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


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


WORST_CASE(?, O(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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 1).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 51 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
(7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(8) CdtProblem
(9) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CdtProblem
(11) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(12) CdtProblem
(13) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CdtProblem
(15) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CdtProblem
(17) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) CdtProblem
(19) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CdtProblem
(21) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(22) CdtProblem
(23) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(24) CdtProblem
(25) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(26) CdtProblem
(27) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(28) CdtProblem
(29) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
(30) CdtProblem
(31) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
(32) CdtProblem
(33) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(34) CdtProblem
(35) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(36) CdtProblem
(37) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
(38) CdtProblem
(39) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
(40) CdtProblem
(41) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
(42) CdtProblem
(43) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
(44) CdtProblem
(45) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 2 ms]
(46) CdtProblem
(47) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(48) CdtProblem
(49) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(50) BOUNDS(1, 1)


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, 1).


The TRS R consists of the following rules:

   f(f(X)) -> f(g(f(g(f(X)))))
   f(g(f(X))) -> f(g(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(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))


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

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


The TRS R consists of the following rules:

   f(f(X)) -> f(g(f(g(f(X)))))
   f(g(f(X))) -> f(g(X))

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(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, 1).


The TRS R consists of the following rules:

   f(f(X)) -> f(g(f(g(f(X)))))
   f(g(f(X))) -> f(g(X))

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(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(g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   f(f(z0)) -> f(g(f(g(f(z0)))))
   f(g(f(z0))) -> f(g(z0))
Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0) -> c2(F(encArg(z0)), ENCARG(z0))
   ENCODE_G(z0) -> c3(ENCARG(z0))
   F(f(z0)) -> c4(F(g(f(g(f(z0))))), F(g(f(z0))), F(z0))
   F(g(f(z0))) -> c5(F(g(z0)))
S tuples:
   F(f(z0)) -> c4(F(g(f(g(f(z0))))), F(g(f(z0))), F(z0))
   F(g(f(z0))) -> c5(F(g(z0)))
K tuples:none
Defined Rule Symbols:   f_1, encArg_1, encode_f_1, encode_g_1

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, ENCODE_G_1, F_1

Compound Symbols:   c_1, c1_2, c2_2, c3_1, c4_3, c5_1


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   ENCODE_G(z0) -> c3(ENCARG(z0))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   f(f(z0)) -> f(g(f(g(f(z0)))))
   f(g(f(z0))) -> f(g(z0))
Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0) -> c2(F(encArg(z0)), ENCARG(z0))
   F(f(z0)) -> c4(F(g(f(g(f(z0))))), F(g(f(z0))), F(z0))
   F(g(f(z0))) -> c5(F(g(z0)))
S tuples:
   F(f(z0)) -> c4(F(g(f(g(f(z0))))), F(g(f(z0))), F(z0))
   F(g(f(z0))) -> c5(F(g(z0)))
K tuples:none
Defined Rule Symbols:   f_1, encArg_1, encode_f_1, encode_g_1

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, F_1

Compound Symbols:   c_1, c1_2, c2_2, c4_3, c5_1


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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   f(f(z0)) -> f(g(f(g(f(z0)))))
   f(g(f(z0))) -> f(g(z0))
Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0))
   F(f(z0)) -> c4(F(g(f(g(f(z0))))), F(g(f(z0))), F(z0))
   F(g(f(z0))) -> c5(F(g(z0)))
   ENCODE_F(z0) -> c3(F(encArg(z0)))
   ENCODE_F(z0) -> c3(ENCARG(z0))
S tuples:
   F(f(z0)) -> c4(F(g(f(g(f(z0))))), F(g(f(z0))), F(z0))
   F(g(f(z0))) -> c5(F(g(z0)))
K tuples:none
Defined Rule Symbols:   f_1, encArg_1, encode_f_1, encode_g_1

Defined Pair Symbols:   ENCARG_1, F_1, ENCODE_F_1

Compound Symbols:   c_1, c1_2, c4_3, c5_1, c3_1


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

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   ENCODE_F(z0) -> c3(ENCARG(z0))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   encode_f(z0) -> f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   f(f(z0)) -> f(g(f(g(f(z0)))))
   f(g(f(z0))) -> f(g(z0))
Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0))
   F(f(z0)) -> c4(F(g(f(g(f(z0))))), F(g(f(z0))), F(z0))
   F(g(f(z0))) -> c5(F(g(z0)))
   ENCODE_F(z0) -> c3(F(encArg(z0)))
S tuples:
   F(f(z0)) -> c4(F(g(f(g(f(z0))))), F(g(f(z0))), F(z0))
   F(g(f(z0))) -> c5(F(g(z0)))
K tuples:none
Defined Rule Symbols:   f_1, encArg_1, encode_f_1, encode_g_1

Defined Pair Symbols:   ENCARG_1, F_1, ENCODE_F_1

Compound Symbols:   c_1, c1_2, c4_3, c5_1, c3_1


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

(13) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_f(z0) -> f(encArg(z0))
   encode_g(z0) -> g(encArg(z0))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   f(f(z0)) -> f(g(f(g(f(z0)))))
   f(g(f(z0))) -> f(g(z0))
Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0))
   F(f(z0)) -> c4(F(g(f(g(f(z0))))), F(g(f(z0))), F(z0))
   F(g(f(z0))) -> c5(F(g(z0)))
   ENCODE_F(z0) -> c3(F(encArg(z0)))
S tuples:
   F(f(z0)) -> c4(F(g(f(g(f(z0))))), F(g(f(z0))), F(z0))
   F(g(f(z0))) -> c5(F(g(z0)))
K tuples:none
Defined Rule Symbols:   encArg_1, f_1

