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


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


KILLED
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 160 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) typed CpxTrs
    (9) OrderProof [LOWER BOUND(ID), 0 ms]
    (10) typed CpxTrs
    (11) RewriteLemmaProof [LOWER BOUND(ID), 322 ms]
    (12) BOUNDS(1, INF)
(13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(14) TRS for Loop Detection
(15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(16) CpxTRS
(17) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(18) CpxRelTRS
    (19) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxRelTRS
    (21) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxWeightedTrs
        (23) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
        (24) CpxTypedWeightedTrs
        (25) CompletionProof [UPPER BOUND(ID), 0 ms]
        (26) CpxTypedWeightedCompleteTrs
            (27) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
            (28) CpxTypedWeightedCompleteTrs
            (29) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
            (30) CpxRNTS
            (31) InliningProof [UPPER BOUND(ID), 353 ms]
            (32) CpxRNTS
            (33) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
            (34) CpxRNTS
            (35) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
            (36) CpxRNTS
            (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
            (38) CpxRNTS
            (39) IntTrsBoundProof [UPPER BOUND(ID), 104 ms]
            (40) CpxRNTS
            (41) IntTrsBoundProof [UPPER BOUND(ID), 19 ms]
            (42) CpxRNTS
            (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
            (44) CpxRNTS
            (45) IntTrsBoundProof [UPPER BOUND(ID), 150 ms]
            (46) CpxRNTS
            (47) IntTrsBoundProof [UPPER BOUND(ID), 32 ms]
            (48) CpxRNTS
            (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
            (50) CpxRNTS
            (51) IntTrsBoundProof [UPPER BOUND(ID), 3548 ms]
            (52) CpxRNTS
            (53) IntTrsBoundProof [UPPER BOUND(ID), 1182 ms]
            (54) CpxRNTS
        (55) CompletionProof [UPPER BOUND(ID), 0 ms]
        (56) CpxTypedWeightedCompleteTrs
            (57) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
            (58) CpxRNTS
    (59) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
    (60) CdtProblem
        (61) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
        (62) CdtProblem
        (63) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (64) CdtProblem
        (65) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
        (66) CdtProblem
        (67) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
        (68) CdtProblem
        (69) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 80 ms]
        (70) CdtProblem
        (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (72) CdtProblem
        (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (74) CdtProblem
        (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (76) CdtProblem
        (77) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (78) CdtProblem
        (79) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 24 ms]
        (80) CdtProblem
        (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (82) CdtProblem
        (83) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (84) CdtProblem
        (85) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (86) CdtProblem
        (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 31 ms]
        (88) CdtProblem
        (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (90) CdtProblem
        (91) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (92) CdtProblem
        (93) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (94) CdtProblem
        (95) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 19 ms]
        (96) CdtProblem
        (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (98) CdtProblem
        (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (100) CdtProblem
        (101) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (102) CdtProblem
        (103) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 14 ms]
        (104) CdtProblem
        (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (106) CdtProblem
        (107) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (108) CdtProblem
        (109) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (110) CdtProblem


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

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


The TRS R consists of the following rules:

   f(x, x) -> f(i(x), g(g(x)))
   f(x, y) -> x
   g(x) -> i(x)
   f(x, i(x)) -> f(x, x)
   f(i(x), i(g(x))) -> a

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(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(i(x_1)) -> i(encArg(x_1))
   encArg(a) -> a
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_i(x_1) -> i(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a


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

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


The TRS R consists of the following rules:

   f(x, x) -> f(i(x), g(g(x)))
   f(x, y) -> x
   g(x) -> i(x)
   f(x, i(x)) -> f(x, x)
   f(i(x), i(g(x))) -> a

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

   encArg(i(x_1)) -> i(encArg(x_1))
   encArg(a) -> a
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_i(x_1) -> i(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a

Rewrite Strategy: FULL
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(x, x) -> f(i(x), g(g(x)))
   f(x, y) -> x
   g(x) -> i(x)
   f(x, i(x)) -> f(x, x)
   f(i(x), i(g(x))) -> a

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

   encArg(i(x_1)) -> i(encArg(x_1))
   encArg(a) -> a
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_i(x_1) -> i(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a

Rewrite Strategy: FULL
----------------------------------------

(5) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(6)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(x, x) -> f(i(x), g(g(x)))
   f(x, y) -> x
   g(x) -> i(x)
   f(x, i(x)) -> f(x, x)
   f(i(x), i(g(x))) -> a

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

   encArg(i(x_1)) -> i(encArg(x_1))
   encArg(a) -> a
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_i(x_1) -> i(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a

Rewrite Strategy: FULL
----------------------------------------

(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(8)
Obligation:
TRS:
Rules:
f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
g(x) -> i(x)
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a
encArg(i(x_1)) -> i(encArg(x_1))
encArg(a) -> a
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(cons_g(x_1)) -> g(encArg(x_1))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_i(x_1) -> i(encArg(x_1))
encode_g(x_1) -> g(encArg(x_1))
encode_a -> a

Types:
f :: i:a:cons_f:cons_g -> i:a:cons_f:cons_g -> i:a:cons_f:cons_g
i :: i:a:cons_f:cons_g -> i:a:cons_f:cons_g
g :: i:a:cons_f:cons_g -> i:a:cons_f:cons_g
a :: i:a:cons_f:cons_g
encArg :: i:a:cons_f:cons_g -> i:a:cons_f:cons_g
cons_f :: i:a:cons_f:cons_g -> i:a:cons_f:cons_g -> i:a:cons_f:cons_g
cons_g :: i:a:cons_f:cons_g -> i:a:cons_f:cons_g
encode_f :: i:a:cons_f:cons_g -> i:a:cons_f:cons_g -> i:a:cons_f:cons_g
encode_i :: i:a:cons_f:cons_g -> i:a:cons_f:cons_g
encode_g :: i:a:cons_f:cons_g -> i:a:cons_f:cons_g
encode_a :: i:a:cons_f:cons_g
hole_i:a:cons_f:cons_g1_0 :: i:a:cons_f:cons_g
gen_i:a:cons_f:cons_g2_0 :: Nat -> i:a:cons_f:cons_g

