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


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


WORST_CASE(?, O(n^2))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 291 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
(7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 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)), 45 ms]
(20) CdtProblem
(21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 286 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^2).


The TRS R consists of the following rules:

   f(0, 1, g(x, y), z) -> f(g(x, y), g(x, y), g(x, y), h(x))
   g(0, 1) -> 0
   g(0, 1) -> 1
   h(g(x, y)) -> h(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(1) -> 1
   encArg(cons_f(x_1, x_2, x_3, x_4)) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_f(x_1, x_2, x_3, x_4) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_0 -> 0
   encode_1 -> 1
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_h(x_1) -> h(encArg(x_1))


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

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


The TRS R consists of the following rules:

   f(0, 1, g(x, y), z) -> f(g(x, y), g(x, y), g(x, y), h(x))
   g(0, 1) -> 0
   g(0, 1) -> 1
   h(g(x, y)) -> h(x)

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

   encArg(0) -> 0
   encArg(1) -> 1
   encArg(cons_f(x_1, x_2, x_3, x_4)) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_f(x_1, x_2, x_3, x_4) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_0 -> 0
   encode_1 -> 1
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_h(x_1) -> h(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^2).


The TRS R consists of the following rules:

   f(0, 1, g(x, y), z) -> f(g(x, y), g(x, y), g(x, y), h(x))
   g(0, 1) -> 0
   g(0, 1) -> 1
   h(g(x, y)) -> h(x)

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

   encArg(0) -> 0
   encArg(1) -> 1
   encArg(cons_f(x_1, x_2, x_3, x_4)) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_f(x_1, x_2, x_3, x_4) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_0 -> 0
   encode_1 -> 1
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_h(x_1) -> h(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(0) -> 0
   encArg(1) -> 1
   encArg(cons_f(z0, z1, z2, z3)) -> f(encArg(z0), encArg(z1), encArg(z2), encArg(z3))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encArg(cons_h(z0)) -> h(encArg(z0))
   encode_f(z0, z1, z2, z3) -> f(encArg(z0), encArg(z1), encArg(z2), encArg(z3))
   encode_0 -> 0
   encode_1 -> 1
   encode_g(z0, z1) -> g(encArg(z0), encArg(z1))
   encode_h(z0) -> h(encArg(z0))
   f(0, 1, g(z0, z1), z2) -> f(g(z0, z1), g(z0, z1), g(z0, z1), h(z0))
   g(0, 1) -> 0
   g(0, 1) -> 1
   h(g(z0, z1)) -> h(z0)
Tuples:
   ENCARG(0) -> c
   ENCARG(1) -> c1
   ENCARG(cons_f(z0, z1, z2, z3)) -> c2(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)), ENCARG(z0), ENCARG(z1), ENCARG(z2), ENCARG(z3))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_h(z0)) -> c4(H(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0, z1, z2, z3) -> c5(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)), ENCARG(z0), ENCARG(z1), ENCARG(z2), ENCARG(z3))
   ENCODE_0 -> c6
   ENCODE_1 -> c7
   ENCODE_G(z0, z1) -> c8(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_H(z0) -> c9(H(encArg(z0)), ENCARG(z0))
   F(0, 1, g(z0, z1), z2) -> c10(F(g(z0, z1), g(z0, z1), g(z0, z1), h(z0)), G(z0, z1), G(z0, z1), G(z0, z1), H(z0))
   G(0, 1) -> c11
   G(0, 1) -> c12
   H(g(z0, z1)) -> c13(H(z0))
S tuples:
   F(0, 1, g(z0, z1), z2) -> c10(F(g(z0, z1), g(z0, z1), g(z0, z1), h(z0)), G(z0, z1), G(z0, z1), G(z0, z1), H(z0))
   G(0, 1) -> c11
   G(0, 1) -> c12
   H(g(z0, z1)) -> c13(H(z0))
K tuples:none
Defined Rule Symbols:   f_4, g_2, h_1, encArg_1, encode_f_4, encode_0, encode_1, encode_g_2, encode_h_1

Defined Pair Symbols:   ENCARG_1, ENCODE_F_4, ENCODE_0, ENCODE_1, ENCODE_G_2, ENCODE_H_1, F_4, G_2, H_1