Defined Pair Symbols:   ENCARG_1, F_1, ENCODE_F_1

Compound Symbols:   c_1, c1_2, c4_3, c5_1, c3_1


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

(15) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0)) by 
   ENCARG(cons_f(g(z0))) -> c1(F(g(encArg(z0))), ENCARG(g(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   f(f(z0)) -> f(g(f(g(f(z0)))))
   f(g(f(z0))) -> f(g(z0))
Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   F(f(z0)) -> c4(F(g(f(g(f(z0))))), F(g(f(z0))), F(z0))
   F(g(f(z0))) -> c5(F(g(z0)))
   ENCODE_F(z0) -> c3(F(encArg(z0)))
   ENCARG(cons_f(g(z0))) -> c1(F(g(encArg(z0))), ENCARG(g(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
S tuples:
   F(f(z0)) -> c4(F(g(f(g(f(z0))))), F(g(f(z0))), F(z0))
   F(g(f(z0))) -> c5(F(g(z0)))
K tuples:none
Defined Rule Symbols:   encArg_1, f_1

Defined Pair Symbols:   ENCARG_1, F_1, ENCODE_F_1

Compound Symbols:   c_1, c4_3, c5_1, c3_1, c1_2


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

(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(f(z0)) -> c4(F(g(f(g(f(z0))))), F(g(f(z0))), F(z0)) by 
   F(f(z0)) -> c4(F(g(f(g(z0)))), F(g(f(z0))), F(z0))

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   f(f(z0)) -> f(g(f(g(f(z0)))))
   f(g(f(z0))) -> f(g(z0))
Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   F(g(f(z0))) -> c5(F(g(z0)))
   ENCODE_F(z0) -> c3(F(encArg(z0)))
   ENCARG(cons_f(g(z0))) -> c1(F(g(encArg(z0))), ENCARG(g(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   F(f(z0)) -> c4(F(g(f(g(z0)))), F(g(f(z0))), F(z0))
S tuples:
   F(g(f(z0))) -> c5(F(g(z0)))
   F(f(z0)) -> c4(F(g(f(g(z0)))), F(g(f(z0))), F(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, f_1

Defined Pair Symbols:   ENCARG_1, F_1, ENCODE_F_1

Compound Symbols:   c_1, c5_1, c3_1, c1_2, c4_3


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

(19) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_F(z0) -> c3(F(encArg(z0))) by 
   ENCODE_F(g(z0)) -> c3(F(g(encArg(z0))))
   ENCODE_F(cons_f(z0)) -> c3(F(f(encArg(z0))))

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   f(f(z0)) -> f(g(f(g(f(z0)))))
   f(g(f(z0))) -> f(g(z0))
Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   F(g(f(z0))) -> c5(F(g(z0)))
   ENCARG(cons_f(g(z0))) -> c1(F(g(encArg(z0))), ENCARG(g(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   F(f(z0)) -> c4(F(g(f(g(z0)))), F(g(f(z0))), F(z0))
   ENCODE_F(g(z0)) -> c3(F(g(encArg(z0))))
   ENCODE_F(cons_f(z0)) -> c3(F(f(encArg(z0))))
S tuples:
   F(g(f(z0))) -> c5(F(g(z0)))
   F(f(z0)) -> c4(F(g(f(g(z0)))), F(g(f(z0))), F(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, f_1

Defined Pair Symbols:   ENCARG_1, F_1, ENCODE_F_1

Compound Symbols:   c_1, c5_1, c1_2, c4_3, c3_1


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

(21) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_f(g(z0))) -> c1(F(g(encArg(z0))), ENCARG(g(z0))) by 
   ENCARG(cons_f(g(g(z0)))) -> c1(F(g(g(encArg(z0)))), ENCARG(g(g(z0))))
   ENCARG(cons_f(g(cons_f(z0)))) -> c1(F(g(f(encArg(z0)))), ENCARG(g(cons_f(z0))))
   ENCARG(cons_f(g(x0))) -> c1(ENCARG(g(x0)))

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   f(f(z0)) -> f(g(f(g(f(z0)))))
   f(g(f(z0))) -> f(g(z0))
Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   F(g(f(z0))) -> c5(F(g(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   F(f(z0)) -> c4(F(g(f(g(z0)))), F(g(f(z0))), F(z0))
   ENCODE_F(g(z0)) -> c3(F(g(encArg(z0))))
   ENCODE_F(cons_f(z0)) -> c3(F(f(encArg(z0))))
   ENCARG(cons_f(g(g(z0)))) -> c1(F(g(g(encArg(z0)))), ENCARG(g(g(z0))))
   ENCARG(cons_f(g(cons_f(z0)))) -> c1(F(g(f(encArg(z0)))), ENCARG(g(cons_f(z0))))
   ENCARG(cons_f(g(x0))) -> c1(ENCARG(g(x0)))
S tuples:
   F(g(f(z0))) -> c5(F(g(z0)))
   F(f(z0)) -> c4(F(g(f(g(z0)))), F(g(f(z0))), F(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, f_1

Defined Pair Symbols:   ENCARG_1, F_1, ENCODE_F_1

Compound Symbols:   c_1, c5_1, c1_2, c4_3, c3_1, c1_1


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

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   f(f(z0)) -> f(g(f(g(f(z0)))))
   f(g(f(z0))) -> f(g(z0))
Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   F(g(f(z0))) -> c5(F(g(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   F(f(z0)) -> c4(F(g(f(g(z0)))), F(g(f(z0))), F(z0))
   ENCODE_F(g(z0)) -> c3(F(g(encArg(z0))))
   ENCODE_F(cons_f(z0)) -> c3(F(f(encArg(z0))))
   ENCARG(cons_f(g(cons_f(z0)))) -> c1(F(g(f(encArg(z0)))), ENCARG(g(cons_f(z0))))
   ENCARG(cons_f(g(x0))) -> c1(ENCARG(g(x0)))
   ENCARG(cons_f(g(g(z0)))) -> c1(ENCARG(g(g(z0))))
S tuples:
   F(g(f(z0))) -> c5(F(g(z0)))
   F(f(z0)) -> c4(F(g(f(g(z0)))), F(g(f(z0))), F(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, f_1