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
f, encArg

They will be analysed ascendingly in the following order:
f < encArg

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

(10)
Obligation:
TRS:
Rules:
f(x, x) -> f(i(x), g(g(x)))
f(x, y) -> x
g(x) -> i(x)
f(x, i(x)) -> f(x, x)
f(i(x), i(g(x))) -> a
encArg(i(x_1)) -> i(encArg(x_1))
encArg(a) -> a
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(cons_g(x_1)) -> g(encArg(x_1))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_i(x_1) -> i(encArg(x_1))
encode_g(x_1) -> g(encArg(x_1))
encode_a -> a

Types:
f :: i:a:cons_f:cons_g -> i:a:cons_f:cons_g -> i:a:cons_f:cons_g
i :: i:a:cons_f:cons_g -> i:a:cons_f:cons_g
g :: i:a:cons_f:cons_g -> i:a:cons_f:cons_g
a :: i:a:cons_f:cons_g
encArg :: i:a:cons_f:cons_g -> i:a:cons_f:cons_g
cons_f :: i:a:cons_f:cons_g -> i:a:cons_f:cons_g -> i:a:cons_f:cons_g
cons_g :: i:a:cons_f:cons_g -> i:a:cons_f:cons_g
encode_f :: i:a:cons_f:cons_g -> i:a:cons_f:cons_g -> i:a:cons_f:cons_g
encode_i :: i:a:cons_f:cons_g -> i:a:cons_f:cons_g
encode_g :: i:a:cons_f:cons_g -> i:a:cons_f:cons_g
encode_a :: i:a:cons_f:cons_g
hole_i:a:cons_f:cons_g1_0 :: i:a:cons_f:cons_g
gen_i:a:cons_f:cons_g2_0 :: Nat -> i:a:cons_f:cons_g


Generator Equations:
gen_i:a:cons_f:cons_g2_0(0) <=> a
gen_i:a:cons_f:cons_g2_0(+(x, 1)) <=> i(gen_i:a:cons_f:cons_g2_0(x))


The following defined symbols remain to be analysed:
f, encArg

They will be analysed ascendingly in the following order:
f < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_i:a:cons_f:cons_g2_0(n815_0)) -> gen_i:a:cons_f:cons_g2_0(n815_0), rt in Omega(0)

Induction Base:
encArg(gen_i:a:cons_f:cons_g2_0(0)) ->_R^Omega(0)
a

Induction Step:
encArg(gen_i:a:cons_f:cons_g2_0(+(n815_0, 1))) ->_R^Omega(0)
i(encArg(gen_i:a:cons_f:cons_g2_0(n815_0))) ->_IH
i(gen_i:a:cons_f:cons_g2_0(c816_0))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(12)
BOUNDS(1, INF)

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

(13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(14)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(x, x) -> f(i(x), g(g(x)))
   f(x, y) -> x
   g(x) -> i(x)
   f(x, i(x)) -> f(x, x)
   f(i(x), i(g(x))) -> a

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

   encArg(i(x_1)) -> i(encArg(x_1))
   encArg(a) -> a
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_i(x_1) -> i(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a

Rewrite Strategy: FULL
----------------------------------------

(15) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(16)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(x, x) -> f(i(x), g(g(x)))
   f(x, y) -> x
   g(x) -> i(x)
   f(x, i(x)) -> f(x, x)
   f(i(x), i(g(x))) -> a
   encArg(i(x_1)) -> i(encArg(x_1))
   encArg(a) -> a
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_i(x_1) -> i(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(17) NonCtorToCtorProof (UPPER BOUND(ID))
transformed non-ctor to ctor-system
----------------------------------------

(18)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(x, x) -> f(i(x), g(g(x)))
   f(x, y) -> x
   g(x) -> i(x)
   f(x, i(x)) -> f(x, x)
   f(i(x), i(c_g(x))) -> a

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

   encArg(i(x_1)) -> i(encArg(x_1))
   encArg(a) -> a
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_i(x_1) -> i(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a
   g(x0) -> c_g(x0)

Rewrite Strategy: FULL
----------------------------------------

(19) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.
----------------------------------------

(20)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(x, x) -> f(i(x), g(g(x)))
   f(x, y) -> x
   g(x) -> i(x)
   f(x, i(x)) -> f(x, x)
   f(i(x), i(c_g(x))) -> a

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

   encArg(i(x_1)) -> i(encArg(x_1))
   encArg(a) -> a
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_i(x_1) -> i(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a
   g(x0) -> c_g(x0)

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

(21) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(22)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(x, x) -> f(i(x), g(g(x))) [1]
   f(x, y) -> x [1]
   g(x) -> i(x) [1]
   f(x, i(x)) -> f(x, x) [1]
   f(i(x), i(c_g(x))) -> a [1]
   encArg(i(x_1)) -> i(encArg(x_1)) [0]
   encArg(a) -> a [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_g(x_1)) -> g(encArg(x_1)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_i(x_1) -> i(encArg(x_1)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_a -> a [0]
   g(x0) -> c_g(x0) [0]

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

(23) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(24)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(x, x) -> f(i(x), g(g(x))) [1]
   f(x, y) -> x [1]
   g(x) -> i(x) [1]
   f(x, i(x)) -> f(x, x) [1]
   f(i(x), i(c_g(x))) -> a [1]
   encArg(i(x_1)) -> i(encArg(x_1)) [0]
   encArg(a) -> a [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_g(x_1)) -> g(encArg(x_1)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_i(x_1) -> i(encArg(x_1)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_a -> a [0]
   g(x0) -> c_g(x0) [0]

The TRS has the following type information:
f :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
i :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
g :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
c_g :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
a :: i:c_g:a:cons_f:cons_g
encArg :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
cons_f :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
cons_g :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
encode_f :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
encode_i :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
encode_g :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
encode_a :: i:c_g:a:cons_f:cons_g

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

(25) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
none

(c) The following functions are completely defined:

   f_2
   encArg_1
   encode_f_2
   encode_i_1
   encode_g_1
   encode_a
   g_1

Due to the following rules being added:

   encArg(v0) -> a [0]
   encode_f(v0, v1) -> a [0]
   encode_i(v0) -> a [0]
   encode_g(v0) -> a [0]
   encode_a -> a [0]
   g(v0) -> a [0]