Compound Symbols:   c, c1, c2_5, c3_3, c4_2, c5_5, c6, c7, c8_3, c9_2, c10_5, c11, c12, c13_1


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

(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   ENCODE_1 -> c7
   ENCARG(0) -> c
   ENCODE_0 -> c6
   ENCARG(1) -> c1

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(1) -> 1
   encArg(cons_f(z0, z1, z2, z3)) -> f(encArg(z0), encArg(z1), encArg(z2), encArg(z3))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encArg(cons_h(z0)) -> h(encArg(z0))
   encode_f(z0, z1, z2, z3) -> f(encArg(z0), encArg(z1), encArg(z2), encArg(z3))
   encode_0 -> 0
   encode_1 -> 1
   encode_g(z0, z1) -> g(encArg(z0), encArg(z1))
   encode_h(z0) -> h(encArg(z0))
   f(0, 1, g(z0, z1), z2) -> f(g(z0, z1), g(z0, z1), g(z0, z1), h(z0))
   g(0, 1) -> 0
   g(0, 1) -> 1
   h(g(z0, z1)) -> h(z0)
Tuples:
   ENCARG(cons_f(z0, z1, z2, z3)) -> c2(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)), ENCARG(z0), ENCARG(z1), ENCARG(z2), ENCARG(z3))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_h(z0)) -> c4(H(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0, z1, z2, z3) -> c5(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)), ENCARG(z0), ENCARG(z1), ENCARG(z2), ENCARG(z3))
   ENCODE_G(z0, z1) -> c8(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_H(z0) -> c9(H(encArg(z0)), ENCARG(z0))
   F(0, 1, g(z0, z1), z2) -> c10(F(g(z0, z1), g(z0, z1), g(z0, z1), h(z0)), G(z0, z1), G(z0, z1), G(z0, z1), H(z0))
   G(0, 1) -> c11
   G(0, 1) -> c12
   H(g(z0, z1)) -> c13(H(z0))
S tuples:
   F(0, 1, g(z0, z1), z2) -> c10(F(g(z0, z1), g(z0, z1), g(z0, z1), h(z0)), G(z0, z1), G(z0, z1), G(z0, z1), H(z0))
   G(0, 1) -> c11
   G(0, 1) -> c12
   H(g(z0, z1)) -> c13(H(z0))
K tuples:none
Defined Rule Symbols:   f_4, g_2, h_1, encArg_1, encode_f_4, encode_0, encode_1, encode_g_2, encode_h_1

Defined Pair Symbols:   ENCARG_1, ENCODE_F_4, ENCODE_G_2, ENCODE_H_1, F_4, G_2, H_1

Compound Symbols:   c2_5, c3_3, c4_2, c5_5, c8_3, c9_2, c10_5, c11, c12, c13_1


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

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(1) -> 1
   encArg(cons_f(z0, z1, z2, z3)) -> f(encArg(z0), encArg(z1), encArg(z2), encArg(z3))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encArg(cons_h(z0)) -> h(encArg(z0))
   encode_f(z0, z1, z2, z3) -> f(encArg(z0), encArg(z1), encArg(z2), encArg(z3))
   encode_0 -> 0
   encode_1 -> 1
   encode_g(z0, z1) -> g(encArg(z0), encArg(z1))
   encode_h(z0) -> h(encArg(z0))
   f(0, 1, g(z0, z1), z2) -> f(g(z0, z1), g(z0, z1), g(z0, z1), h(z0))
   g(0, 1) -> 0
   g(0, 1) -> 1
   h(g(z0, z1)) -> h(z0)
Tuples:
   ENCARG(cons_f(z0, z1, z2, z3)) -> c2(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)), ENCARG(z0), ENCARG(z1), ENCARG(z2), ENCARG(z3))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_h(z0)) -> c4(H(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0, z1, z2, z3) -> c5(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)), ENCARG(z0), ENCARG(z1), ENCARG(z2), ENCARG(z3))
   ENCODE_G(z0, z1) -> c8(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_H(z0) -> c9(H(encArg(z0)), ENCARG(z0))
   G(0, 1) -> c11
   G(0, 1) -> c12
   H(g(z0, z1)) -> c13(H(z0))
   F(0, 1, g(z0, z1), z2) -> c10(G(z0, z1), G(z0, z1), G(z0, z1), H(z0))
S tuples:
   G(0, 1) -> c11
   G(0, 1) -> c12
   H(g(z0, z1)) -> c13(H(z0))
   F(0, 1, g(z0, z1), z2) -> c10(G(z0, z1), G(z0, z1), G(z0, z1), H(z0))
K tuples:none
Defined Rule Symbols:   f_4, g_2, h_1, encArg_1, encode_f_4, encode_0, encode_1, encode_g_2, encode_h_1

