/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 65 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)), 70 ms]
(18) CdtProblem
(19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 24 ms]
(20) CdtProblem
(21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 23 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:

   h(X) -> g(X)
   g(a) -> f(b)
   f(X) -> h(a)
   a -> b

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(b) -> b
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_a) -> a
   encode_h(x_1) -> h(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a
   encode_f(x_1) -> f(encArg(x_1))
   encode_b -> b


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

(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(X) -> g(X)
   g(a) -> f(b)
   f(X) -> h(a)
   a -> b

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

   encArg(b) -> b
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_a) -> a
   encode_h(x_1) -> h(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a
   encode_f(x_1) -> f(encArg(x_1))
   encode_b -> b

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(X) -> g(X)
   g(a) -> f(b)
   f(X) -> h(a)
   a -> b

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

   encArg(b) -> b
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_a) -> a
   encode_h(x_1) -> h(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a
   encode_f(x_1) -> f(encArg(x_1))
   encode_b -> b

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   encode_h(z0) -> h(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_a -> a
   encode_f(z0) -> f(encArg(z0))
   encode_b -> b
   h(z0) -> g(z0)
   g(a) -> f(b)
   f(z0) -> h(a)
   a -> b
Tuples:
   ENCARG(b) -> c
   ENCARG(cons_h(z0)) -> c1(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   ENCODE_H(z0) -> c5(H(encArg(z0)), ENCARG(z0))
   ENCODE_G(z0) -> c6(G(encArg(z0)), ENCARG(z0))
   ENCODE_A -> c7(A)
   ENCODE_F(z0) -> c8(F(encArg(z0)), ENCARG(z0))
   ENCODE_B -> c9
   H(z0) -> c10(G(z0))
   G(a) -> c11(F(b))
   F(z0) -> c12(H(a), A)
   A -> c13
S tuples:
   H(z0) -> c10(G(z0))
   G(a) -> c11(F(b))
   F(z0) -> c12(H(a), A)
   A -> c13
K tuples:none
Defined Rule Symbols:   h_1, g_1, f_1, a, encArg_1, encode_h_1, encode_g_1, encode_a, encode_f_1, encode_b

Defined Pair Symbols:   ENCARG_1, ENCODE_H_1, ENCODE_G_1, ENCODE_A, ENCODE_F_1, ENCODE_B, H_1, G_1, F_1, A

Compound Symbols:   c, c1_2, c2_2, c3_2, c4_1, c5_2, c6_2, c7_1, c8_2, c9, c10_1, c11_1, c12_2, c13


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_A -> c7(A)
   G(a) -> c11(F(b))
Removed 2 trailing nodes:
   ENCODE_B -> c9
   ENCARG(b) -> c

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   encode_h(z0) -> h(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_a -> a
   encode_f(z0) -> f(encArg(z0))
   encode_b -> b
   h(z0) -> g(z0)
   g(a) -> f(b)
   f(z0) -> h(a)
   a -> b
Tuples:
   ENCARG(cons_h(z0)) -> c1(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_g(z0)) -> c2(G(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   ENCODE_H(z0) -> c5(H(encArg(z0)), ENCARG(z0))
   ENCODE_G(z0) -> c6(G(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0) -> c8(F(encArg(z0)), ENCARG(z0))
   H(z0) -> c10(G(z0))
   F(z0) -> c12(H(a), A)
   A -> c13
S tuples:
   H(z0) -> c10(G(z0))
   F(z0) -> c12(H(a), A)
   A -> c13
K tuples:none
Defined Rule Symbols:   h_1, g_1, f_1, a, encArg_1, encode_h_1, encode_g_1, encode_a, encode_f_1, encode_b

Defined Pair Symbols:   ENCARG_1, ENCODE_H_1, ENCODE_G_1, ENCODE_F_1, H_1, F_1, A

Compound Symbols:   c1_2, c2_2, c3_2, c4_1, c5_2, c6_2, c8_2, c10_1, c12_2, c13


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

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing tuple parts
----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   encode_h(z0) -> h(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_a -> a
   encode_f(z0) -> f(encArg(z0))
   encode_b -> b
   h(z0) -> g(z0)
   g(a) -> f(b)
   f(z0) -> h(a)
   a -> b
Tuples:
   ENCARG(cons_h(z0)) -> c1(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   ENCODE_H(z0) -> c5(H(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0) -> c8(F(encArg(z0)), ENCARG(z0))
   F(z0) -> c12(H(a), A)
   A -> c13
   ENCARG(cons_g(z0)) -> c2(ENCARG(z0))
   ENCODE_G(z0) -> c6(ENCARG(z0))
   H(z0) -> c10
S tuples:
   F(z0) -> c12(H(a), A)
   A -> c13
   H(z0) -> c10
K tuples:none
Defined Rule Symbols:   h_1, g_1, f_1, a, encArg_1, encode_h_1, encode_g_1, encode_a, encode_f_1, encode_b