And the following fresh constants: none

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

(26)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(x, x) -> f(i(x), g(g(x))) [1]
   f(x, y) -> x [1]
   g(x) -> i(x) [1]
   f(x, i(x)) -> f(x, x) [1]
   f(i(x), i(c_g(x))) -> a [1]
   encArg(i(x_1)) -> i(encArg(x_1)) [0]
   encArg(a) -> a [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_g(x_1)) -> g(encArg(x_1)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_i(x_1) -> i(encArg(x_1)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_a -> a [0]
   g(x0) -> c_g(x0) [0]
   encArg(v0) -> a [0]
   encode_f(v0, v1) -> a [0]
   encode_i(v0) -> a [0]
   encode_g(v0) -> a [0]
   encode_a -> a [0]
   g(v0) -> a [0]

The TRS has the following type information:
f :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
i :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
g :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
c_g :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
a :: i:c_g:a:cons_f:cons_g
encArg :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
cons_f :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
cons_g :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
encode_f :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
encode_i :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
encode_g :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
encode_a :: i:c_g:a:cons_f:cons_g

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

(27) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(28)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(x, x) -> f(i(x), g(i(x))) [2]
   f(x, x) -> f(i(x), g(c_g(x))) [1]
   f(x, x) -> f(i(x), g(a)) [1]
   f(x, y) -> x [1]
   g(x) -> i(x) [1]
   f(x, i(x)) -> f(x, x) [1]
   f(i(x), i(c_g(x))) -> a [1]
   encArg(i(x_1)) -> i(encArg(x_1)) [0]
   encArg(a) -> a [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_g(i(x_117))) -> g(i(encArg(x_117))) [0]
   encArg(cons_g(a)) -> g(a) [0]
   encArg(cons_g(cons_f(x_118, x_25))) -> g(f(encArg(x_118), encArg(x_25))) [0]
   encArg(cons_g(cons_g(x_119))) -> g(g(encArg(x_119))) [0]
   encArg(cons_g(x_1)) -> g(a) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_i(x_1) -> i(encArg(x_1)) [0]
   encode_g(i(x_138)) -> g(i(encArg(x_138))) [0]
   encode_g(a) -> g(a) [0]
   encode_g(cons_f(x_139, x_212)) -> g(f(encArg(x_139), encArg(x_212))) [0]
   encode_g(cons_g(x_140)) -> g(g(encArg(x_140))) [0]
   encode_g(x_1) -> g(a) [0]
   encode_a -> a [0]
   g(x0) -> c_g(x0) [0]
   encArg(v0) -> a [0]
   encode_f(v0, v1) -> a [0]
   encode_i(v0) -> a [0]
   encode_g(v0) -> a [0]
   encode_a -> a [0]
   g(v0) -> a [0]

The TRS has the following type information:
f :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
i :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
g :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
c_g :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
a :: i:c_g:a:cons_f:cons_g
encArg :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
cons_f :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
cons_g :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
encode_f :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
encode_i :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
encode_g :: i:c_g:a:cons_f:cons_g -> i:c_g:a:cons_f:cons_g
encode_a :: i:c_g:a:cons_f:cons_g

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

(29) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   a => 0

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

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> g(g(encArg(x_119))) :|: z = 1 + (1 + x_119), x_119 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_118), encArg(x_25))) :|: z = 1 + (1 + x_118 + x_25), x_25 >= 0, x_118 >= 0
   encArg(z) -{ 0 }-> g(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> g(0) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> g(1 + encArg(x_117)) :|: z = 1 + (1 + x_117), x_117 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_g(z) -{ 0 }-> g(g(encArg(x_140))) :|: z = 1 + x_140, x_140 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_139), encArg(x_212))) :|: z = 1 + x_139 + x_212, x_212 >= 0, x_139 >= 0
   encode_g(z) -{ 0 }-> g(0) :|: z = 0
   encode_g(z) -{ 0 }-> g(0) :|: x_1 >= 0, z = x_1
   encode_g(z) -{ 0 }-> g(1 + encArg(x_138)) :|: x_138 >= 0, z = 1 + x_138
   encode_g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_i(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_i(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   f(z, z') -{ 1 }-> x :|: x >= 0, y >= 0, z = x, z' = y
   f(z, z') -{ 1 }-> f(x, x) :|: z' = 1 + x, x >= 0, z = x
   f(z, z') -{ 1 }-> f(1 + x, g(0)) :|: z' = x, x >= 0, z = x
   f(z, z') -{ 2 }-> f(1 + x, g(1 + x)) :|: z' = x, x >= 0, z = x
   f(z, z') -{ 1 }-> f(1 + x, g(1 + x)) :|: z' = x, x >= 0, z = x
   f(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 1 + (1 + x)
   g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   g(z) -{ 1 }-> 1 + x :|: x >= 0, z = x
   g(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0


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

(31) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   g(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0
   g(z) -{ 1 }-> 1 + x :|: x >= 0, z = x
   g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

