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


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


WORST_CASE(?, O(n^1))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 183 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)), 67 ms]
(18) CdtProblem
(19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 37 ms]
(20) CdtProblem
(21) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(22) 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:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(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(0) -> 0
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_0 -> 0
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))


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

(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:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

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

   encArg(0) -> 0
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_0 -> 0
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))

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:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

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

   encArg(0) -> 0
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_0 -> 0
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_s(z0)) -> s(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_h(z0, z1)) -> h(encArg(z0), encArg(z1))
   encode_s(z0) -> s(encArg(z0))
   encode_0 -> 0
   encode_f(z0) -> f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_h(z0, z1) -> h(encArg(z0), encArg(z1))
   s(s(0)) -> f(0)
   s(s(s(0))) -> f(s(0))
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   s(z0) -> h(z0, 0)
   s(z0) -> h(0, z0)
   f(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   f(g(z0)) -> g(g(f(z0)))
   g(z0) -> h(z0, z0)
   g(s(z0)) -> s(s(g(z0)))
   h(f(z0), g(z0)) -> f(s(z0))
Tuples:
   ENCARG(0) -> c
   ENCARG(cons_s(z0)) -> c1(S(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c2(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   ENCARG(cons_h(z0, z1)) -> c4(H(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_S(z0) -> c5(S(encArg(z0)), ENCARG(z0))
   ENCODE_0 -> c6
   ENCODE_F(z0) -> c7(F(encArg(z0)), ENCARG(z0))
   ENCODE_G(z0) -> c8(G(encArg(z0)), ENCARG(z0))
   ENCODE_H(z0, z1) -> c9(H(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   S(s(0)) -> c10(F(0))
   S(s(s(0))) -> c11(F(s(0)), S(0))
   S(s(s(s(s(s(s(s(0)))))))) -> c12(F(s(s(0))), S(s(0)), S(0))
   S(z0) -> c13(H(z0, 0))
   S(z0) -> c14(H(0, z0))
   F(0) -> c15
   F(s(0)) -> c16(S(0))
   F(s(s(0))) -> c17(S(s(s(s(s(s(0)))))), S(s(s(s(s(0))))), S(s(s(s(0)))), S(s(s(0))), S(s(0)), S(0))
   F(g(z0)) -> c18(G(g(f(z0))), G(f(z0)), F(z0))
   G(z0) -> c19(H(z0, z0))
   G(s(z0)) -> c20(S(s(g(z0))), S(g(z0)), G(z0))
   H(f(z0), g(z0)) -> c21(F(s(z0)), S(z0))
S tuples:
   S(s(0)) -> c10(F(0))
   S(s(s(0))) -> c11(F(s(0)), S(0))
   S(s(s(s(s(s(s(s(0)))))))) -> c12(F(s(s(0))), S(s(0)), S(0))
   S(z0) -> c13(H(z0, 0))
   S(z0) -> c14(H(0, z0))
   F(0) -> c15
   F(s(0)) -> c16(S(0))
   F(s(s(0))) -> c17(S(s(s(s(s(s(0)))))), S(s(s(s(s(0))))), S(s(s(s(0)))), S(s(s(0))), S(s(0)), S(0))
   F(g(z0)) -> c18(G(g(f(z0))), G(f(z0)), F(z0))
   G(z0) -> c19(H(z0, z0))
   G(s(z0)) -> c20(S(s(g(z0))), S(g(z0)), G(z0))
   H(f(z0), g(z0)) -> c21(F(s(z0)), S(z0))
K tuples:none
Defined Rule Symbols:   s_1, f_1, g_1, h_2, encArg_1, encode_s_1, encode_0, encode_f_1, encode_g_1, encode_h_2

Defined Pair Symbols:   ENCARG_1, ENCODE_S_1, ENCODE_0, ENCODE_F_1, ENCODE_G_1, ENCODE_H_2, S_1, F_1, G_1, H_2

Compound Symbols:   c, c1_2, c2_2, c3_2, c4_3, c5_2, c6, c7_2, c8_2, c9_3, c10_1, c11_2, c12_3, c13_1, c14_1, c15, c16_1, c17_6, c18_3, c19_1, c20_3, c21_2


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 8 leading nodes:
   S(s(0)) -> c10(F(0))
   S(s(s(0))) -> c11(F(s(0)), S(0))
   S(s(s(s(s(s(s(s(0)))))))) -> c12(F(s(s(0))), S(s(0)), S(0))
   F(s(0)) -> c16(S(0))
   F(s(s(0))) -> c17(S(s(s(s(s(s(0)))))), S(s(s(s(s(0))))), S(s(s(s(0)))), S(s(s(0))), S(s(0)), S(0))
   H(f(z0), g(z0)) -> c21(F(s(z0)), S(z0))
   F(g(z0)) -> c18(G(g(f(z0))), G(f(z0)), F(z0))
   G(s(z0)) -> c20(S(s(g(z0))), S(g(z0)), G(z0))
Removed 2 trailing nodes:
   ENCARG(0) -> c
   ENCODE_0 -> c6

