/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 199 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)), 97 ms]
(18) CdtProblem
(19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 42 ms]
(20) CdtProblem
(21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 45 ms]
(22) CdtProblem
(23) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(24) 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:

   f(g(x), g(y)) -> f(p(f(g(x), s(y))), g(s(p(x))))
   p(0) -> g(0)
   g(s(p(x))) -> p(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(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_s(x_1) -> s(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:

   f(g(x), g(y)) -> f(p(f(g(x), s(y))), g(s(p(x))))
   p(0) -> g(0)
   g(s(p(x))) -> p(x)

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_s(x_1) -> s(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:

   f(g(x), g(y)) -> f(p(f(g(x), s(y))), g(s(p(x))))
   p(0) -> g(0)
   g(s(p(x))) -> p(x)

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_s(x_1) -> s(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(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_g(z0) -> g(encArg(z0))
   encode_p(z0) -> p(encArg(z0))
   encode_s(z0) -> s(encArg(z0))
   encode_0 -> 0
   f(g(z0), g(z1)) -> f(p(f(g(z0), s(z1))), g(s(p(z0))))
   p(0) -> g(0)
   g(s(p(z0))) -> p(z0)
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(0) -> c1
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_p(z0)) -> c3(P(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c4(G(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0, z1) -> c5(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_G(z0) -> c6(G(encArg(z0)), ENCARG(z0))
   ENCODE_P(z0) -> c7(P(encArg(z0)), ENCARG(z0))
   ENCODE_S(z0) -> c8(ENCARG(z0))
   ENCODE_0 -> c9
   F(g(z0), g(z1)) -> c10(F(p(f(g(z0), s(z1))), g(s(p(z0)))), P(f(g(z0), s(z1))), F(g(z0), s(z1)), G(z0), G(s(p(z0))), P(z0))
   P(0) -> c11(G(0))
   G(s(p(z0))) -> c12(P(z0))
S tuples:
   F(g(z0), g(z1)) -> c10(F(p(f(g(z0), s(z1))), g(s(p(z0)))), P(f(g(z0), s(z1))), F(g(z0), s(z1)), G(z0), G(s(p(z0))), P(z0))
   P(0) -> c11(G(0))
   G(s(p(z0))) -> c12(P(z0))
K tuples:none
Defined Rule Symbols:   f_2, p_1, g_1, encArg_1, encode_f_2, encode_g_1, encode_p_1, encode_s_1, encode_0

Defined Pair Symbols:   ENCARG_1, ENCODE_F_2, ENCODE_G_1, ENCODE_P_1, ENCODE_S_1, ENCODE_0, F_2, P_1, G_1

Compound Symbols:   c_1, c1, c2_3, c3_2, c4_2, c5_3, c6_2, c7_2, c8_1, c9, c10_6, c11_1, c12_1


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   ENCODE_S(z0) -> c8(ENCARG(z0))
Removed 2 trailing nodes:
   ENCARG(0) -> c1
   ENCODE_0 -> c9

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_g(z0) -> g(encArg(z0))
   encode_p(z0) -> p(encArg(z0))
   encode_s(z0) -> s(encArg(z0))
   encode_0 -> 0
   f(g(z0), g(z1)) -> f(p(f(g(z0), s(z1))), g(s(p(z0))))
   p(0) -> g(0)
   g(s(p(z0))) -> p(z0)
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_p(z0)) -> c3(P(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c4(G(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0, z1) -> c5(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_G(z0) -> c6(G(encArg(z0)), ENCARG(z0))
   ENCODE_P(z0) -> c7(P(encArg(z0)), ENCARG(z0))
   F(g(z0), g(z1)) -> c10(F(p(f(g(z0), s(z1))), g(s(p(z0)))), P(f(g(z0), s(z1))), F(g(z0), s(z1)), G(z0), G(s(p(z0))), P(z0))
   P(0) -> c11(G(0))
   G(s(p(z0))) -> c12(P(z0))
S tuples:
   F(g(z0), g(z1)) -> c10(F(p(f(g(z0), s(z1))), g(s(p(z0)))), P(f(g(z0), s(z1))), F(g(z0), s(z1)), G(z0), G(s(p(z0))), P(z0))
   P(0) -> c11(G(0))
   G(s(p(z0))) -> c12(P(z0))
K tuples:none
Defined Rule Symbols:   f_2, p_1, g_1, encArg_1, encode_f_2, encode_g_1, encode_p_1, encode_s_1, encode_0

Defined Pair Symbols:   ENCARG_1, ENCODE_F_2, ENCODE_G_1, ENCODE_P_1, F_2, P_1, G_1