Defined Pair Symbols:   ENCARG_1, F_1, ENCODE_F_1

Compound Symbols:   c_1, c5_1, c1_2, c4_3, c3_1, c1_1


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

(25) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_F(g(z0)) -> c3(F(g(encArg(z0)))) by 
   ENCODE_F(g(g(z0))) -> c3(F(g(g(encArg(z0)))))
   ENCODE_F(g(cons_f(z0))) -> c3(F(g(f(encArg(z0)))))

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

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   f(f(z0)) -> f(g(f(g(f(z0)))))
   f(g(f(z0))) -> f(g(z0))
Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   F(g(f(z0))) -> c5(F(g(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   F(f(z0)) -> c4(F(g(f(g(z0)))), F(g(f(z0))), F(z0))
   ENCODE_F(cons_f(z0)) -> c3(F(f(encArg(z0))))
   ENCARG(cons_f(g(cons_f(z0)))) -> c1(F(g(f(encArg(z0)))), ENCARG(g(cons_f(z0))))
   ENCARG(cons_f(g(x0))) -> c1(ENCARG(g(x0)))
   ENCARG(cons_f(g(g(z0)))) -> c1(ENCARG(g(g(z0))))
   ENCODE_F(g(g(z0))) -> c3(F(g(g(encArg(z0)))))
   ENCODE_F(g(cons_f(z0))) -> c3(F(g(f(encArg(z0)))))
S tuples:
   F(g(f(z0))) -> c5(F(g(z0)))
   F(f(z0)) -> c4(F(g(f(g(z0)))), F(g(f(z0))), F(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, f_1

Defined Pair Symbols:   ENCARG_1, F_1, ENCODE_F_1

Compound Symbols:   c_1, c5_1, c1_2, c4_3, c3_1, c1_1


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

(27) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   ENCODE_F(g(g(z0))) -> c3(F(g(g(encArg(z0)))))

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

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   f(f(z0)) -> f(g(f(g(f(z0)))))
   f(g(f(z0))) -> f(g(z0))
Tuples:
   ENCARG(g(z0)) -> c(ENCARG(z0))
   F(g(f(z0))) -> c5(F(g(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   F(f(z0)) -> c4(F(g(f(g(z0)))), F(g(f(z0))), F(z0))
   ENCODE_F(cons_f(z0)) -> c3(F(f(encArg(z0))))
   ENCARG(cons_f(g(cons_f(z0)))) -> c1(F(g(f(encArg(z0)))), ENCARG(g(cons_f(z0))))
   ENCARG(cons_f(g(x0))) -> c1(ENCARG(g(x0)))
   ENCARG(cons_f(g(g(z0)))) -> c1(ENCARG(g(g(z0))))
   ENCODE_F(g(cons_f(z0))) -> c3(F(g(f(encArg(z0)))))
S tuples:
   F(g(f(z0))) -> c5(F(g(z0)))
   F(f(z0)) -> c4(F(g(f(g(z0)))), F(g(f(z0))), F(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, f_1

Defined Pair Symbols:   ENCARG_1, F_1, ENCODE_F_1

Compound Symbols:   c_1, c5_1, c1_2, c4_3, c3_1, c1_1


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

(29) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ENCARG(g(z0)) -> c(ENCARG(z0)) by 
   ENCARG(g(g(y0))) -> c(ENCARG(g(y0)))
   ENCARG(g(cons_f(cons_f(y0)))) -> c(ENCARG(cons_f(cons_f(y0))))
   ENCARG(g(cons_f(g(cons_f(y0))))) -> c(ENCARG(cons_f(g(cons_f(y0)))))
   ENCARG(g(cons_f(g(y0)))) -> c(ENCARG(cons_f(g(y0))))
   ENCARG(g(cons_f(g(g(y0))))) -> c(ENCARG(cons_f(g(g(y0)))))

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

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   f(f(z0)) -> f(g(f(g(f(z0)))))
   f(g(f(z0))) -> f(g(z0))
Tuples:
   F(g(f(z0))) -> c5(F(g(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   F(f(z0)) -> c4(F(g(f(g(z0)))), F(g(f(z0))), F(z0))
   ENCODE_F(cons_f(z0)) -> c3(F(f(encArg(z0))))
   ENCARG(cons_f(g(cons_f(z0)))) -> c1(F(g(f(encArg(z0)))), ENCARG(g(cons_f(z0))))
   ENCARG(cons_f(g(x0))) -> c1(ENCARG(g(x0)))
   ENCARG(cons_f(g(g(z0)))) -> c1(ENCARG(g(g(z0))))
   ENCODE_F(g(cons_f(z0))) -> c3(F(g(f(encArg(z0)))))
   ENCARG(g(g(y0))) -> c(ENCARG(g(y0)))
   ENCARG(g(cons_f(cons_f(y0)))) -> c(ENCARG(cons_f(cons_f(y0))))
   ENCARG(g(cons_f(g(cons_f(y0))))) -> c(ENCARG(cons_f(g(cons_f(y0)))))
   ENCARG(g(cons_f(g(y0)))) -> c(ENCARG(cons_f(g(y0))))
   ENCARG(g(cons_f(g(g(y0))))) -> c(ENCARG(cons_f(g(g(y0)))))
S tuples:
   F(g(f(z0))) -> c5(F(g(z0)))
   F(f(z0)) -> c4(F(g(f(g(z0)))), F(g(f(z0))), F(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, f_1

Defined Pair Symbols:   F_1, ENCARG_1, ENCODE_F_1

Compound Symbols:   c5_1, c1_2, c4_3, c3_1, c1_1, c_1


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

(31) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(g(f(z0))) -> c5(F(g(z0))) by 
   F(g(f(f(y0)))) -> c5(F(g(f(y0))))