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_s(z0)) -> s(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_h(z0, z1)) -> h(encArg(z0), encArg(z1))
   encode_s(z0) -> s(encArg(z0))
   encode_0 -> 0
   encode_f(z0) -> f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_h(z0, z1) -> h(encArg(z0), encArg(z1))
   s(s(0)) -> f(0)
   s(s(s(0))) -> f(s(0))
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   s(z0) -> h(z0, 0)
   s(z0) -> h(0, z0)
   f(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   f(g(z0)) -> g(g(f(z0)))
   g(z0) -> h(z0, z0)
   g(s(z0)) -> s(s(g(z0)))
   h(f(z0), g(z0)) -> f(s(z0))
Tuples:
   ENCARG(cons_s(z0)) -> c1(S(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c2(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   ENCARG(cons_h(z0, z1)) -> c4(H(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_S(z0) -> c5(S(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0) -> c7(F(encArg(z0)), ENCARG(z0))
   ENCODE_G(z0) -> c8(G(encArg(z0)), ENCARG(z0))
   ENCODE_H(z0, z1) -> c9(H(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   S(z0) -> c13(H(z0, 0))
   S(z0) -> c14(H(0, z0))
   F(0) -> c15
   G(z0) -> c19(H(z0, z0))
S tuples:
   S(z0) -> c13(H(z0, 0))
   S(z0) -> c14(H(0, z0))
   F(0) -> c15
   G(z0) -> c19(H(z0, z0))
K tuples:none
Defined Rule Symbols:   s_1, f_1, g_1, h_2, encArg_1, encode_s_1, encode_0, encode_f_1, encode_g_1, encode_h_2

Defined Pair Symbols:   ENCARG_1, ENCODE_S_1, ENCODE_F_1, ENCODE_G_1, ENCODE_H_2, S_1, F_1, G_1

Compound Symbols:   c1_2, c2_2, c3_2, c4_3, c5_2, c7_2, c8_2, c9_3, c13_1, c14_1, c15, c19_1


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

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing tuple parts
----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_s(z0)) -> s(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_h(z0, z1)) -> h(encArg(z0), encArg(z1))
   encode_s(z0) -> s(encArg(z0))
   encode_0 -> 0
   encode_f(z0) -> f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_h(z0, z1) -> h(encArg(z0), encArg(z1))
   s(s(0)) -> f(0)
   s(s(s(0))) -> f(s(0))
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   s(z0) -> h(z0, 0)
   s(z0) -> h(0, z0)
   f(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   f(g(z0)) -> g(g(f(z0)))
   g(z0) -> h(z0, z0)
   g(s(z0)) -> s(s(g(z0)))
   h(f(z0), g(z0)) -> f(s(z0))
Tuples:
   ENCARG(cons_s(z0)) -> c1(S(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c2(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   ENCODE_S(z0) -> c5(S(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0) -> c7(F(encArg(z0)), ENCARG(z0))
   ENCODE_G(z0) -> c8(G(encArg(z0)), ENCARG(z0))
   F(0) -> c15
   ENCARG(cons_h(z0, z1)) -> c4(ENCARG(z0), ENCARG(z1))
   ENCODE_H(z0, z1) -> c9(ENCARG(z0), ENCARG(z1))
   S(z0) -> c13
   S(z0) -> c14
   G(z0) -> c19
S tuples:
   F(0) -> c15
   S(z0) -> c13
   S(z0) -> c14
   G(z0) -> c19
K tuples:none
Defined Rule Symbols:   s_1, f_1, g_1, h_2, encArg_1, encode_s_1, encode_0, encode_f_1, encode_g_1, encode_h_2

Defined Pair Symbols:   ENCARG_1, ENCODE_S_1, ENCODE_F_1, ENCODE_G_1, F_1, ENCODE_H_2, S_1, G_1

Compound Symbols:   c1_2, c2_2, c3_2, c5_2, c7_2, c8_2, c15, c4_2, c9_2, c13, c14, c19