Defined Pair Symbols:   ENCARG_1, ENCODE_H_1, ENCODE_F_1, F_1, A, ENCODE_G_1, H_1

Compound Symbols:   c1_2, c3_2, c4_1, c5_2, c8_2, c12_2, c13, c2_1, c6_1, c10


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

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   encode_h(z0) -> h(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_a -> a
   encode_f(z0) -> f(encArg(z0))
   encode_b -> b
   h(z0) -> g(z0)
   g(a) -> f(b)
   f(z0) -> h(a)
   a -> b
Tuples:
   ENCARG(cons_h(z0)) -> c1(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   A -> c13
   ENCARG(cons_g(z0)) -> c2(ENCARG(z0))
   ENCODE_G(z0) -> c6(ENCARG(z0))
   H(z0) -> c10
   ENCODE_H(z0) -> c(H(encArg(z0)))
   ENCODE_H(z0) -> c(ENCARG(z0))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCODE_F(z0) -> c(ENCARG(z0))
   F(z0) -> c(H(a))
   F(z0) -> c(A)
S tuples:
   A -> c13
   H(z0) -> c10
   F(z0) -> c(H(a))
   F(z0) -> c(A)
K tuples:none
Defined Rule Symbols:   h_1, g_1, f_1, a, encArg_1, encode_h_1, encode_g_1, encode_a, encode_f_1, encode_b

Defined Pair Symbols:   ENCARG_1, A, ENCODE_G_1, H_1, ENCODE_H_1, ENCODE_F_1, F_1

Compound Symbols:   c1_2, c3_2, c4_1, c13, c2_1, c6_1, c10, c_1


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

(13) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 3 leading nodes:
   ENCODE_G(z0) -> c6(ENCARG(z0))
   ENCODE_H(z0) -> c(ENCARG(z0))
   ENCODE_F(z0) -> c(ENCARG(z0))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   encode_h(z0) -> h(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_a -> a
   encode_f(z0) -> f(encArg(z0))
   encode_b -> b
   h(z0) -> g(z0)
   g(a) -> f(b)
   f(z0) -> h(a)
   a -> b
Tuples:
   ENCARG(cons_h(z0)) -> c1(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   A -> c13
   ENCARG(cons_g(z0)) -> c2(ENCARG(z0))
   H(z0) -> c10
   ENCODE_H(z0) -> c(H(encArg(z0)))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   F(z0) -> c(H(a))
   F(z0) -> c(A)
S tuples:
   A -> c13
   H(z0) -> c10
   F(z0) -> c(H(a))
   F(z0) -> c(A)
K tuples:none
Defined Rule Symbols:   h_1, g_1, f_1, a, encArg_1, encode_h_1, encode_g_1, encode_a, encode_f_1, encode_b

Defined Pair Symbols:   ENCARG_1, A, H_1, ENCODE_H_1, ENCODE_F_1, F_1

Compound Symbols:   c1_2, c3_2, c4_1, c13, c2_1, c10, c_1


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

(15) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_h(z0) -> h(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_a -> a
   encode_f(z0) -> f(encArg(z0))
   encode_b -> b
   g(a) -> f(b)

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   h(z0) -> g(z0)
   f(z0) -> h(a)
   a -> b
Tuples:
   ENCARG(cons_h(z0)) -> c1(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   A -> c13
   ENCARG(cons_g(z0)) -> c2(ENCARG(z0))
   H(z0) -> c10
   ENCODE_H(z0) -> c(H(encArg(z0)))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   F(z0) -> c(H(a))
   F(z0) -> c(A)
S tuples:
   A -> c13
   H(z0) -> c10
   F(z0) -> c(H(a))
   F(z0) -> c(A)
K tuples:none
Defined Rule Symbols:   encArg_1, h_1, f_1, a

Defined Pair Symbols:   ENCARG_1, A, H_1, ENCODE_H_1, ENCODE_F_1, F_1

Compound Symbols:   c1_2, c3_2, c4_1, c13, c2_1, c10, 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.
   A -> c13
   F(z0) -> c(H(a))
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons_h(z0)) -> c1(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   A -> c13
   ENCARG(cons_g(z0)) -> c2(ENCARG(z0))
   H(z0) -> c10
   ENCODE_H(z0) -> c(H(encArg(z0)))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   F(z0) -> c(H(a))
   F(z0) -> c(A)
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A) = [2]
   POL(ENCARG(x_1)) = [2] + x_1
   POL(ENCODE_F(x_1)) = [2] + x_1
   POL(ENCODE_H(x_1)) = x_1
   POL(F(x_1)) = [2]
   POL(H(x_1)) = 0
   POL(a) = 0
   POL(b) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10) = 0
   POL(c13) = 0
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(cons_a) = 0
   POL(cons_f(x_1)) = [2] + x_1
   POL(cons_g(x_1)) = [2] + x_1
   POL(cons_h(x_1)) = x_1
   POL(encArg(x_1)) = [3]x_1
   POL(f(x_1)) = x_1
   POL(g(x_1)) = 0
   POL(h(x_1)) = 0