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   f(f(z0)) -> f(g(f(g(f(z0)))))
   f(g(f(z0))) -> f(g(z0))
Tuples:
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   F(f(z0)) -> c4(F(g(f(g(z0)))), F(g(f(z0))), F(z0))
   ENCODE_F(cons_f(z0)) -> c3(F(f(encArg(z0))))
   ENCARG(cons_f(g(cons_f(z0)))) -> c1(F(g(f(encArg(z0)))), ENCARG(g(cons_f(z0))))
   ENCARG(cons_f(g(x0))) -> c1(ENCARG(g(x0)))
   ENCARG(cons_f(g(g(z0)))) -> c1(ENCARG(g(g(z0))))
   ENCODE_F(g(cons_f(z0))) -> c3(F(g(f(encArg(z0)))))
   ENCARG(g(g(y0))) -> c(ENCARG(g(y0)))
   ENCARG(g(cons_f(cons_f(y0)))) -> c(ENCARG(cons_f(cons_f(y0))))
   ENCARG(g(cons_f(g(cons_f(y0))))) -> c(ENCARG(cons_f(g(cons_f(y0)))))
   ENCARG(g(cons_f(g(y0)))) -> c(ENCARG(cons_f(g(y0))))
   ENCARG(g(cons_f(g(g(y0))))) -> c(ENCARG(cons_f(g(g(y0)))))
   F(g(f(f(y0)))) -> c5(F(g(f(y0))))
S tuples:
   F(f(z0)) -> c4(F(g(f(g(z0)))), F(g(f(z0))), F(z0))
   F(g(f(f(y0)))) -> c5(F(g(f(y0))))
K tuples:none
Defined Rule Symbols:   encArg_1, f_1

Defined Pair Symbols:   ENCARG_1, F_1, ENCODE_F_1

Compound Symbols:   c1_2, c4_3, c3_1, c1_1, c_1, c5_1


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

(33) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   ENCODE_F(g(cons_f(z0))) -> c3(F(g(f(encArg(z0)))))
   F(g(f(f(y0)))) -> c5(F(g(f(y0))))

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   f(f(z0)) -> f(g(f(g(f(z0)))))
   f(g(f(z0))) -> f(g(z0))
Tuples:
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   F(f(z0)) -> c4(F(g(f(g(z0)))), F(g(f(z0))), F(z0))
   ENCODE_F(cons_f(z0)) -> c3(F(f(encArg(z0))))
   ENCARG(cons_f(g(cons_f(z0)))) -> c1(F(g(f(encArg(z0)))), ENCARG(g(cons_f(z0))))
   ENCARG(cons_f(g(x0))) -> c1(ENCARG(g(x0)))
   ENCARG(cons_f(g(g(z0)))) -> c1(ENCARG(g(g(z0))))
   ENCARG(g(g(y0))) -> c(ENCARG(g(y0)))
   ENCARG(g(cons_f(cons_f(y0)))) -> c(ENCARG(cons_f(cons_f(y0))))
   ENCARG(g(cons_f(g(cons_f(y0))))) -> c(ENCARG(cons_f(g(cons_f(y0)))))
   ENCARG(g(cons_f(g(y0)))) -> c(ENCARG(cons_f(g(y0))))
   ENCARG(g(cons_f(g(g(y0))))) -> c(ENCARG(cons_f(g(g(y0)))))
S tuples:
   F(f(z0)) -> c4(F(g(f(g(z0)))), F(g(f(z0))), F(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, f_1

Defined Pair Symbols:   ENCARG_1, F_1, ENCODE_F_1

Compound Symbols:   c1_2, c4_3, c3_1, c1_1, c_1


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

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   f(f(z0)) -> f(g(f(g(f(z0)))))
   f(g(f(z0))) -> f(g(z0))
Tuples:
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCODE_F(cons_f(z0)) -> c3(F(f(encArg(z0))))
   ENCARG(cons_f(g(x0))) -> c1(ENCARG(g(x0)))
   ENCARG(cons_f(g(g(z0)))) -> c1(ENCARG(g(g(z0))))
   ENCARG(g(g(y0))) -> c(ENCARG(g(y0)))
   ENCARG(g(cons_f(cons_f(y0)))) -> c(ENCARG(cons_f(cons_f(y0))))
   ENCARG(g(cons_f(g(cons_f(y0))))) -> c(ENCARG(cons_f(g(cons_f(y0)))))
   ENCARG(g(cons_f(g(y0)))) -> c(ENCARG(cons_f(g(y0))))
   ENCARG(g(cons_f(g(g(y0))))) -> c(ENCARG(cons_f(g(g(y0)))))
   F(f(z0)) -> c4(F(z0))
   ENCARG(cons_f(g(cons_f(z0)))) -> c1(ENCARG(g(cons_f(z0))))
S tuples:
   F(f(z0)) -> c4(F(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, f_1

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, F_1

Compound Symbols:   c1_2, c3_1, c1_1, c_1, c4_1


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

(37) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ENCARG(cons_f(g(x0))) -> c1(ENCARG(g(x0))) by 
   ENCARG(cons_f(g(g(y0)))) -> c1(ENCARG(g(g(y0))))
   ENCARG(cons_f(g(cons_f(cons_f(y0))))) -> c1(ENCARG(g(cons_f(cons_f(y0)))))
   ENCARG(cons_f(g(cons_f(g(cons_f(y0)))))) -> c1(ENCARG(g(cons_f(g(cons_f(y0))))))
   ENCARG(cons_f(g(cons_f(g(y0))))) -> c1(ENCARG(g(cons_f(g(y0)))))
   ENCARG(cons_f(g(cons_f(g(g(y0)))))) -> c1(ENCARG(g(cons_f(g(g(y0))))))