Defined Pair Symbols:   ENCARG_1, ENCODE_F_4, ENCODE_G_2, ENCODE_H_1, G_2, H_1, F_4

Compound Symbols:   c2_5, c3_3, c4_2, c5_5, c8_3, c9_2, c11, c12, c13_1, c10_4


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

(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(1) -> 1
   encArg(cons_f(z0, z1, z2, z3)) -> f(encArg(z0), encArg(z1), encArg(z2), encArg(z3))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encArg(cons_h(z0)) -> h(encArg(z0))
   encode_f(z0, z1, z2, z3) -> f(encArg(z0), encArg(z1), encArg(z2), encArg(z3))
   encode_0 -> 0
   encode_1 -> 1
   encode_g(z0, z1) -> g(encArg(z0), encArg(z1))
   encode_h(z0) -> h(encArg(z0))
   f(0, 1, g(z0, z1), z2) -> f(g(z0, z1), g(z0, z1), g(z0, z1), h(z0))
   g(0, 1) -> 0
   g(0, 1) -> 1
   h(g(z0, z1)) -> h(z0)
Tuples:
   ENCARG(cons_f(z0, z1, z2, z3)) -> c2(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)), ENCARG(z0), ENCARG(z1), ENCARG(z2), ENCARG(z3))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_h(z0)) -> c4(H(encArg(z0)), ENCARG(z0))
   G(0, 1) -> c11
   G(0, 1) -> c12
   H(g(z0, z1)) -> c13(H(z0))
   ENCODE_F(z0, z1, z2, z3) -> c(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)))
   ENCODE_F(z0, z1, z2, z3) -> c(ENCARG(z0))
   ENCODE_F(z0, z1, z2, z3) -> c(ENCARG(z1))
   ENCODE_F(z0, z1, z2, z3) -> c(ENCARG(z2))
   ENCODE_F(z0, z1, z2, z3) -> c(ENCARG(z3))
   ENCODE_G(z0, z1) -> c(G(encArg(z0), encArg(z1)))
   ENCODE_G(z0, z1) -> c(ENCARG(z0))
   ENCODE_G(z0, z1) -> c(ENCARG(z1))
   ENCODE_H(z0) -> c(H(encArg(z0)))
   ENCODE_H(z0) -> c(ENCARG(z0))
   F(0, 1, g(z0, z1), z2) -> c(G(z0, z1))
   F(0, 1, g(z0, z1), z2) -> c(H(z0))
S tuples:
   G(0, 1) -> c11
   G(0, 1) -> c12
   H(g(z0, z1)) -> c13(H(z0))
   F(0, 1, g(z0, z1), z2) -> c(G(z0, z1))
   F(0, 1, g(z0, z1), z2) -> c(H(z0))
K tuples:none
Defined Rule Symbols:   f_4, g_2, h_1, encArg_1, encode_f_4, encode_0, encode_1, encode_g_2, encode_h_1

Defined Pair Symbols:   ENCARG_1, G_2, H_1, ENCODE_F_4, ENCODE_G_2, ENCODE_H_1, F_4

