c(t(z3), z3)))
   h(c(s(z0), c(s(0), z1)), z2, t(z0)) -> h(z1, c(s(0), c(z0, z2)), t(t(c(z0, s(z0)))))
   t(t(z0)) -> t(c(t(z0), z0))
   t(z0) -> z0
   t(z0) -> c(0, c(0, c(0, c(0, c(0, z0)))))
Tuples:
   ENCARG(c(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(s(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_t(z0)) -> c5(T(encArg(z0)), ENCARG(z0))
   T(z0) -> c15
   T(z0) -> c16
   ENCARG(cons_h(z0, z1, z2)) -> c4(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCODE_T(z0) -> c3(T(encArg(z0)))
   ENCODE_T(z0) -> c3(ENCARG(z0))
   ENCODE_H(z0, z1, z2) -> c3(ENCARG(z0))
   ENCODE_H(z0, z1, z2) -> c3(ENCARG(z1))
   ENCODE_H(z0, z1, z2) -> c3(ENCARG(z2))
S tuples:
   T(z0) -> c15
   T(z0) -> c16
K tuples:none
Defined Rule Symbols:   h_3, t_1, encArg_1, encode_h_3, encode_c_2, encode_s_1, encode_t_1, encode_0

Defined Pair Symbols:   ENCARG_1, T_1, ENCODE_T_1, ENCODE_H_3

Compound Symbols:   c1_2, c2_1, c5_2, c15, c16, c4_3, c3_1


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

(13) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 4 leading nodes:
   ENCODE_T(z0) -> c3(ENCARG(z0))
   ENCODE_H(z0, z1, z2) -> c3(ENCARG(z0))
   ENCODE_H(z0, z1, z2) -> c3(ENCARG(z1))
   ENCODE_H(z0, z1, z2) -> c3(ENCARG(z2))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0, z1)) -> c(encArg(z0), encArg(z1))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_h(z0, z1, z2)) -> h(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_t(z0)) -> t(encArg(z0))
   encode_h(z0, z1, z2) -> h(encArg(z0), encArg(z1), encArg(z2))
   encode_c(z0, z1) -> c(encArg(z0), encArg(z1))
   encode_s(z0) -> s(encArg(z0))
   encode_t(z0) -> t(encArg(z0))
   encode_0 -> 0
   h(c(z0, z1), c(s(z2), z2), t(z3)) -> h(z2, c(z1, z0), t(t(c(z0, c(z1, t(z3))))))
   h(z0, c(z1, z2), t(z3)) -> h(c(s(z1), z0), z2, t(c(t(z3), z3)))
   h(c(s(z0), c(s(0), z1)), z2, t(z0)) -> h(z1, c(s(0), c(z0, z2)), t(t(c(z0, s(z0)))))
   t(t(z0)) -> t(c(t(z0), z0))
   t(z0) -> z0
   t(z0) -> c(0, c(0, c(0, c(0, c(0, z0)))))
Tuples:
   ENCARG(c(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(s(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_t(z0)) -> c5(T(encArg(z0)), ENCARG(z0))
   T(z0) -> c15
   T(z0) -> c16
   ENCARG(cons_h(z0, z1, z2)) -> c4(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCODE_T(z0) -> c3(T(encArg(z0)))
S tuples:
   T(z0) -> c15
   T(z0) -> c16
K tuples:none
Defined Rule Symbols:   h_3, t_1, encArg_1, encode_h_3, encode_c_2, encode_s_1, encode_t_1, encode_0

Defined Pair Symbols:   ENCARG_1, T_1, ENCODE_T_1

Compound Symbols:   c1_2, c2_1, c5_2, c15, c16, c4_3, c3_1


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

(15) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_h(z0, z1, z2) -> h(encArg(z0), encArg(z1), encArg(z2))
   encode_c(z0, z1) -> c(encArg(z0), encArg(z1))
   encode_s(z0) -> s(encArg(z0))
   encode_t(z0) -> t(encArg(z0))
   encode_0 -> 0
   h(c(z0, z1), c(s(z2), z2), t(z3)) -> h(z2, c(z1, z0), t(t(c(z0, c(z1, t(z3))))))
   h(z0, c(z1, z2), t(z3)) -> h(c(s(z1), z0), z2, t(c(t(z3), z3)))
   h(c(s(z0), c(s(0), z1)), z2, t(z0)) -> h(z1, c(s(0), c(z0, z2)), t(t(c(z0, s(z0)))))
   t(t(z0)) -> t(c(t(z0), z0))

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(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0, z1)) -> c(encArg(z0), encArg(z1))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_h(z0, z1, z2)) -> h(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_t(z0)) -> t(encArg(z0))
   t(z0) -> z0
   t(z0) -> c(0, c(0, c(0, c(0, c(0, z0)))))
Tuples:
   ENCARG(c(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(s(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_t(z0)) -> c5(T(encArg(z0)), ENCARG(z0))
   T(z0) -> c15
   T(z0) -> c16
   ENCARG(cons_h(z0, z1, z2)) -> c4(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCODE_T(z0) -> c3(T(encArg(z0)))
S tuples:
   T(z0) -> c15
   T(z0) -> c16
K tuples:none
Defined Rule Symbols:   encArg_1, t_1

Defined Pair Symbols:   ENCARG_1, T_1, ENCODE_T_1

Compound Symbols:   c1_2, c2_1, c5_2, c15, c16, c4_3, c3_1


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(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   T(z0) -> c15
   T(z0) -> c16
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(c(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(s(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_t(z0)) -> c5(T(encArg(z0)), ENCARG(z0))
   T(z0) -> c15
   T(z0) -> c16
   ENCARG(cons_h(z0, z1, z2)) -> c4(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCODE_T(z0) -> c3(T(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_T(x_1)) = [1] + x_1
   POL(T(x_1)) = [1]
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c15) = 0
   POL(c16) = 0
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(cons_h(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(cons_t(x_1)) = [1] + x_1
   POL(encArg(x_1)) = x_1
   POL(h(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(s(x_1)) = [1] + x_1
   POL(t(x_1)) = [1] + x_1

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(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(c(z0, z1)) -> c(encArg(z0), encArg(z1))
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_h(z0, z1, z2)) -> h(encArg(z0), encArg(z1), encArg(z2))
   encArg(cons_t(z0)) -> t(encArg(z0))
   t(z0) -> z0
   t(z0) -> c(0, c(0, c(0, c(0, c(0, z0)))))
Tuples:
   ENCARG(c(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1))
   ENCARG(s(z0)) -> c2(ENCARG(z0))
   ENCARG(cons_t(z0)) -> c5(T(encArg(z0)), ENCARG(z0))
   T(z0) -> c15
   T(z0) -> c16
   ENCARG(cons_h(z0, z1, z2)) -> c4(ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCODE_T(z0) -> c3(T(encArg(z0)))
S tuples:none
K tuples:
   T(z0) -> c15
   T(z0) -> c16
Defined Rule Symbols:   encArg_1, t_1

Defined Pair Symbols:   ENCARG_1, T_1, ENCODE_T_1

Compound Symbols:   c1_2, c2_1, c5_2, c15, c16, c4_3, c3_1


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(19) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(20)
BOUNDS(1, 1)