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   f(f(z0)) -> f(g(f(g(f(z0)))))
   f(g(f(z0))) -> f(g(z0))
Tuples:
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCODE_F(cons_f(z0)) -> c3(F(f(encArg(z0))))
   ENCARG(cons_f(g(g(z0)))) -> c1(ENCARG(g(g(z0))))
   ENCARG(g(g(y0))) -> c(ENCARG(g(y0)))
   ENCARG(g(cons_f(cons_f(y0)))) -> c(ENCARG(cons_f(cons_f(y0))))
   ENCARG(g(cons_f(g(cons_f(y0))))) -> c(ENCARG(cons_f(g(cons_f(y0)))))
   ENCARG(g(cons_f(g(y0)))) -> c(ENCARG(cons_f(g(y0))))
   ENCARG(g(cons_f(g(g(y0))))) -> c(ENCARG(cons_f(g(g(y0)))))
   F(f(z0)) -> c4(F(z0))
   ENCARG(cons_f(g(cons_f(z0)))) -> c1(ENCARG(g(cons_f(z0))))
   ENCARG(cons_f(g(cons_f(cons_f(y0))))) -> c1(ENCARG(g(cons_f(cons_f(y0)))))
   ENCARG(cons_f(g(cons_f(g(cons_f(y0)))))) -> c1(ENCARG(g(cons_f(g(cons_f(y0))))))
   ENCARG(cons_f(g(cons_f(g(y0))))) -> c1(ENCARG(g(cons_f(g(y0)))))
   ENCARG(cons_f(g(cons_f(g(g(y0)))))) -> c1(ENCARG(g(cons_f(g(g(y0))))))
S tuples:
   F(f(z0)) -> c4(F(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, f_1

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, F_1

Compound Symbols:   c1_2, c3_1, c1_1, c_1, c4_1


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

(39) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ENCARG(g(g(y0))) -> c(ENCARG(g(y0))) by 
   ENCARG(g(g(g(y0)))) -> c(ENCARG(g(g(y0))))
   ENCARG(g(g(cons_f(cons_f(y0))))) -> c(ENCARG(g(cons_f(cons_f(y0)))))
   ENCARG(g(g(cons_f(g(cons_f(y0)))))) -> c(ENCARG(g(cons_f(g(cons_f(y0))))))
   ENCARG(g(g(cons_f(g(y0))))) -> c(ENCARG(g(cons_f(g(y0)))))
   ENCARG(g(g(cons_f(g(g(y0)))))) -> c(ENCARG(g(cons_f(g(g(y0))))))

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   f(f(z0)) -> f(g(f(g(f(z0)))))
   f(g(f(z0))) -> f(g(z0))
Tuples:
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCODE_F(cons_f(z0)) -> c3(F(f(encArg(z0))))
   ENCARG(cons_f(g(g(z0)))) -> c1(ENCARG(g(g(z0))))
   ENCARG(g(cons_f(cons_f(y0)))) -> c(ENCARG(cons_f(cons_f(y0))))
   ENCARG(g(cons_f(g(cons_f(y0))))) -> c(ENCARG(cons_f(g(cons_f(y0)))))
   ENCARG(g(cons_f(g(y0)))) -> c(ENCARG(cons_f(g(y0))))
   ENCARG(g(cons_f(g(g(y0))))) -> c(ENCARG(cons_f(g(g(y0)))))
   F(f(z0)) -> c4(F(z0))
   ENCARG(cons_f(g(cons_f(z0)))) -> c1(ENCARG(g(cons_f(z0))))
   ENCARG(cons_f(g(cons_f(cons_f(y0))))) -> c1(ENCARG(g(cons_f(cons_f(y0)))))
   ENCARG(cons_f(g(cons_f(g(cons_f(y0)))))) -> c1(ENCARG(g(cons_f(g(cons_f(y0))))))
   ENCARG(cons_f(g(cons_f(g(y0))))) -> c1(ENCARG(g(cons_f(g(y0)))))
   ENCARG(cons_f(g(cons_f(g(g(y0)))))) -> c1(ENCARG(g(cons_f(g(g(y0))))))
   ENCARG(g(g(g(y0)))) -> c(ENCARG(g(g(y0))))
   ENCARG(g(g(cons_f(cons_f(y0))))) -> c(ENCARG(g(cons_f(cons_f(y0)))))
   ENCARG(g(g(cons_f(g(cons_f(y0)))))) -> c(ENCARG(g(cons_f(g(cons_f(y0))))))
   ENCARG(g(g(cons_f(g(y0))))) -> c(ENCARG(g(cons_f(g(y0)))))
   ENCARG(g(g(cons_f(g(g(y0)))))) -> c(ENCARG(g(cons_f(g(g(y0))))))
S tuples:
   F(f(z0)) -> c4(F(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, f_1

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, F_1

Compound Symbols:   c1_2, c3_1, c1_1, c_1, c4_1


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

(41) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ENCARG(cons_f(g(g(z0)))) -> c1(ENCARG(g(g(z0)))) by 
   ENCARG(cons_f(g(g(g(y0))))) -> c1(ENCARG(g(g(g(y0)))))
   ENCARG(cons_f(g(g(cons_f(cons_f(y0)))))) -> c1(ENCARG(g(g(cons_f(cons_f(y0))))))
   ENCARG(cons_f(g(g(cons_f(g(cons_f(y0))))))) -> c1(ENCARG(g(g(cons_f(g(cons_f(y0)))))))
   ENCARG(cons_f(g(g(cons_f(g(y0)))))) -> c1(ENCARG(g(g(cons_f(g(y0))))))
   ENCARG(cons_f(g(g(cons_f(g(g(y0))))))) -> c1(ENCARG(g(g(cons_f(g(g(y0)))))))