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

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_s(z0)) -> s(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_h(z0, z1)) -> h(encArg(z0), encArg(z1))
   encode_s(z0) -> s(encArg(z0))
   encode_0 -> 0
   encode_f(z0) -> f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_h(z0, z1) -> h(encArg(z0), encArg(z1))
   s(s(0)) -> f(0)
   s(s(s(0))) -> f(s(0))
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   s(z0) -> h(z0, 0)
   s(z0) -> h(0, z0)
   f(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   f(g(z0)) -> g(g(f(z0)))
   g(z0) -> h(z0, z0)
   g(s(z0)) -> s(s(g(z0)))
   h(f(z0), g(z0)) -> f(s(z0))
Tuples:
   ENCARG(cons_s(z0)) -> c1(S(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c2(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(0) -> c15
   ENCARG(cons_h(z0, z1)) -> c4(ENCARG(z0), ENCARG(z1))
   S(z0) -> c13
   S(z0) -> c14
   G(z0) -> c19
   ENCODE_S(z0) -> c(S(encArg(z0)))
   ENCODE_S(z0) -> c(ENCARG(z0))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCODE_F(z0) -> c(ENCARG(z0))
   ENCODE_G(z0) -> c(G(encArg(z0)))
   ENCODE_G(z0) -> c(ENCARG(z0))
   ENCODE_H(z0, z1) -> c(ENCARG(z0))
   ENCODE_H(z0, z1) -> c(ENCARG(z1))
S tuples:
   F(0) -> c15
   S(z0) -> c13
   S(z0) -> c14
   G(z0) -> c19
K tuples:none
Defined Rule Symbols:   s_1, f_1, g_1, h_2, encArg_1, encode_s_1, encode_0, encode_f_1, encode_g_1, encode_h_2

Defined Pair Symbols:   ENCARG_1, F_1, S_1, G_1, ENCODE_S_1, ENCODE_F_1, ENCODE_G_1, ENCODE_H_2

Compound Symbols:   c1_2, c2_2, c3_2, c15, c4_2, c13, c14, c19, c_1


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

(13) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 5 leading nodes:
   ENCODE_S(z0) -> c(ENCARG(z0))
   ENCODE_F(z0) -> c(ENCARG(z0))
   ENCODE_G(z0) -> c(ENCARG(z0))
   ENCODE_H(z0, z1) -> c(ENCARG(z0))
   ENCODE_H(z0, z1) -> c(ENCARG(z1))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_s(z0)) -> s(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_h(z0, z1)) -> h(encArg(z0), encArg(z1))
   encode_s(z0) -> s(encArg(z0))
   encode_0 -> 0
   encode_f(z0) -> f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_h(z0, z1) -> h(encArg(z0), encArg(z1))
   s(s(0)) -> f(0)
   s(s(s(0))) -> f(s(0))
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   s(z0) -> h(z0, 0)
   s(z0) -> h(0, z0)
   f(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   f(g(z0)) -> g(g(f(z0)))
   g(z0) -> h(z0, z0)
   g(s(z0)) -> s(s(g(z0)))
   h(f(z0), g(z0)) -> f(s(z0))
Tuples:
   ENCARG(cons_s(z0)) -> c1(S(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c2(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(0) -> c15
   ENCARG(cons_h(z0, z1)) -> c4(ENCARG(z0), ENCARG(z1))
   S(z0) -> c13
   S(z0) -> c14
   G(z0) -> c19
   ENCODE_S(z0) -> c(S(encArg(z0)))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCODE_G(z0) -> c(G(encArg(z0)))
S tuples:
   F(0) -> c15
   S(z0) -> c13
   S(z0) -> c14
   G(z0) -> c19
K tuples:none
Defined Rule Symbols:   s_1, f_1, g_1, h_2, encArg_1, encode_s_1, encode_0, encode_f_1, encode_g_1, encode_h_2

Defined Pair Symbols:   ENCARG_1, F_1, S_1, G_1, ENCODE_S_1, ENCODE_F_1, ENCODE_G_1

Compound Symbols:   c1_2, c2_2, c3_2, c15, c4_2, c13, c14, c19, c_1


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

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

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_s(z0)) -> s(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_h(z0, z1)) -> h(encArg(z0), encArg(z1))
   s(z0) -> h(z0, 0)
   s(z0) -> h(0, z0)
   f(0) -> 0
   g(z0) -> h(z0, z0)
Tuples:
   ENCARG(cons_s(z0)) -> c1(S(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c2(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(0) -> c15
   ENCARG(cons_h(z0, z1)) -> c4(ENCARG(z0), ENCARG(z1))
   S(z0) -> c13
   S(z0) -> c14
   G(z0) -> c19
   ENCODE_S(z0) -> c(S(encArg(z0)))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCODE_G(z0) -> c(G(encArg(z0)))
S tuples:
   F(0) -> c15
   S(z0) -> c13
   S(z0) -> c14
   G(z0) -> c19
K tuples:none
Defined Rule Symbols:   encArg_1, s_1, f_1, g_1

Defined Pair Symbols:   ENCARG_1, F_1, S_1, G_1, ENCODE_S_1, ENCODE_F_1, ENCODE_G_1