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   h(z0) -> g(z0)
   f(z0) -> h(a)
   a -> b
Tuples:
   ENCARG(cons_h(z0)) -> c1(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   A -> c13
   ENCARG(cons_g(z0)) -> c2(ENCARG(z0))
   H(z0) -> c10
   ENCODE_H(z0) -> c(H(encArg(z0)))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   F(z0) -> c(H(a))
   F(z0) -> c(A)
S tuples:
   H(z0) -> c10
   F(z0) -> c(A)
K tuples:
   A -> c13
   F(z0) -> c(H(a))
Defined Rule Symbols:   encArg_1, h_1, f_1, a

Defined Pair Symbols:   ENCARG_1, A, H_1, ENCODE_H_1, ENCODE_F_1, F_1

Compound Symbols:   c1_2, c3_2, c4_1, c13, c2_1, c10, 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.
   H(z0) -> c10
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons_h(z0)) -> c1(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   A -> c13
   ENCARG(cons_g(z0)) -> c2(ENCARG(z0))
   H(z0) -> c10
   ENCODE_H(z0) -> c(H(encArg(z0)))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   F(z0) -> c(H(a))
   F(z0) -> c(A)
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A) = [1]
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_F(x_1)) = [1] + x_1
   POL(ENCODE_H(x_1)) = [1]
   POL(F(x_1)) = [1]
   POL(H(x_1)) = [1]
   POL(a) = [1]
   POL(b) = [1]
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10) = 0
   POL(c13) = 0
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(cons_a) = [1]
   POL(cons_f(x_1)) = [1] + x_1
   POL(cons_g(x_1)) = x_1
   POL(cons_h(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(f(x_1)) = [1] + x_1
   POL(g(x_1)) = [1]
   POL(h(x_1)) = [1]

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   h(z0) -> g(z0)
   f(z0) -> h(a)
   a -> b
Tuples:
   ENCARG(cons_h(z0)) -> c1(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   A -> c13
   ENCARG(cons_g(z0)) -> c2(ENCARG(z0))
   H(z0) -> c10
   ENCODE_H(z0) -> c(H(encArg(z0)))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   F(z0) -> c(H(a))
   F(z0) -> c(A)
S tuples:
   F(z0) -> c(A)
K tuples:
   A -> c13
   F(z0) -> c(H(a))
   H(z0) -> c10
Defined Rule Symbols:   encArg_1, h_1, f_1, a

Defined Pair Symbols:   ENCARG_1, A, H_1, ENCODE_H_1, ENCODE_F_1, F_1

Compound Symbols:   c1_2, c3_2, c4_1, c13, c2_1, c10, c_1


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(21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(z0) -> c(A)
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons_h(z0)) -> c1(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   A -> c13
   ENCARG(cons_g(z0)) -> c2(ENCARG(z0))
   H(z0) -> c10
   ENCODE_H(z0) -> c(H(encArg(z0)))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   F(z0) -> c(H(a))
   F(z0) -> c(A)
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A) = 0
   POL(ENCARG(x_1)) = [1] + x_1
   POL(ENCODE_F(x_1)) = [1]
   POL(ENCODE_H(x_1)) = [1]
   POL(F(x_1)) = [1]
   POL(H(x_1)) = [1]
   POL(a) = 0
   POL(b) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10) = 0
   POL(c13) = 0
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(cons_a) = 0
   POL(cons_f(x_1)) = [1] + x_1
   POL(cons_g(x_1)) = [1] + x_1
   POL(cons_h(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(f(x_1)) = [1] + x_1
   POL(g(x_1)) = [1] + x_1
   POL(h(x_1)) = [1] + x_1

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(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   h(z0) -> g(z0)
   f(z0) -> h(a)
   a -> b
Tuples:
   ENCARG(cons_h(z0)) -> c1(H(encArg(z0)), ENCARG(z0))
   ENCARG(cons_f(z0)) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c4(A)
   A -> c13
   ENCARG(cons_g(z0)) -> c2(ENCARG(z0))
   H(z0) -> c10
   ENCODE_H(z0) -> c(H(encArg(z0)))
   ENCODE_F(z0) -> c(F(encArg(z0)))
   F(z0) -> c(H(a))
   F(z0) -> c(A)
S tuples:none
K tuples:
   A -> c13
   F(z0) -> c(H(a))
   H(z0) -> c10
   F(z0) -> c(A)
Defined Rule Symbols:   encArg_1, h_1, f_1, a

Defined Pair Symbols:   ENCARG_1, A, H_1, ENCODE_H_1, ENCODE_F_1, F_1

Compound Symbols:   c1_2, c3_2, c4_1, c13, c2_1, c10, c_1


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(23) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(24)
BOUNDS(1, 1)