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

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> g(g(encArg(x_119))) :|: z = 1 + (1 + x_119), x_119 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_118), encArg(x_25))) :|: z = 1 + (1 + x_118 + x_25), x_25 >= 0, x_118 >= 0
   encArg(z) -{ 0 }-> g(1 + encArg(x_117)) :|: z = 1 + (1 + x_117), x_117 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + x_1, x_1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 1 }-> 1 + x :|: z = 1 + 0, x >= 0, 0 = x
   encArg(z) -{ 1 }-> 1 + x :|: z = 1 + x_1, x_1 >= 0, x >= 0, 0 = x
   encArg(z) -{ 0 }-> 1 + x0 :|: z = 1 + 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + x0 :|: z = 1 + x_1, x_1 >= 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_g(z) -{ 0 }-> g(g(encArg(x_140))) :|: z = 1 + x_140, x_140 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_139), encArg(x_212))) :|: z = 1 + x_139 + x_212, x_212 >= 0, x_139 >= 0
   encode_g(z) -{ 0 }-> g(1 + encArg(x_138)) :|: x_138 >= 0, z = 1 + x_138
   encode_g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_g(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 0 }-> 0 :|: x_1 >= 0, z = x_1, v0 >= 0, 0 = v0
   encode_g(z) -{ 1 }-> 1 + x :|: z = 0, x >= 0, 0 = x
   encode_g(z) -{ 1 }-> 1 + x :|: x_1 >= 0, z = x_1, x >= 0, 0 = x
   encode_g(z) -{ 0 }-> 1 + x0 :|: z = 0, 0 = x0, x0 >= 0
   encode_g(z) -{ 0 }-> 1 + x0 :|: x_1 >= 0, z = x_1, 0 = x0, x0 >= 0
   encode_i(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_i(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   f(z, z') -{ 1 }-> x :|: x >= 0, y >= 0, z = x, z' = y
   f(z, z') -{ 1 }-> f(x, x) :|: z' = 1 + x, x >= 0, z = x
   f(z, z') -{ 2 }-> f(1 + x, 0) :|: z' = x, x >= 0, z = x, v0 >= 0, 1 + x = v0
   f(z, z') -{ 1 }-> f(1 + x, 0) :|: z' = x, x >= 0, z = x, v0 >= 0, 1 + x = v0
   f(z, z') -{ 1 }-> f(1 + x, 0) :|: z' = x, x >= 0, z = x, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(1 + x, 1 + x') :|: z' = x, x >= 0, z = x, x' >= 0, 1 + x = x'
   f(z, z') -{ 2 }-> f(1 + x, 1 + x') :|: z' = x, x >= 0, z = x, x' >= 0, 1 + x = x'
   f(z, z') -{ 2 }-> f(1 + x, 1 + x') :|: z' = x, x >= 0, z = x, x' >= 0, 0 = x'
   f(z, z') -{ 2 }-> f(1 + x, 1 + x0) :|: z' = x, x >= 0, z = x, 1 + x = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + x, 1 + x0) :|: z' = x, x >= 0, z = x, 1 + x = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + x, 1 + x0) :|: z' = x, x >= 0, z = x, 0 = x0, x0 >= 0
   f(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 1 + (1 + x)
   g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   g(z) -{ 1 }-> 1 + x :|: x >= 0, z = x
   g(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0


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

(33) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> g(g(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_118), encArg(x_25))) :|: z = 1 + (1 + x_118 + x_25), x_25 >= 0, x_118 >= 0
   encArg(z) -{ 0 }-> g(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 1 }-> 1 + x :|: z = 1 + 0, x >= 0, 0 = x
   encArg(z) -{ 1 }-> 1 + x :|: z - 1 >= 0, x >= 0, 0 = x
   encArg(z) -{ 0 }-> 1 + x0 :|: z = 1 + 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + x0 :|: z - 1 >= 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(z - 1))) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_139), encArg(x_212))) :|: z = 1 + x_139 + x_212, x_212 >= 0, x_139 >= 0
   encode_g(z) -{ 0 }-> g(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 1 }-> 1 + x :|: z = 0, x >= 0, 0 = x
   encode_g(z) -{ 1 }-> 1 + x :|: z >= 0, x >= 0, 0 = x
   encode_g(z) -{ 0 }-> 1 + x0 :|: z = 0, 0 = x0, x0 >= 0
   encode_g(z) -{ 0 }-> 1 + x0 :|: z >= 0, 0 = x0, x0 >= 0
   encode_i(z) -{ 0 }-> 0 :|: z >= 0
   encode_i(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1
   f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1))
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 1 }-> 1 + z :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0


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

(35) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { encode_a }
   { g }
   { f }
   { encArg }
   { encode_i }
   { encode_f }
   { encode_g }

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

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> g(g(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_118), encArg(x_25))) :|: z = 1 + (1 + x_118 + x_25), x_25 >= 0, x_118 >= 0
   encArg(z) -{ 0 }-> g(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 1 }-> 1 + x :|: z = 1 + 0, x >= 0, 0 = x
   encArg(z) -{ 1 }-> 1 + x :|: z - 1 >= 0, x >= 0, 0 = x
   encArg(z) -{ 0 }-> 1 + x0 :|: z = 1 + 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + x0 :|: z - 1 >= 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(z - 1))) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_139), encArg(x_212))) :|: z = 1 + x_139 + x_212, x_212 >= 0, x_139 >= 0
   encode_g(z) -{ 0 }-> g(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 1 }-> 1 + x :|: z = 0, x >= 0, 0 = x
   encode_g(z) -{ 1 }-> 1 + x :|: z >= 0, x >= 0, 0 = x
   encode_g(z) -{ 0 }-> 1 + x0 :|: z = 0, 0 = x0, x0 >= 0
   encode_g(z) -{ 0 }-> 1 + x0 :|: z >= 0, 0 = x0, x0 >= 0
   encode_i(z) -{ 0 }-> 0 :|: z >= 0
   encode_i(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1
   f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1))
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 1 }-> 1 + z :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_a}, {g}, {f}, {encArg}, {encode_i}, {encode_f}, {encode_g}

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

(37) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> g(g(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_118), encArg(x_25))) :|: z = 1 + (1 + x_118 + x_25), x_25 >= 0, x_118 >= 0
   encArg(z) -{ 0 }-> g(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 1 }-> 1 + x :|: z = 1 + 0, x >= 0, 0 = x
   encArg(z) -{ 1 }-> 1 + x :|: z - 1 >= 0, x >= 0, 0 = x
   encArg(z) -{ 0 }-> 1 + x0 :|: z = 1 + 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + x0 :|: z - 1 >= 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(z - 1))) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_139), encArg(x_212))) :|: z = 1 + x_139 + x_212, x_212 >= 0, x_139 >= 0
   encode_g(z) -{ 0 }-> g(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 1 }-> 1 + x :|: z = 0, x >= 0, 0 = x
   encode_g(z) -{ 1 }-> 1 + x :|: z >= 0, x >= 0, 0 = x
   encode_g(z) -{ 0 }-> 1 + x0 :|: z = 0, 0 = x0, x0 >= 0
   encode_g(z) -{ 0 }-> 1 + x0 :|: z >= 0, 0 = x0, x0 >= 0
   encode_i(z) -{ 0 }-> 0 :|: z >= 0
   encode_i(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1
   f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1))
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 1 }-> 1 + z :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_a}, {g}, {f}, {encArg}, {encode_i}, {encode_f}, {encode_g}