Compound Symbols:   c_1, c2_3, c3_2, c4_2, c5_3, c6_2, c7_2, c10_6, c11_1, c12_1


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

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing tuple parts
----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_g(z0) -> g(encArg(z0))
   encode_p(z0) -> p(encArg(z0))
   encode_s(z0) -> s(encArg(z0))
   encode_0 -> 0
   f(g(z0), g(z1)) -> f(p(f(g(z0), s(z1))), g(s(p(z0))))
   p(0) -> g(0)
   g(s(p(z0))) -> p(z0)
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_p(z0)) -> c3(P(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c4(G(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0, z1) -> c5(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_G(z0) -> c6(G(encArg(z0)), ENCARG(z0))
   ENCODE_P(z0) -> c7(P(encArg(z0)), ENCARG(z0))
   G(s(p(z0))) -> c12(P(z0))
   F(g(z0), g(z1)) -> c10(G(z0), G(s(p(z0))), P(z0))
   P(0) -> c11
S tuples:
   G(s(p(z0))) -> c12(P(z0))
   F(g(z0), g(z1)) -> c10(G(z0), G(s(p(z0))), P(z0))
   P(0) -> c11
K tuples:none
Defined Rule Symbols:   f_2, p_1, g_1, encArg_1, encode_f_2, encode_g_1, encode_p_1, encode_s_1, encode_0

Defined Pair Symbols:   ENCARG_1, ENCODE_F_2, ENCODE_G_1, ENCODE_P_1, G_1, F_2, P_1

Compound Symbols:   c_1, c2_3, c3_2, c4_2, c5_3, c6_2, c7_2, c12_1, c10_3, c11


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

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_g(z0) -> g(encArg(z0))
   encode_p(z0) -> p(encArg(z0))
   encode_s(z0) -> s(encArg(z0))
   encode_0 -> 0
   f(g(z0), g(z1)) -> f(p(f(g(z0), s(z1))), g(s(p(z0))))
   p(0) -> g(0)
   g(s(p(z0))) -> p(z0)
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_p(z0)) -> c3(P(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c4(G(encArg(z0)), ENCARG(z0))
   G(s(p(z0))) -> c12(P(z0))
   P(0) -> c11
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_F(z0, z1) -> c1(ENCARG(z0))
   ENCODE_F(z0, z1) -> c1(ENCARG(z1))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   ENCODE_G(z0) -> c1(ENCARG(z0))
   ENCODE_P(z0) -> c1(P(encArg(z0)))
   ENCODE_P(z0) -> c1(ENCARG(z0))
   F(g(z0), g(z1)) -> c1(G(z0))
   F(g(z0), g(z1)) -> c1(G(s(p(z0))))
   F(g(z0), g(z1)) -> c1(P(z0))
S tuples:
   G(s(p(z0))) -> c12(P(z0))
   P(0) -> c11
   F(g(z0), g(z1)) -> c1(G(z0))
   F(g(z0), g(z1)) -> c1(G(s(p(z0))))
   F(g(z0), g(z1)) -> c1(P(z0))
K tuples:none
Defined Rule Symbols:   f_2, p_1, g_1, encArg_1, encode_f_2, encode_g_1, encode_p_1, encode_s_1, encode_0

Defined Pair Symbols:   ENCARG_1, G_1, P_1, ENCODE_F_2, ENCODE_G_1, ENCODE_P_1, F_2

Compound Symbols:   c_1, c2_3, c3_2, c4_2, c12_1, c11, c1_1


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

(13) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 4 leading nodes:
   ENCODE_F(z0, z1) -> c1(ENCARG(z0))
   ENCODE_F(z0, z1) -> c1(ENCARG(z1))
   ENCODE_G(z0) -> c1(ENCARG(z0))
   ENCODE_P(z0) -> c1(ENCARG(z0))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_g(z0) -> g(encArg(z0))
   encode_p(z0) -> p(encArg(z0))
   encode_s(z0) -> s(encArg(z0))
   encode_0 -> 0
   f(g(z0), g(z1)) -> f(p(f(g(z0), s(z1))), g(s(p(z0))))
   p(0) -> g(0)
   g(s(p(z0))) -> p(z0)
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_p(z0)) -> c3(P(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c4(G(encArg(z0)), ENCARG(z0))
   G(s(p(z0))) -> c12(P(z0))
   P(0) -> c11
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   ENCODE_P(z0) -> c1(P(encArg(z0)))
   F(g(z0), g(z1)) -> c1(G(z0))
   F(g(z0), g(z1)) -> c1(G(s(p(z0))))
   F(g(z0), g(z1)) -> c1(P(z0))
S tuples:
   G(s(p(z0))) -> c12(P(z0))
   P(0) -> c11
   F(g(z0), g(z1)) -> c1(G(z0))
   F(g(z0), g(z1)) -> c1(G(s(p(z0))))
   F(g(z0), g(z1)) -> c1(P(z0))
K tuples:none
Defined Rule Symbols:   f_2, p_1, g_1, encArg_1, encode_f_2, encode_g_1, encode_p_1, encode_s_1, encode_0