Compound Symbols:   c2_5, c3_3, c4_2, c11, c12, c13_1, c_1


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

(13) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 7 leading nodes:
   ENCODE_F(z0, z1, z2, z3) -> c(ENCARG(z0))
   ENCODE_F(z0, z1, z2, z3) -> c(ENCARG(z1))
   ENCODE_F(z0, z1, z2, z3) -> c(ENCARG(z2))
   ENCODE_F(z0, z1, z2, z3) -> c(ENCARG(z3))
   ENCODE_G(z0, z1) -> c(ENCARG(z0))
   ENCODE_G(z0, z1) -> c(ENCARG(z1))
   ENCODE_H(z0) -> c(ENCARG(z0))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(1) -> 1
   encArg(cons_f(z0, z1, z2, z3)) -> f(encArg(z0), encArg(z1), encArg(z2), encArg(z3))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encArg(cons_h(z0)) -> h(encArg(z0))
   encode_f(z0, z1, z2, z3) -> f(encArg(z0), encArg(z1), encArg(z2), encArg(z3))
   encode_0 -> 0
   encode_1 -> 1
   encode_g(z0, z1) -> g(encArg(z0), encArg(z1))
   encode_h(z0) -> h(encArg(z0))
   f(0, 1, g(z0, z1), z2) -> f(g(z0, z1), g(z0, z1), g(z0, z1), h(z0))
   g(0, 1) -> 0
   g(0, 1) -> 1
   h(g(z0, z1)) -> h(z0)
Tuples:
   ENCARG(cons_f(z0, z1, z2, z3)) -> c2(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)), ENCARG(z0), ENCARG(z1), ENCARG(z2), ENCARG(z3))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_h(z0)) -> c4(H(encArg(z0)), ENCARG(z0))
   G(0, 1) -> c11
   G(0, 1) -> c12
   H(g(z0, z1)) -> c13(H(z0))
   ENCODE_F(z0, z1, z2, z3) -> c(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)))
   ENCODE_G(z0, z1) -> c(G(encArg(z0), encArg(z1)))
   ENCODE_H(z0) -> c(H(encArg(z0)))
   F(0, 1, g(z0, z1), z2) -> c(G(z0, z1))
   F(0, 1, g(z0, z1), z2) -> c(H(z0))
S tuples:
   G(0, 1) -> c11
   G(0, 1) -> c12
   H(g(z0, z1)) -> c13(H(z0))
   F(0, 1, g(z0, z1), z2) -> c(G(z0, z1))
   F(0, 1, g(z0, z1), z2) -> c(H(z0))
K tuples:none
Defined Rule Symbols:   f_4, g_2, h_1, encArg_1, encode_f_4, encode_0, encode_1, encode_g_2, encode_h_1

Defined Pair Symbols:   ENCARG_1, G_2, H_1, ENCODE_F_4, ENCODE_G_2, ENCODE_H_1, F_4

Compound Symbols:   c2_5, c3_3, c4_2, c11, c12, c13_1, c_1


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

(15) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_f(z0, z1, z2, z3) -> f(encArg(z0), encArg(z1), encArg(z2), encArg(z3))
   encode_0 -> 0
   encode_1 -> 1
   encode_g(z0, z1) -> g(encArg(z0), encArg(z1))
   encode_h(z0) -> h(encArg(z0))

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(1) -> 1
   encArg(cons_f(z0, z1, z2, z3)) -> f(encArg(z0), encArg(z1), encArg(z2), encArg(z3))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encArg(cons_h(z0)) -> h(encArg(z0))
   f(0, 1, g(z0, z1), z2) -> f(g(z0, z1), g(z0, z1), g(z0, z1), h(z0))
   g(0, 1) -> 0
   g(0, 1) -> 1
   h(g(z0, z1)) -> h(z0)
Tuples:
   ENCARG(cons_f(z0, z1, z2, z3)) -> c2(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)), ENCARG(z0), ENCARG(z1), ENCARG(z2), ENCARG(z3))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_h(z0)) -> c4(H(encArg(z0)), ENCARG(z0))
   G(0, 1) -> c11
   G(0, 1) -> c12
   H(g(z0, z1)) -> c13(H(z0))
   ENCODE_F(z0, z1, z2, z3) -> c(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)))
   ENCODE_G(z0, z1) -> c(G(encArg(z0), encArg(z1)))
   ENCODE_H(z0) -> c(H(encArg(z0)))
   F(0, 1, g(z0, z1), z2) -> c(G(z0, z1))
   F(0, 1, g(z0, z1), z2) -> c(H(z0))
S tuples:
   G(0, 1) -> c11
   G(0, 1) -> c12
   H(g(z0, z1)) -> c13(H(z0))
   F(0, 1, g(z0, z1), z2) -> c(G(z0, z1))
   F(0, 1, g(z0, z1), z2) -> c(H(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, f_4, g_2, h_1