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   f(f(z0)) -> f(g(f(g(f(z0)))))
   f(g(f(z0))) -> f(g(z0))
Tuples:
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCODE_F(cons_f(z0)) -> c3(F(f(encArg(z0))))
   ENCARG(cons_f(g(g(z0)))) -> c1(ENCARG(g(g(z0))))
   ENCARG(g(cons_f(cons_f(y0)))) -> c(ENCARG(cons_f(cons_f(y0))))
   ENCARG(g(cons_f(g(cons_f(y0))))) -> c(ENCARG(cons_f(g(cons_f(y0)))))
   ENCARG(g(cons_f(g(y0)))) -> c(ENCARG(cons_f(g(y0))))
   ENCARG(g(cons_f(g(g(y0))))) -> c(ENCARG(cons_f(g(g(y0)))))
   F(f(z0)) -> c4(F(z0))
   ENCARG(cons_f(g(cons_f(z0)))) -> c1(ENCARG(g(cons_f(z0))))
   ENCARG(cons_f(g(cons_f(cons_f(y0))))) -> c1(ENCARG(g(cons_f(cons_f(y0)))))
   ENCARG(cons_f(g(cons_f(g(cons_f(y0)))))) -> c1(ENCARG(g(cons_f(g(cons_f(y0))))))
   ENCARG(cons_f(g(cons_f(g(y0))))) -> c1(ENCARG(g(cons_f(g(y0)))))
   ENCARG(cons_f(g(cons_f(g(g(y0)))))) -> c1(ENCARG(g(cons_f(g(g(y0))))))
   ENCARG(g(g(g(y0)))) -> c(ENCARG(g(g(y0))))
   ENCARG(g(g(cons_f(cons_f(y0))))) -> c(ENCARG(g(cons_f(cons_f(y0)))))
   ENCARG(g(g(cons_f(g(cons_f(y0)))))) -> c(ENCARG(g(cons_f(g(cons_f(y0))))))
   ENCARG(g(g(cons_f(g(y0))))) -> c(ENCARG(g(cons_f(g(y0)))))
   ENCARG(g(g(cons_f(g(g(y0)))))) -> c(ENCARG(g(cons_f(g(g(y0))))))
   ENCARG(cons_f(g(g(g(y0))))) -> c1(ENCARG(g(g(g(y0)))))
   ENCARG(cons_f(g(g(cons_f(cons_f(y0)))))) -> c1(ENCARG(g(g(cons_f(cons_f(y0))))))
   ENCARG(cons_f(g(g(cons_f(g(cons_f(y0))))))) -> c1(ENCARG(g(g(cons_f(g(cons_f(y0)))))))
   ENCARG(cons_f(g(g(cons_f(g(y0)))))) -> c1(ENCARG(g(g(cons_f(g(y0))))))
   ENCARG(cons_f(g(g(cons_f(g(g(y0))))))) -> c1(ENCARG(g(g(cons_f(g(g(y0)))))))
S tuples:
   F(f(z0)) -> c4(F(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, f_1

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, F_1

Compound Symbols:   c1_2, c3_1, c1_1, c_1, c4_1


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

(43) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ENCARG(g(cons_f(g(y0)))) -> c(ENCARG(cons_f(g(y0)))) by 
   ENCARG(g(cons_f(g(cons_f(y0))))) -> c(ENCARG(cons_f(g(cons_f(y0)))))
   ENCARG(g(cons_f(g(g(y0))))) -> c(ENCARG(cons_f(g(g(y0)))))
   ENCARG(g(cons_f(g(cons_f(cons_f(y0)))))) -> c(ENCARG(cons_f(g(cons_f(cons_f(y0))))))
   ENCARG(g(cons_f(g(cons_f(g(cons_f(y0))))))) -> c(ENCARG(cons_f(g(cons_f(g(cons_f(y0)))))))
   ENCARG(g(cons_f(g(cons_f(g(y0)))))) -> c(ENCARG(cons_f(g(cons_f(g(y0))))))
   ENCARG(g(cons_f(g(cons_f(g(g(y0))))))) -> c(ENCARG(cons_f(g(cons_f(g(g(y0)))))))
   ENCARG(g(cons_f(g(g(g(y0)))))) -> c(ENCARG(cons_f(g(g(g(y0))))))
   ENCARG(g(cons_f(g(g(cons_f(cons_f(y0))))))) -> c(ENCARG(cons_f(g(g(cons_f(cons_f(y0)))))))
   ENCARG(g(cons_f(g(g(cons_f(g(cons_f(y0)))))))) -> c(ENCARG(cons_f(g(g(cons_f(g(cons_f(y0))))))))
   ENCARG(g(cons_f(g(g(cons_f(g(y0))))))) -> c(ENCARG(cons_f(g(g(cons_f(g(y0)))))))
   ENCARG(g(cons_f(g(g(cons_f(g(g(y0)))))))) -> c(ENCARG(cons_f(g(g(cons_f(g(g(y0))))))))