Compound Symbols:   c1_2, c2_2, c3_2, c15, c4_2, c13, c14, c19, c_1


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

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(0) -> c15
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons_s(z0)) -> c1(S(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c2(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(0) -> c15
   ENCARG(cons_h(z0, z1)) -> c4(ENCARG(z0), ENCARG(z1))
   S(z0) -> c13
   S(z0) -> c14
   G(z0) -> c19
   ENCODE_S(z0) -> c(S(encArg(z0)))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCODE_G(z0) -> c(G(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_F(x_1)) = [1]
   POL(ENCODE_G(x_1)) = 0
   POL(ENCODE_S(x_1)) = [3] + x_1
   POL(F(x_1)) = [1]
   POL(G(x_1)) = 0
   POL(S(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c13) = 0
   POL(c14) = 0
   POL(c15) = 0
   POL(c19) = 0
   POL(c2(x_1, x_2)) = x_1 + x_2
   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)) = [1] + x_1
   POL(cons_g(x_1)) = x_1
   POL(cons_h(x_1, x_2)) = x_1 + x_2
   POL(cons_s(x_1)) = x_1
   POL(encArg(x_1)) = [2]x_1
   POL(f(x_1)) = 0
   POL(g(x_1)) = x_1
   POL(h(x_1, x_2)) = 0
   POL(s(x_1)) = x_1

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_s(z0)) -> s(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_h(z0, z1)) -> h(encArg(z0), encArg(z1))
   s(z0) -> h(z0, 0)
   s(z0) -> h(0, z0)
   f(0) -> 0
   g(z0) -> h(z0, z0)
Tuples:
   ENCARG(cons_s(z0)) -> c1(S(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c2(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(0) -> c15
   ENCARG(cons_h(z0, z1)) -> c4(ENCARG(z0), ENCARG(z1))
   S(z0) -> c13
   S(z0) -> c14
   G(z0) -> c19
   ENCODE_S(z0) -> c(S(encArg(z0)))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCODE_G(z0) -> c(G(encArg(z0)))
S tuples:
   S(z0) -> c13
   S(z0) -> c14
   G(z0) -> c19
K tuples:
   F(0) -> c15
Defined Rule Symbols:   encArg_1, s_1, f_1, g_1

Defined Pair Symbols:   ENCARG_1, F_1, S_1, G_1, ENCODE_S_1, ENCODE_F_1, ENCODE_G_1

Compound Symbols:   c1_2, c2_2, c3_2, c15, c4_2, c13, c14, c19, c_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.
   S(z0) -> c13
   S(z0) -> c14
   G(z0) -> c19
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons_s(z0)) -> c1(S(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c2(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(0) -> c15
   ENCARG(cons_h(z0, z1)) -> c4(ENCARG(z0), ENCARG(z1))
   S(z0) -> c13
   S(z0) -> c14
   G(z0) -> c19
   ENCODE_S(z0) -> c(S(encArg(z0)))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCODE_G(z0) -> c(G(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(ENCARG(x_1)) = [1] + x_1
   POL(ENCODE_F(x_1)) = x_1
   POL(ENCODE_G(x_1)) = [1]
   POL(ENCODE_S(x_1)) = [1] + x_1
   POL(F(x_1)) = 0
   POL(G(x_1)) = [1]
   POL(S(x_1)) = [1]
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c13) = 0
   POL(c14) = 0
   POL(c15) = 0
   POL(c19) = 0
   POL(c2(x_1, x_2)) = x_1 + x_2
   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_1
   POL(cons_g(x_1)) = [1] + x_1
   POL(cons_h(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_s(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(f(x_1)) = [1]
   POL(g(x_1)) = [1] + x_1
   POL(h(x_1, x_2)) = [1] + x_2
   POL(s(x_1)) = [1] + x_1

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(cons_s(z0)) -> s(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_h(z0, z1)) -> h(encArg(z0), encArg(z1))
   s(z0) -> h(z0, 0)
   s(z0) -> h(0, z0)
   f(0) -> 0
   g(z0) -> h(z0, z0)
Tuples:
   ENCARG(cons_s(z0)) -> c1(S(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c2(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   F(0) -> c15
   ENCARG(cons_h(z0, z1)) -> c4(ENCARG(z0), ENCARG(z1))
   S(z0) -> c13
   S(z0) -> c14
   G(z0) -> c19
   ENCODE_S(z0) -> c(S(encArg(z0)))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCODE_G(z0) -> c(G(encArg(z0)))
S tuples:none
K tuples:
   F(0) -> c15
   S(z0) -> c13
   S(z0) -> c14
   G(z0) -> c19
Defined Rule Symbols:   encArg_1, s_1, f_1, g_1

Defined Pair Symbols:   ENCARG_1, F_1, S_1, G_1, ENCODE_S_1, ENCODE_F_1, ENCODE_G_1

Compound Symbols:   c1_2, c2_2, c3_2, c15, c4_2, c13, c14, c19, c_1


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

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

(22)
BOUNDS(1, 1)