Defined Pair Symbols:   ENCARG_1, G_2, H_1, ENCODE_F_4, ENCODE_G_2, ENCODE_H_1, F_4

Compound Symbols:   c2_5, c3_3, c4_2, c11, c12, c13_1, 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.
   G(0, 1) -> c11
   G(0, 1) -> c12
   F(0, 1, g(z0, z1), z2) -> c(H(z0))
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons_f(z0, z1, z2, z3)) -> c2(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)), ENCARG(z0), ENCARG(z1), ENCARG(z2), ENCARG(z3))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_h(z0)) -> c4(H(encArg(z0)), ENCARG(z0))
   G(0, 1) -> c11
   G(0, 1) -> c12
   H(g(z0, z1)) -> c13(H(z0))
   ENCODE_F(z0, z1, z2, z3) -> c(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)))
   ENCODE_G(z0, z1) -> c(G(encArg(z0), encArg(z1)))
   ENCODE_H(z0) -> c(H(encArg(z0)))
   F(0, 1, g(z0, z1), z2) -> c(G(z0, z1))
   F(0, 1, g(z0, z1), z2) -> c(H(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(1) = [1]
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_F(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3
   POL(ENCODE_G(x_1, x_2)) = [1]
   POL(ENCODE_H(x_1)) = x_1
   POL(F(x_1, x_2, x_3, x_4)) = [1]
   POL(G(x_1, x_2)) = [1]
   POL(H(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c11) = 0
   POL(c12) = 0
   POL(c13(x_1)) = x_1
   POL(c2(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_2 + x_3 + x_4 + x_5
   POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(cons_f(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(cons_g(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_h(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(f(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4
   POL(g(x_1, x_2)) = x_2
   POL(h(x_1)) = [1] + x_1

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(1) -> 1
   encArg(cons_f(z0, z1, z2, z3)) -> f(encArg(z0), encArg(z1), encArg(z2), encArg(z3))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encArg(cons_h(z0)) -> h(encArg(z0))
   f(0, 1, g(z0, z1), z2) -> f(g(z0, z1), g(z0, z1), g(z0, z1), h(z0))
   g(0, 1) -> 0
   g(0, 1) -> 1
   h(g(z0, z1)) -> h(z0)
Tuples:
   ENCARG(cons_f(z0, z1, z2, z3)) -> c2(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)), ENCARG(z0), ENCARG(z1), ENCARG(z2), ENCARG(z3))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_h(z0)) -> c4(H(encArg(z0)), ENCARG(z0))
   G(0, 1) -> c11
   G(0, 1) -> c12
   H(g(z0, z1)) -> c13(H(z0))
   ENCODE_F(z0, z1, z2, z3) -> c(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)))
   ENCODE_G(z0, z1) -> c(G(encArg(z0), encArg(z1)))
   ENCODE_H(z0) -> c(H(encArg(z0)))
   F(0, 1, g(z0, z1), z2) -> c(G(z0, z1))
   F(0, 1, g(z0, z1), z2) -> c(H(z0))
S tuples:
   H(g(z0, z1)) -> c13(H(z0))
   F(0, 1, g(z0, z1), z2) -> c(G(z0, z1))
K tuples:
   G(0, 1) -> c11
   G(0, 1) -> c12
   F(0, 1, g(z0, z1), z2) -> c(H(z0))
Defined Rule Symbols:   encArg_1, f_4, g_2, h_1

Defined Pair Symbols:   ENCARG_1, G_2, H_1, ENCODE_F_4, ENCODE_G_2, ENCODE_H_1, F_4