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_a
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> g(g(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_118), encArg(x_25))) :|: z = 1 + (1 + x_118 + x_25), x_25 >= 0, x_118 >= 0
   encArg(z) -{ 0 }-> g(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 1 }-> 1 + x :|: z = 1 + 0, x >= 0, 0 = x
   encArg(z) -{ 1 }-> 1 + x :|: z - 1 >= 0, x >= 0, 0 = x
   encArg(z) -{ 0 }-> 1 + x0 :|: z = 1 + 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + x0 :|: z - 1 >= 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(z - 1))) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_139), encArg(x_212))) :|: z = 1 + x_139 + x_212, x_212 >= 0, x_139 >= 0
   encode_g(z) -{ 0 }-> g(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 1 }-> 1 + x :|: z = 0, x >= 0, 0 = x
   encode_g(z) -{ 1 }-> 1 + x :|: z >= 0, x >= 0, 0 = x
   encode_g(z) -{ 0 }-> 1 + x0 :|: z = 0, 0 = x0, x0 >= 0
   encode_g(z) -{ 0 }-> 1 + x0 :|: z >= 0, 0 = x0, x0 >= 0
   encode_i(z) -{ 0 }-> 0 :|: z >= 0
   encode_i(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1
   f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1))
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 1 }-> 1 + z :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_a}, {g}, {f}, {encArg}, {encode_i}, {encode_f}, {encode_g}
Previous analysis results are:
  encode_a: runtime: ?, size: O(1) [0]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: encode_a
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> g(g(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_118), encArg(x_25))) :|: z = 1 + (1 + x_118 + x_25), x_25 >= 0, x_118 >= 0
   encArg(z) -{ 0 }-> g(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 1 }-> 1 + x :|: z = 1 + 0, x >= 0, 0 = x
   encArg(z) -{ 1 }-> 1 + x :|: z - 1 >= 0, x >= 0, 0 = x
   encArg(z) -{ 0 }-> 1 + x0 :|: z = 1 + 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + x0 :|: z - 1 >= 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(z - 1))) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_139), encArg(x_212))) :|: z = 1 + x_139 + x_212, x_212 >= 0, x_139 >= 0
   encode_g(z) -{ 0 }-> g(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 1 }-> 1 + x :|: z = 0, x >= 0, 0 = x
   encode_g(z) -{ 1 }-> 1 + x :|: z >= 0, x >= 0, 0 = x
   encode_g(z) -{ 0 }-> 1 + x0 :|: z = 0, 0 = x0, x0 >= 0
   encode_g(z) -{ 0 }-> 1 + x0 :|: z >= 0, 0 = x0, x0 >= 0
   encode_i(z) -{ 0 }-> 0 :|: z >= 0
   encode_i(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1
   f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1))
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 1 }-> 1 + z :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {g}, {f}, {encArg}, {encode_i}, {encode_f}, {encode_g}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]

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

(43) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> g(g(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_118), encArg(x_25))) :|: z = 1 + (1 + x_118 + x_25), x_25 >= 0, x_118 >= 0
   encArg(z) -{ 0 }-> g(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 1 }-> 1 + x :|: z = 1 + 0, x >= 0, 0 = x
   encArg(z) -{ 1 }-> 1 + x :|: z - 1 >= 0, x >= 0, 0 = x
   encArg(z) -{ 0 }-> 1 + x0 :|: z = 1 + 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + x0 :|: z - 1 >= 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(z - 1))) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_139), encArg(x_212))) :|: z = 1 + x_139 + x_212, x_212 >= 0, x_139 >= 0
   encode_g(z) -{ 0 }-> g(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 1 }-> 1 + x :|: z = 0, x >= 0, 0 = x
   encode_g(z) -{ 1 }-> 1 + x :|: z >= 0, x >= 0, 0 = x
   encode_g(z) -{ 0 }-> 1 + x0 :|: z = 0, 0 = x0, x0 >= 0
   encode_g(z) -{ 0 }-> 1 + x0 :|: z >= 0, 0 = x0, x0 >= 0
   encode_i(z) -{ 0 }-> 0 :|: z >= 0
   encode_i(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1
   f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1))
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 1 }-> 1 + z :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {g}, {f}, {encArg}, {encode_i}, {encode_f}, {encode_g}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: g
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z

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

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> g(g(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_118), encArg(x_25))) :|: z = 1 + (1 + x_118 + x_25), x_25 >= 0, x_118 >= 0
   encArg(z) -{ 0 }-> g(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 1 }-> 1 + x :|: z = 1 + 0, x >= 0, 0 = x
   encArg(z) -{ 1 }-> 1 + x :|: z - 1 >= 0, x >= 0, 0 = x
   encArg(z) -{ 0 }-> 1 + x0 :|: z = 1 + 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + x0 :|: z - 1 >= 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(z - 1))) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_139), encArg(x_212))) :|: z = 1 + x_139 + x_212, x_212 >= 0, x_139 >= 0
   encode_g(z) -{ 0 }-> g(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 1 }-> 1 + x :|: z = 0, x >= 0, 0 = x
   encode_g(z) -{ 1 }-> 1 + x :|: z >= 0, x >= 0, 0 = x
   encode_g(z) -{ 0 }-> 1 + x0 :|: z = 0, 0 = x0, x0 >= 0
   encode_g(z) -{ 0 }-> 1 + x0 :|: z >= 0, 0 = x0, x0 >= 0
   encode_i(z) -{ 0 }-> 0 :|: z >= 0
   encode_i(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1
   f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1))
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 1 }-> 1 + z :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {g}, {f}, {encArg}, {encode_i}, {encode_f}, {encode_g}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  g: runtime: ?, size: O(n^1) [1 + z]