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   f(f(z0)) -> f(g(f(g(f(z0)))))
   f(g(f(z0))) -> f(g(z0))
Tuples:
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCODE_F(cons_f(z0)) -> c3(F(f(encArg(z0))))
   ENCARG(cons_f(g(g(z0)))) -> c1(ENCARG(g(g(z0))))
   ENCARG(g(cons_f(cons_f(y0)))) -> c(ENCARG(cons_f(cons_f(y0))))
   ENCARG(g(cons_f(g(cons_f(y0))))) -> c(ENCARG(cons_f(g(cons_f(y0)))))
   ENCARG(g(cons_f(g(g(y0))))) -> c(ENCARG(cons_f(g(g(y0)))))
   F(f(z0)) -> c4(F(z0))
   ENCARG(cons_f(g(cons_f(z0)))) -> c1(ENCARG(g(cons_f(z0))))
   ENCARG(cons_f(g(cons_f(cons_f(y0))))) -> c1(ENCARG(g(cons_f(cons_f(y0)))))
   ENCARG(cons_f(g(cons_f(g(cons_f(y0)))))) -> c1(ENCARG(g(cons_f(g(cons_f(y0))))))
   ENCARG(cons_f(g(cons_f(g(y0))))) -> c1(ENCARG(g(cons_f(g(y0)))))
   ENCARG(cons_f(g(cons_f(g(g(y0)))))) -> c1(ENCARG(g(cons_f(g(g(y0))))))
   ENCARG(g(g(g(y0)))) -> c(ENCARG(g(g(y0))))
   ENCARG(g(g(cons_f(cons_f(y0))))) -> c(ENCARG(g(cons_f(cons_f(y0)))))
   ENCARG(g(g(cons_f(g(cons_f(y0)))))) -> c(ENCARG(g(cons_f(g(cons_f(y0))))))
   ENCARG(g(g(cons_f(g(y0))))) -> c(ENCARG(g(cons_f(g(y0)))))
   ENCARG(g(g(cons_f(g(g(y0)))))) -> c(ENCARG(g(cons_f(g(g(y0))))))
   ENCARG(cons_f(g(g(g(y0))))) -> c1(ENCARG(g(g(g(y0)))))
   ENCARG(cons_f(g(g(cons_f(cons_f(y0)))))) -> c1(ENCARG(g(g(cons_f(cons_f(y0))))))
   ENCARG(cons_f(g(g(cons_f(g(cons_f(y0))))))) -> c1(ENCARG(g(g(cons_f(g(cons_f(y0)))))))
   ENCARG(cons_f(g(g(cons_f(g(y0)))))) -> c1(ENCARG(g(g(cons_f(g(y0))))))
   ENCARG(cons_f(g(g(cons_f(g(g(y0))))))) -> c1(ENCARG(g(g(cons_f(g(g(y0)))))))
   ENCARG(g(cons_f(g(cons_f(cons_f(y0)))))) -> c(ENCARG(cons_f(g(cons_f(cons_f(y0))))))
   ENCARG(g(cons_f(g(cons_f(g(cons_f(y0))))))) -> c(ENCARG(cons_f(g(cons_f(g(cons_f(y0)))))))
   ENCARG(g(cons_f(g(cons_f(g(y0)))))) -> c(ENCARG(cons_f(g(cons_f(g(y0))))))
   ENCARG(g(cons_f(g(cons_f(g(g(y0))))))) -> c(ENCARG(cons_f(g(cons_f(g(g(y0)))))))
   ENCARG(g(cons_f(g(g(g(y0)))))) -> c(ENCARG(cons_f(g(g(g(y0))))))
   ENCARG(g(cons_f(g(g(cons_f(cons_f(y0))))))) -> c(ENCARG(cons_f(g(g(cons_f(cons_f(y0)))))))
   ENCARG(g(cons_f(g(g(cons_f(g(cons_f(y0)))))))) -> c(ENCARG(cons_f(g(g(cons_f(g(cons_f(y0))))))))
   ENCARG(g(cons_f(g(g(cons_f(g(y0))))))) -> c(ENCARG(cons_f(g(g(cons_f(g(y0)))))))
   ENCARG(g(cons_f(g(g(cons_f(g(g(y0)))))))) -> c(ENCARG(cons_f(g(g(cons_f(g(g(y0))))))))
S tuples:
   F(f(z0)) -> c4(F(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, f_1

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, F_1

Compound Symbols:   c1_2, c3_1, c1_1, c_1, c4_1


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

(45) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(f(z0)) -> c4(F(z0)) by 
   F(f(f(y0))) -> c4(F(f(y0)))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   f(f(z0)) -> f(g(f(g(f(z0)))))
   f(g(f(z0))) -> f(g(z0))
Tuples:
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCODE_F(cons_f(z0)) -> c3(F(f(encArg(z0))))
   ENCARG(cons_f(g(g(z0)))) -> c1(ENCARG(g(g(z0))))
   ENCARG(g(cons_f(cons_f(y0)))) -> c(ENCARG(cons_f(cons_f(y0))))
   ENCARG(g(cons_f(g(cons_f(y0))))) -> c(ENCARG(cons_f(g(cons_f(y0)))))
   ENCARG(g(cons_f(g(g(y0))))) -> c(ENCARG(cons_f(g(g(y0)))))
   ENCARG(cons_f(g(cons_f(z0)))) -> c1(ENCARG(g(cons_f(z0))))
   ENCARG(cons_f(g(cons_f(cons_f(y0))))) -> c1(ENCARG(g(cons_f(cons_f(y0)))))
   ENCARG(cons_f(g(cons_f(g(cons_f(y0)))))) -> c1(ENCARG(g(cons_f(g(cons_f(y0))))))
   ENCARG(cons_f(g(cons_f(g(y0))))) -> c1(ENCARG(g(cons_f(g(y0)))))
   ENCARG(cons_f(g(cons_f(g(g(y0)))))) -> c1(ENCARG(g(cons_f(g(g(y0))))))
   ENCARG(g(g(g(y0)))) -> c(ENCARG(g(g(y0))))
   ENCARG(g(g(cons_f(cons_f(y0))))) -> c(ENCARG(g(cons_f(cons_f(y0)))))
   ENCARG(g(g(cons_f(g(cons_f(y0)))))) -> c(ENCARG(g(cons_f(g(cons_f(y0))))))
   ENCARG(g(g(cons_f(g(y0))))) -> c(ENCARG(g(cons_f(g(y0)))))
   ENCARG(g(g(cons_f(g(g(y0)))))) -> c(ENCARG(g(cons_f(g(g(y0))))))
   ENCARG(cons_f(g(g(g(y0))))) -> c1(ENCARG(g(g(g(y0)))))
   ENCARG(cons_f(g(g(cons_f(cons_f(y0)))))) -> c1(ENCARG(g(g(cons_f(cons_f(y0))))))
   ENCARG(cons_f(g(g(cons_f(g(cons_f(y0))))))) -> c1(ENCARG(g(g(cons_f(g(cons_f(y0)))))))
   ENCARG(cons_f(g(g(cons_f(g(y0)))))) -> c1(ENCARG(g(g(cons_f(g(y0))))))
   ENCARG(cons_f(g(g(cons_f(g(g(y0))))))) -> c1(ENCARG(g(g(cons_f(g(g(y0)))))))
   ENCARG(g(cons_f(g(cons_f(cons_f(y0)))))) -> c(ENCARG(cons_f(g(cons_f(cons_f(y0))))))
   ENCARG(g(cons_f(g(cons_f(g(cons_f(y0))))))) -> c(ENCARG(cons_f(g(cons_f(g(cons_f(y0)))))))
   ENCARG(g(cons_f(g(cons_f(g(y0)))))) -> c(ENCARG(cons_f(g(cons_f(g(y0))))))
   ENCARG(g(cons_f(g(cons_f(g(g(y0))))))) -> c(ENCARG(cons_f(g(cons_f(g(g(y0)))))))
   ENCARG(g(cons_f(g(g(g(y0)))))) -> c(ENCARG(cons_f(g(g(g(y0))))))
   ENCARG(g(cons_f(g(g(cons_f(cons_f(y0))))))) -> c(ENCARG(cons_f(g(g(cons_f(cons_f(y0)))))))
   ENCARG(g(cons_f(g(g(cons_f(g(cons_f(y0)))))))) -> c(ENCARG(cons_f(g(g(cons_f(g(cons_f(y0))))))))
   ENCARG(g(cons_f(g(g(cons_f(g(y0))))))) -> c(ENCARG(cons_f(g(g(cons_f(g(y0)))))))
   ENCARG(g(cons_f(g(g(cons_f(g(g(y0)))))))) -> c(ENCARG(cons_f(g(g(cons_f(g(g(y0))))))))
   F(f(f(y0))) -> c4(F(f(y0)))
S tuples:
   F(f(f(y0))) -> c4(F(f(y0)))
K tuples:none
Defined Rule Symbols:   encArg_1, f_1