Compound Symbols:   c2_5, c3_3, c4_2, c11, c12, c13_1, 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.
   F(0, 1, g(z0, z1), z2) -> c(G(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons_f(z0, z1, z2, z3)) -> c2(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)), ENCARG(z0), ENCARG(z1), ENCARG(z2), ENCARG(z3))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_h(z0)) -> c4(H(encArg(z0)), ENCARG(z0))
   G(0, 1) -> c11
   G(0, 1) -> c12
   H(g(z0, z1)) -> c13(H(z0))
   ENCODE_F(z0, z1, z2, z3) -> c(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)))
   ENCODE_G(z0, z1) -> c(G(encArg(z0), encArg(z1)))
   ENCODE_H(z0) -> c(H(encArg(z0)))
   F(0, 1, g(z0, z1), z2) -> c(G(z0, z1))
   F(0, 1, g(z0, z1), z2) -> c(H(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(1) = [1]
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_F(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4
   POL(ENCODE_G(x_1, x_2)) = 0
   POL(ENCODE_H(x_1)) = [1]
   POL(F(x_1, x_2, x_3, x_4)) = [1]
   POL(G(x_1, x_2)) = 0
   POL(H(x_1)) = [1]
   POL(c(x_1)) = x_1
   POL(c11) = 0
   POL(c12) = 0
   POL(c13(x_1)) = x_1
   POL(c2(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_2 + x_3 + x_4 + x_5
   POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(cons_f(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(cons_g(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_h(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(f(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4
   POL(g(x_1, x_2)) = x_2
   POL(h(x_1)) = [1] + x_1

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(1) -> 1
   encArg(cons_f(z0, z1, z2, z3)) -> f(encArg(z0), encArg(z1), encArg(z2), encArg(z3))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encArg(cons_h(z0)) -> h(encArg(z0))
   f(0, 1, g(z0, z1), z2) -> f(g(z0, z1), g(z0, z1), g(z0, z1), h(z0))
   g(0, 1) -> 0
   g(0, 1) -> 1
   h(g(z0, z1)) -> h(z0)
Tuples:
   ENCARG(cons_f(z0, z1, z2, z3)) -> c2(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)), ENCARG(z0), ENCARG(z1), ENCARG(z2), ENCARG(z3))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_h(z0)) -> c4(H(encArg(z0)), ENCARG(z0))
   G(0, 1) -> c11
   G(0, 1) -> c12
   H(g(z0, z1)) -> c13(H(z0))
   ENCODE_F(z0, z1, z2, z3) -> c(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)))
   ENCODE_G(z0, z1) -> c(G(encArg(z0), encArg(z1)))
   ENCODE_H(z0) -> c(H(encArg(z0)))
   F(0, 1, g(z0, z1), z2) -> c(G(z0, z1))
   F(0, 1, g(z0, z1), z2) -> c(H(z0))
S tuples:
   H(g(z0, z1)) -> c13(H(z0))
K tuples:
   G(0, 1) -> c11
   G(0, 1) -> c12
   F(0, 1, g(z0, z1), z2) -> c(H(z0))
   F(0, 1, g(z0, z1), z2) -> c(G(z0, z1))
Defined Rule Symbols:   encArg_1, f_4, g_2, h_1

Defined Pair Symbols:   ENCARG_1, G_2, H_1, ENCODE_F_4, ENCODE_G_2, ENCODE_H_1, F_4