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: g
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> g(g(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_118), encArg(x_25))) :|: z = 1 + (1 + x_118 + x_25), x_25 >= 0, x_118 >= 0
   encArg(z) -{ 0 }-> g(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 1 }-> 1 + x :|: z = 1 + 0, x >= 0, 0 = x
   encArg(z) -{ 1 }-> 1 + x :|: z - 1 >= 0, x >= 0, 0 = x
   encArg(z) -{ 0 }-> 1 + x0 :|: z = 1 + 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + x0 :|: z - 1 >= 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(z - 1))) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_139), encArg(x_212))) :|: z = 1 + x_139 + x_212, x_212 >= 0, x_139 >= 0
   encode_g(z) -{ 0 }-> g(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 1 }-> 1 + x :|: z = 0, x >= 0, 0 = x
   encode_g(z) -{ 1 }-> 1 + x :|: z >= 0, x >= 0, 0 = x
   encode_g(z) -{ 0 }-> 1 + x0 :|: z = 0, 0 = x0, x0 >= 0
   encode_g(z) -{ 0 }-> 1 + x0 :|: z >= 0, 0 = x0, x0 >= 0
   encode_i(z) -{ 0 }-> 0 :|: z >= 0
   encode_i(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1
   f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1))
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 1 }-> 1 + z :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {f}, {encArg}, {encode_i}, {encode_f}, {encode_g}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  g: runtime: O(1) [1], size: O(n^1) [1 + z]

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

(49) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> g(g(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_118), encArg(x_25))) :|: z = 1 + (1 + x_118 + x_25), x_25 >= 0, x_118 >= 0
   encArg(z) -{ 0 }-> g(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 1 }-> 1 + x :|: z = 1 + 0, x >= 0, 0 = x
   encArg(z) -{ 1 }-> 1 + x :|: z - 1 >= 0, x >= 0, 0 = x
   encArg(z) -{ 0 }-> 1 + x0 :|: z = 1 + 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + x0 :|: z - 1 >= 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(z - 1))) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_139), encArg(x_212))) :|: z = 1 + x_139 + x_212, x_212 >= 0, x_139 >= 0
   encode_g(z) -{ 0 }-> g(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 1 }-> 1 + x :|: z = 0, x >= 0, 0 = x
   encode_g(z) -{ 1 }-> 1 + x :|: z >= 0, x >= 0, 0 = x
   encode_g(z) -{ 0 }-> 1 + x0 :|: z = 0, 0 = x0, x0 >= 0
   encode_g(z) -{ 0 }-> 1 + x0 :|: z >= 0, 0 = x0, x0 >= 0
   encode_i(z) -{ 0 }-> 0 :|: z >= 0
   encode_i(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1
   f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1))
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 1 }-> 1 + z :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {f}, {encArg}, {encode_i}, {encode_f}, {encode_g}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  g: runtime: O(1) [1], size: O(n^1) [1 + z]

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

(52)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> g(g(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_118), encArg(x_25))) :|: z = 1 + (1 + x_118 + x_25), x_25 >= 0, x_118 >= 0
   encArg(z) -{ 0 }-> g(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 1 }-> 1 + x :|: z = 1 + 0, x >= 0, 0 = x
   encArg(z) -{ 1 }-> 1 + x :|: z - 1 >= 0, x >= 0, 0 = x
   encArg(z) -{ 0 }-> 1 + x0 :|: z = 1 + 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + x0 :|: z - 1 >= 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(z - 1))) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_139), encArg(x_212))) :|: z = 1 + x_139 + x_212, x_212 >= 0, x_139 >= 0
   encode_g(z) -{ 0 }-> g(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 1 }-> 1 + x :|: z = 0, x >= 0, 0 = x
   encode_g(z) -{ 1 }-> 1 + x :|: z >= 0, x >= 0, 0 = x
   encode_g(z) -{ 0 }-> 1 + x0 :|: z = 0, 0 = x0, x0 >= 0
   encode_g(z) -{ 0 }-> 1 + x0 :|: z >= 0, 0 = x0, x0 >= 0
   encode_i(z) -{ 0 }-> 0 :|: z >= 0
   encode_i(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1
   f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1))
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 1 }-> 1 + z :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {f}, {encArg}, {encode_i}, {encode_f}, {encode_g}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  g: runtime: O(1) [1], size: O(n^1) [1 + z]
  f: runtime: ?, size: INF

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

(54)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> g(g(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_118), encArg(x_25))) :|: z = 1 + (1 + x_118 + x_25), x_25 >= 0, x_118 >= 0
   encArg(z) -{ 0 }-> g(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 1 }-> 1 + x :|: z = 1 + 0, x >= 0, 0 = x
   encArg(z) -{ 1 }-> 1 + x :|: z - 1 >= 0, x >= 0, 0 = x
   encArg(z) -{ 0 }-> 1 + x0 :|: z = 1 + 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + x0 :|: z - 1 >= 0, 0 = x0, x0 >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(z - 1))) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_139), encArg(x_212))) :|: z = 1 + x_139 + x_212, x_212 >= 0, x_139 >= 0
   encode_g(z) -{ 0 }-> g(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_g(z) -{ 1 }-> 1 + x :|: z = 0, x >= 0, 0 = x
   encode_g(z) -{ 1 }-> 1 + x :|: z >= 0, x >= 0, 0 = x
   encode_g(z) -{ 0 }-> 1 + x0 :|: z = 0, 0 = x0, x0 >= 0
   encode_g(z) -{ 0 }-> 1 + x0 :|: z >= 0, 0 = x0, x0 >= 0
   encode_i(z) -{ 0 }-> 0 :|: z >= 0
   encode_i(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1
   f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0
   f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x'
   f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0
   f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1))
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 1 }-> 1 + z :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {f}, {encArg}, {encode_i}, {encode_f}, {encode_g}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  g: runtime: O(1) [1], size: O(n^1) [1 + z]
  f: runtime: INF, size: INF

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

(55) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1) -> null_encode_f [0]
   encode_i(v0) -> null_encode_i [0]
   encode_g(v0) -> null_encode_g [0]
   encode_a -> null_encode_a [0]
   g(v0) -> null_g [0]