Defined Pair Symbols:   ENCARG_1, G_1, P_1, ENCODE_F_2, ENCODE_G_1, ENCODE_P_1, F_2

Compound Symbols:   c_1, c2_3, c3_2, c4_2, c12_1, c11, c1_1


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

(15) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_g(z0) -> g(encArg(z0))
   encode_p(z0) -> p(encArg(z0))
   encode_s(z0) -> s(encArg(z0))
   encode_0 -> 0

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(g(z0), g(z1)) -> f(p(f(g(z0), s(z1))), g(s(p(z0))))
   p(0) -> g(0)
   g(s(p(z0))) -> p(z0)
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_p(z0)) -> c3(P(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c4(G(encArg(z0)), ENCARG(z0))
   G(s(p(z0))) -> c12(P(z0))
   P(0) -> c11
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   ENCODE_P(z0) -> c1(P(encArg(z0)))
   F(g(z0), g(z1)) -> c1(G(z0))
   F(g(z0), g(z1)) -> c1(G(s(p(z0))))
   F(g(z0), g(z1)) -> c1(P(z0))
S tuples:
   G(s(p(z0))) -> c12(P(z0))
   P(0) -> c11
   F(g(z0), g(z1)) -> c1(G(z0))
   F(g(z0), g(z1)) -> c1(G(s(p(z0))))
   F(g(z0), g(z1)) -> c1(P(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, f_2, p_1, g_1

Defined Pair Symbols:   ENCARG_1, G_1, P_1, ENCODE_F_2, ENCODE_G_1, ENCODE_P_1, F_2

Compound Symbols:   c_1, c2_3, c3_2, c4_2, c12_1, c11, c1_1


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

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(g(z0), g(z1)) -> c1(G(z0))
   F(g(z0), g(z1)) -> c1(G(s(p(z0))))
   F(g(z0), g(z1)) -> c1(P(z0))
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_p(z0)) -> c3(P(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c4(G(encArg(z0)), ENCARG(z0))
   G(s(p(z0))) -> c12(P(z0))
   P(0) -> c11
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   ENCODE_P(z0) -> c1(P(encArg(z0)))
   F(g(z0), g(z1)) -> c1(G(z0))
   F(g(z0), g(z1)) -> c1(G(s(p(z0))))
   F(g(z0), g(z1)) -> c1(P(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_F(x_1, x_2)) = [1]
   POL(ENCODE_G(x_1)) = 0
   POL(ENCODE_P(x_1)) = x_1
   POL(F(x_1, x_2)) = [1]
   POL(G(x_1)) = 0
   POL(P(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11) = 0
   POL(c12(x_1)) = x_1
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(cons_f(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_g(x_1)) = [1] + x_1
   POL(cons_p(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(f(x_1, x_2)) = [1]
   POL(g(x_1)) = 0
   POL(p(x_1)) = 0
   POL(s(x_1)) = [1] + x_1

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(g(z0), g(z1)) -> f(p(f(g(z0), s(z1))), g(s(p(z0))))
   p(0) -> g(0)
   g(s(p(z0))) -> p(z0)
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_p(z0)) -> c3(P(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c4(G(encArg(z0)), ENCARG(z0))
   G(s(p(z0))) -> c12(P(z0))
   P(0) -> c11
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   ENCODE_P(z0) -> c1(P(encArg(z0)))
   F(g(z0), g(z1)) -> c1(G(z0))
   F(g(z0), g(z1)) -> c1(G(s(p(z0))))
   F(g(z0), g(z1)) -> c1(P(z0))
S tuples:
   G(s(p(z0))) -> c12(P(z0))
   P(0) -> c11
K tuples:
   F(g(z0), g(z1)) -> c1(G(z0))
   F(g(z0), g(z1)) -> c1(G(s(p(z0))))
   F(g(z0), g(z1)) -> c1(P(z0))
Defined Rule Symbols:   encArg_1, f_2, p_1, g_1

Defined Pair Symbols:   ENCARG_1, G_1, P_1, ENCODE_F_2, ENCODE_G_1, ENCODE_P_1, F_2