Compound Symbols:   c2_5, c3_3, c4_2, c11, c12, c13_1, c_1


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

(21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   H(g(z0, z1)) -> c13(H(z0))
We considered the (Usable) Rules:
   encArg(cons_f(z0, z1, z2, z3)) -> f(encArg(z0), encArg(z1), encArg(z2), encArg(z3))
   f(0, 1, g(z0, z1), z2) -> f(g(z0, z1), g(z0, z1), g(z0, z1), h(z0))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encArg(cons_h(z0)) -> h(encArg(z0))
   encArg(0) -> 0
   encArg(1) -> 1
   g(0, 1) -> 0
   g(0, 1) -> 1
   h(g(z0, z1)) -> h(z0)
And the Tuples:
   ENCARG(cons_f(z0, z1, z2, z3)) -> c2(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)), ENCARG(z0), ENCARG(z1), ENCARG(z2), ENCARG(z3))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_h(z0)) -> c4(H(encArg(z0)), ENCARG(z0))
   G(0, 1) -> c11
   G(0, 1) -> c12
   H(g(z0, z1)) -> c13(H(z0))
   ENCODE_F(z0, z1, z2, z3) -> c(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)))
   ENCODE_G(z0, z1) -> c(G(encArg(z0), encArg(z1)))
   ENCODE_H(z0) -> c(H(encArg(z0)))
   F(0, 1, g(z0, z1), z2) -> c(G(z0, z1))
   F(0, 1, g(z0, z1), z2) -> c(H(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(1) = 0
   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_F(x_1, x_2, x_3, x_4)) = [2] + x_1 + [2]x_2 + [2]x_3 + x_4 + [2]x_4^2 + [2]x_3*x_4 + [2]x_2*x_4 + [2]x_1*x_4 + [2]x_1^2 + [2]x_1*x_2 + [2]x_1*x_3 + [2]x_3^2 + x_2*x_3 + [2]x_2^2
   POL(ENCODE_G(x_1, x_2)) = [2] + [2]x_1 + x_2^2 + x_1*x_2 + x_1^2
   POL(ENCODE_H(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(F(x_1, x_2, x_3, x_4)) = [2]x_3
   POL(G(x_1, x_2)) = 0
   POL(H(x_1)) = [2]x_1
   POL(c(x_1)) = x_1
   POL(c11) = 0
   POL(c12) = 0
   POL(c13(x_1)) = x_1
   POL(c2(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_2 + x_3 + x_4 + x_5
   POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(cons_f(x_1, x_2, x_3, x_4)) = [2] + x_1 + x_2 + x_3 + x_4
   POL(cons_g(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_h(x_1)) = [1] + x_1
   POL(encArg(x_1)) = x_1
   POL(f(x_1, x_2, x_3, x_4)) = x_3 + x_4
   POL(g(x_1, x_2)) = [1] + x_1
   POL(h(x_1)) = 0

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(0) -> 0
   encArg(1) -> 1
   encArg(cons_f(z0, z1, z2, z3)) -> f(encArg(z0), encArg(z1), encArg(z2), encArg(z3))
   encArg(cons_g(z0, z1)) -> g(encArg(z0), encArg(z1))
   encArg(cons_h(z0)) -> h(encArg(z0))
   f(0, 1, g(z0, z1), z2) -> f(g(z0, z1), g(z0, z1), g(z0, z1), h(z0))
   g(0, 1) -> 0
   g(0, 1) -> 1
   h(g(z0, z1)) -> h(z0)
Tuples:
   ENCARG(cons_f(z0, z1, z2, z3)) -> c2(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)), ENCARG(z0), ENCARG(z1), ENCARG(z2), ENCARG(z3))
   ENCARG(cons_g(z0, z1)) -> c3(G(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_h(z0)) -> c4(H(encArg(z0)), ENCARG(z0))
   G(0, 1) -> c11
   G(0, 1) -> c12
   H(g(z0, z1)) -> c13(H(z0))
   ENCODE_F(z0, z1, z2, z3) -> c(F(encArg(z0), encArg(z1), encArg(z2), encArg(z3)))
   ENCODE_G(z0, z1) -> c(G(encArg(z0), encArg(z1)))
   ENCODE_H(z0) -> c(H(encArg(z0)))
   F(0, 1, g(z0, z1), z2) -> c(G(z0, z1))
   F(0, 1, g(z0, z1), z2) -> c(H(z0))
S tuples:none
K tuples:
   G(0, 1) -> c11
   G(0, 1) -> c12
   F(0, 1, g(z0, z1), z2) -> c(H(z0))
   F(0, 1, g(z0, z1), z2) -> c(G(z0, z1))
   H(g(z0, z1)) -> c13(H(z0))
Defined Rule Symbols:   encArg_1, f_4, g_2, h_1

Defined Pair Symbols:   ENCARG_1, G_2, H_1, ENCODE_F_4, ENCODE_G_2, ENCODE_H_1, F_4

Compound Symbols:   c2_5, c3_3, c4_2, c11, c12, c13_1, c_1


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

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

(24)
BOUNDS(1, 1)