And the following fresh constants:    null_encArg, null_encode_f, null_encode_i, null_encode_g, null_encode_a, null_g

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

(56)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(x, x) -> f(i(x), g(g(x))) [1]
   f(x, y) -> x [1]
   g(x) -> i(x) [1]
   f(x, i(x)) -> f(x, x) [1]
   f(i(x), i(c_g(x))) -> a [1]
   encArg(i(x_1)) -> i(encArg(x_1)) [0]
   encArg(a) -> a [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_g(x_1)) -> g(encArg(x_1)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_i(x_1) -> i(encArg(x_1)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_a -> a [0]
   g(x0) -> c_g(x0) [0]
   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1) -> null_encode_f [0]
   encode_i(v0) -> null_encode_i [0]
   encode_g(v0) -> null_encode_g [0]
   encode_a -> null_encode_a [0]
   g(v0) -> null_g [0]

The TRS has the following type information:
f :: i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g -> i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g -> i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g
i :: i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g -> i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g
g :: i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g -> i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g
c_g :: i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g -> i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g
a :: i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g
encArg :: i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g -> i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g
cons_f :: i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g -> i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g -> i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g
cons_g :: i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g -> i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g
encode_f :: i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g -> i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g -> i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g
encode_i :: i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g -> i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g
encode_g :: i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g -> i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g
encode_a :: i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g
null_encArg :: i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g
null_encode_f :: i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g
null_encode_i :: i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g
null_encode_g :: i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g
null_encode_a :: i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g
null_g :: i:c_g:a:cons_f:cons_g:null_encArg:null_encode_f:null_encode_i:null_encode_g:null_encode_a:null_g

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

(57) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   a => 0
   null_encArg => 0
   null_encode_f => 0
   null_encode_i => 0
   null_encode_g => 0
   null_encode_a => 0
   null_g => 0

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

(58)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> g(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_g(z) -{ 0 }-> g(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_i(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_i(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   f(z, z') -{ 1 }-> x :|: x >= 0, y >= 0, z = x, z' = y
   f(z, z') -{ 1 }-> f(x, x) :|: z' = 1 + x, x >= 0, z = x
   f(z, z') -{ 1 }-> f(1 + x, g(g(x))) :|: z' = x, x >= 0, z = x
   f(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 1 + (1 + x)
   g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   g(z) -{ 1 }-> 1 + x :|: x >= 0, z = x
   g(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0

Only complete derivations are relevant for the runtime complexity.

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

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_i(z0) -> i(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_a -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(a) -> c1
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0, z1) -> c4(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_I(z0) -> c5(ENCARG(z0))
   ENCODE_G(z0) -> c6(G(encArg(z0)), ENCARG(z0))
   ENCODE_A -> c7
   G(z0) -> c8
   G(z0) -> c9
   F(z0, z0) -> c10(F(i(z0), g(g(z0))), G(g(z0)), G(z0))
   F(z0, z1) -> c11
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(i(z0), i(c_g(z0))) -> c13
S tuples:
   G(z0) -> c9
   F(z0, z0) -> c10(F(i(z0), g(g(z0))), G(g(z0)), G(z0))
   F(z0, z1) -> c11
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(i(z0), i(c_g(z0))) -> c13
K tuples:none
Defined Rule Symbols:   f_2, g_1, encArg_1, encode_f_2, encode_i_1, encode_g_1, encode_a

Defined Pair Symbols:   ENCARG_1, ENCODE_F_2, ENCODE_I_1, ENCODE_G_1, ENCODE_A, G_1, F_2

Compound Symbols:   c_1, c1, c2_3, c3_2, c4_3, c5_1, c6_2, c7, c8, c9, c10_3, c11, c12_1, c13


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

(61) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   ENCODE_I(z0) -> c5(ENCARG(z0))
Removed 3 trailing nodes:
   G(z0) -> c8
   ENCODE_A -> c7
   ENCARG(a) -> c1

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_i(z0) -> i(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_a -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   ENCODE_F(z0, z1) -> c4(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_G(z0) -> c6(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, z0) -> c10(F(i(z0), g(g(z0))), G(g(z0)), G(z0))
   F(z0, z1) -> c11
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(i(z0), i(c_g(z0))) -> c13
S tuples:
   G(z0) -> c9
   F(z0, z0) -> c10(F(i(z0), g(g(z0))), G(g(z0)), G(z0))
   F(z0, z1) -> c11
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(i(z0), i(c_g(z0))) -> c13
K tuples:none
Defined Rule Symbols:   f_2, g_1, encArg_1, encode_f_2, encode_i_1, encode_g_1, encode_a

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

Compound Symbols:   c_1, c2_3, c3_2, c4_3, c6_2, c9, c10_3, c11, c12_1, c13


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

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_i(z0) -> i(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_a -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, z0) -> c10(F(i(z0), g(g(z0))), G(g(z0)), G(z0))
   F(z0, z1) -> c11
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(i(z0), i(c_g(z0))) -> c13
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_F(z0, z1) -> c1(ENCARG(z0))
   ENCODE_F(z0, z1) -> c1(ENCARG(z1))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   ENCODE_G(z0) -> c1(ENCARG(z0))
S tuples:
   G(z0) -> c9
   F(z0, z0) -> c10(F(i(z0), g(g(z0))), G(g(z0)), G(z0))
   F(z0, z1) -> c11
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(i(z0), i(c_g(z0))) -> c13
K tuples:none
Defined Rule Symbols:   f_2, g_1, encArg_1, encode_f_2, encode_i_1, encode_g_1, encode_a