Compound Symbols:   c_1, c2_3, c3_2, c4_2, c12_1, c11, c1_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.
   P(0) -> c11
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_p(z0)) -> c3(P(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c4(G(encArg(z0)), ENCARG(z0))
   G(s(p(z0))) -> c12(P(z0))
   P(0) -> c11
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   ENCODE_P(z0) -> c1(P(encArg(z0)))
   F(g(z0), g(z1)) -> c1(G(z0))
   F(g(z0), g(z1)) -> c1(G(s(p(z0))))
   F(g(z0), g(z1)) -> c1(P(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_F(x_1, x_2)) = [1]
   POL(ENCODE_G(x_1)) = [1]
   POL(ENCODE_P(x_1)) = [1] + x_1
   POL(F(x_1, x_2)) = [1]
   POL(G(x_1)) = [1]
   POL(P(x_1)) = [1]
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11) = 0
   POL(c12(x_1)) = x_1
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(cons_f(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_g(x_1)) = [1] + x_1
   POL(cons_p(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(f(x_1, x_2)) = [1]
   POL(g(x_1)) = 0
   POL(p(x_1)) = 0
   POL(s(x_1)) = [1] + x_1

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(g(z0), g(z1)) -> f(p(f(g(z0), s(z1))), g(s(p(z0))))
   p(0) -> g(0)
   g(s(p(z0))) -> p(z0)
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_p(z0)) -> c3(P(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c4(G(encArg(z0)), ENCARG(z0))
   G(s(p(z0))) -> c12(P(z0))
   P(0) -> c11
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   ENCODE_P(z0) -> c1(P(encArg(z0)))
   F(g(z0), g(z1)) -> c1(G(z0))
   F(g(z0), g(z1)) -> c1(G(s(p(z0))))
   F(g(z0), g(z1)) -> c1(P(z0))
S tuples:
   G(s(p(z0))) -> c12(P(z0))
K tuples:
   F(g(z0), g(z1)) -> c1(G(z0))
   F(g(z0), g(z1)) -> c1(G(s(p(z0))))
   F(g(z0), g(z1)) -> c1(P(z0))
   P(0) -> c11
Defined Rule Symbols:   encArg_1, f_2, p_1, g_1

Defined Pair Symbols:   ENCARG_1, G_1, P_1, ENCODE_F_2, ENCODE_G_1, ENCODE_P_1, F_2

Compound Symbols:   c_1, c2_3, c3_2, c4_2, c12_1, c11, c1_1


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

(21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G(s(p(z0))) -> c12(P(z0))
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_p(z0)) -> c3(P(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c4(G(encArg(z0)), ENCARG(z0))
   G(s(p(z0))) -> c12(P(z0))
   P(0) -> c11
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   ENCODE_P(z0) -> c1(P(encArg(z0)))
   F(g(z0), g(z1)) -> c1(G(z0))
   F(g(z0), g(z1)) -> c1(G(s(p(z0))))
   F(g(z0), g(z1)) -> c1(P(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_F(x_1, x_2)) = [1]
   POL(ENCODE_G(x_1)) = [1]
   POL(ENCODE_P(x_1)) = 0
   POL(F(x_1, x_2)) = [1]
   POL(G(x_1)) = [1]
   POL(P(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11) = 0
   POL(c12(x_1)) = x_1
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(cons_f(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_g(x_1)) = [1] + x_1
   POL(cons_p(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(f(x_1, x_2)) = [1]
   POL(g(x_1)) = 0
   POL(p(x_1)) = 0
   POL(s(x_1)) = [1] + x_1

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(s(z0)) -> s(encArg(z0))
   encArg(0) -> 0
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(g(z0), g(z1)) -> f(p(f(g(z0), s(z1))), g(s(p(z0))))
   p(0) -> g(0)
   g(s(p(z0))) -> p(z0)
Tuples:
   ENCARG(s(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_p(z0)) -> c3(P(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c4(G(encArg(z0)), ENCARG(z0))
   G(s(p(z0))) -> c12(P(z0))
   P(0) -> c11
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   ENCODE_P(z0) -> c1(P(encArg(z0)))
   F(g(z0), g(z1)) -> c1(G(z0))
   F(g(z0), g(z1)) -> c1(G(s(p(z0))))
   F(g(z0), g(z1)) -> c1(P(z0))
S tuples:none
K tuples:
   F(g(z0), g(z1)) -> c1(G(z0))
   F(g(z0), g(z1)) -> c1(G(s(p(z0))))
   F(g(z0), g(z1)) -> c1(P(z0))
   P(0) -> c11
   G(s(p(z0))) -> c12(P(z0))
Defined Rule Symbols:   encArg_1, f_2, p_1, g_1

Defined Pair Symbols:   ENCARG_1, G_1, P_1, ENCODE_F_2, ENCODE_G_1, ENCODE_P_1, F_2

Compound Symbols:   c_1, c2_3, c3_2, c4_2, c12_1, c11, c1_1


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

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

(24)
BOUNDS(1, 1)