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, F_1

Compound Symbols:   c1_2, c3_1, c1_1, c_1, c4_1


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

(47) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   F(f(f(y0))) -> c4(F(f(y0)))
   ENCODE_F(cons_f(z0)) -> c3(F(f(encArg(z0))))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(g(z0)) -> g(encArg(z0))
   encArg(cons_f(z0)) -> f(encArg(z0))
   f(f(z0)) -> f(g(f(g(f(z0)))))
   f(g(f(z0))) -> f(g(z0))
Tuples:
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(g(cons_f(cons_f(y0)))) -> c(ENCARG(cons_f(cons_f(y0))))
   ENCARG(g(cons_f(g(cons_f(y0))))) -> c(ENCARG(cons_f(g(cons_f(y0)))))
   ENCARG(g(cons_f(g(g(y0))))) -> c(ENCARG(cons_f(g(g(y0)))))
   ENCARG(cons_f(g(cons_f(z0)))) -> c1(ENCARG(g(cons_f(z0))))
   ENCARG(cons_f(g(g(y0)))) -> c1(ENCARG(g(g(y0))))
   ENCARG(cons_f(g(cons_f(cons_f(y0))))) -> c1(ENCARG(g(cons_f(cons_f(y0)))))
   ENCARG(cons_f(g(cons_f(g(cons_f(y0)))))) -> c1(ENCARG(g(cons_f(g(cons_f(y0))))))
   ENCARG(cons_f(g(cons_f(g(y0))))) -> c1(ENCARG(g(cons_f(g(y0)))))
   ENCARG(cons_f(g(cons_f(g(g(y0)))))) -> c1(ENCARG(g(cons_f(g(g(y0))))))
   ENCARG(g(g(g(y0)))) -> c(ENCARG(g(g(y0))))
   ENCARG(g(g(cons_f(cons_f(y0))))) -> c(ENCARG(g(cons_f(cons_f(y0)))))
   ENCARG(g(g(cons_f(g(cons_f(y0)))))) -> c(ENCARG(g(cons_f(g(cons_f(y0))))))
   ENCARG(g(g(cons_f(g(y0))))) -> c(ENCARG(g(cons_f(g(y0)))))
   ENCARG(g(g(cons_f(g(g(y0)))))) -> c(ENCARG(g(cons_f(g(g(y0))))))
   ENCARG(cons_f(g(g(g(y0))))) -> c1(ENCARG(g(g(g(y0)))))
   ENCARG(cons_f(g(g(cons_f(cons_f(y0)))))) -> c1(ENCARG(g(g(cons_f(cons_f(y0))))))
   ENCARG(cons_f(g(g(cons_f(g(cons_f(y0))))))) -> c1(ENCARG(g(g(cons_f(g(cons_f(y0)))))))
   ENCARG(cons_f(g(g(cons_f(g(y0)))))) -> c1(ENCARG(g(g(cons_f(g(y0))))))
   ENCARG(cons_f(g(g(cons_f(g(g(y0))))))) -> c1(ENCARG(g(g(cons_f(g(g(y0)))))))
   ENCARG(g(cons_f(g(cons_f(cons_f(y0)))))) -> c(ENCARG(cons_f(g(cons_f(cons_f(y0))))))
   ENCARG(g(cons_f(g(cons_f(g(cons_f(y0))))))) -> c(ENCARG(cons_f(g(cons_f(g(cons_f(y0)))))))
   ENCARG(g(cons_f(g(cons_f(g(y0)))))) -> c(ENCARG(cons_f(g(cons_f(g(y0))))))
   ENCARG(g(cons_f(g(cons_f(g(g(y0))))))) -> c(ENCARG(cons_f(g(cons_f(g(g(y0)))))))
   ENCARG(g(cons_f(g(g(g(y0)))))) -> c(ENCARG(cons_f(g(g(g(y0))))))
   ENCARG(g(cons_f(g(g(cons_f(cons_f(y0))))))) -> c(ENCARG(cons_f(g(g(cons_f(cons_f(y0)))))))
   ENCARG(g(cons_f(g(g(cons_f(g(cons_f(y0)))))))) -> c(ENCARG(cons_f(g(g(cons_f(g(cons_f(y0))))))))
   ENCARG(g(cons_f(g(g(cons_f(g(y0))))))) -> c(ENCARG(cons_f(g(g(cons_f(g(y0)))))))
   ENCARG(g(cons_f(g(g(cons_f(g(g(y0)))))))) -> c(ENCARG(cons_f(g(g(cons_f(g(g(y0))))))))
S tuples:none
K tuples:none
Defined Rule Symbols:   encArg_1, f_1

Defined Pair Symbols:   ENCARG_1

Compound Symbols:   c1_2, c_1, c1_1


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

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

(50)
BOUNDS(1, 1)