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

Compound Symbols:   c_1, c2_3, c3_2, c9, c10_3, c11, c12_1, c13, c1_1


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

(65) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 3 leading nodes:
   ENCODE_F(z0, z1) -> c1(ENCARG(z0))
   ENCODE_F(z0, z1) -> c1(ENCARG(z1))
   ENCODE_G(z0) -> c1(ENCARG(z0))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_i(z0) -> i(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_a -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, z0) -> c10(F(i(z0), g(g(z0))), G(g(z0)), G(z0))
   F(z0, z1) -> c11
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(i(z0), i(c_g(z0))) -> c13
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
S tuples:
   G(z0) -> c9
   F(z0, z0) -> c10(F(i(z0), g(g(z0))), G(g(z0)), G(z0))
   F(z0, z1) -> c11
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(i(z0), i(c_g(z0))) -> c13
K tuples:none
Defined Rule Symbols:   f_2, g_1, encArg_1, encode_f_2, encode_i_1, encode_g_1, encode_a

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

Compound Symbols:   c_1, c2_3, c3_2, c9, c10_3, c11, c12_1, c13, c1_1


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

(67) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   encode_i(z0) -> i(encArg(z0))
   encode_g(z0) -> g(encArg(z0))
   encode_a -> a

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, z0) -> c10(F(i(z0), g(g(z0))), G(g(z0)), G(z0))
   F(z0, z1) -> c11
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(i(z0), i(c_g(z0))) -> c13
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
S tuples:
   G(z0) -> c9
   F(z0, z0) -> c10(F(i(z0), g(g(z0))), G(g(z0)), G(z0))
   F(z0, z1) -> c11
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(i(z0), i(c_g(z0))) -> c13
K tuples:none
Defined Rule Symbols:   encArg_1, f_2, g_1

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

Compound Symbols:   c_1, c2_3, c3_2, c9, c10_3, c11, c12_1, c13, c1_1


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

(69) 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, z1) -> c11
   F(i(z0), i(c_g(z0))) -> c13
We considered the (Usable) Rules:
   g(z0) -> c_g(z0)
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   encArg(i(z0)) -> i(encArg(z0))
   g(z0) -> i(z0)
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
And the Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, z0) -> c10(F(i(z0), g(g(z0))), G(g(z0)), G(z0))
   F(z0, z1) -> c11
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(i(z0), i(c_g(z0))) -> c13
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_F(x_1, x_2)) = [1] + x_1
   POL(ENCODE_G(x_1)) = x_1
   POL(F(x_1, x_2)) = [1] + x_2
   POL(G(x_1)) = 0
   POL(a) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c11) = 0
   POL(c12(x_1)) = x_1
   POL(c13) = 0
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c9) = 0
   POL(c_g(x_1)) = x_1
   POL(cons_f(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_g(x_1)) = [1] + x_1
   POL(encArg(x_1)) = 0
   POL(f(x_1, x_2)) = x_1 + x_2
   POL(g(x_1)) = x_1
   POL(i(x_1)) = x_1

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, z0) -> c10(F(i(z0), g(g(z0))), G(g(z0)), G(z0))
   F(z0, z1) -> c11
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(i(z0), i(c_g(z0))) -> c13
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
S tuples:
   G(z0) -> c9
   F(z0, z0) -> c10(F(i(z0), g(g(z0))), G(g(z0)), G(z0))
   F(z0, i(z0)) -> c12(F(z0, z0))
K tuples:
   F(z0, z1) -> c11
   F(i(z0), i(c_g(z0))) -> c13
Defined Rule Symbols:   encArg_1, f_2, g_1

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

Compound Symbols:   c_1, c2_3, c3_2, c9, c10_3, c11, c12_1, c13, c1_1


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

(71) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(z0, z0) -> c10(F(i(z0), g(g(z0))), G(g(z0)), G(z0)) by 
   F(x0, x0) -> c10(F(i(x0), c_g(g(x0))), G(g(x0)), G(x0))
   F(x0, x0) -> c10(F(i(x0), i(g(x0))), G(g(x0)), G(x0))
   F(z0, z0) -> c10(F(i(z0), g(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, z1) -> c11
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(i(z0), i(c_g(z0))) -> c13
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(x0, x0) -> c10(F(i(x0), c_g(g(x0))), G(g(x0)), G(x0))
   F(x0, x0) -> c10(F(i(x0), i(g(x0))), G(g(x0)), G(x0))
   F(z0, z0) -> c10(F(i(z0), g(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
S tuples:
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(x0, x0) -> c10(F(i(x0), c_g(g(x0))), G(g(x0)), G(x0))
   F(x0, x0) -> c10(F(i(x0), i(g(x0))), G(g(x0)), G(x0))
   F(z0, z0) -> c10(F(i(z0), g(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
K tuples:
   F(z0, z1) -> c11
   F(i(z0), i(c_g(z0))) -> c13
Defined Rule Symbols:   encArg_1, f_2, g_1

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

Compound Symbols:   c_1, c2_3, c3_2, c9, c11, c12_1, c13, c1_1, c10_3


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

(73) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   F(i(z0), i(c_g(z0))) -> c13
   F(z0, z1) -> c11

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(x0, x0) -> c10(F(i(x0), c_g(g(x0))), G(g(x0)), G(x0))
   F(x0, x0) -> c10(F(i(x0), i(g(x0))), G(g(x0)), G(x0))
   F(z0, z0) -> c10(F(i(z0), g(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
S tuples:
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(x0, x0) -> c10(F(i(x0), c_g(g(x0))), G(g(x0)), G(x0))
   F(x0, x0) -> c10(F(i(x0), i(g(x0))), G(g(x0)), G(x0))
   F(z0, z0) -> c10(F(i(z0), g(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, f_2, g_1

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

Compound Symbols:   c_1, c2_3, c3_2, c9, c12_1, c1_1, c10_3


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

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(x0, x0) -> c10(F(i(x0), i(g(x0))), G(g(x0)), G(x0))
   F(z0, z0) -> c10(F(i(z0), g(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)), G(x0))
S tuples:
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(x0, x0) -> c10(F(i(x0), i(g(x0))), G(g(x0)), G(x0))
   F(z0, z0) -> c10(F(i(z0), g(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)), G(x0))
K tuples:none
Defined Rule Symbols:   encArg_1, f_2, g_1

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

Compound Symbols:   c_1, c2_3, c3_2, c9, c12_1, c1_1, c10_3, c10_2


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

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(x0, x0) -> c10(F(i(x0), i(g(x0))), G(g(x0)), G(x0))
   F(z0, z0) -> c10(F(i(z0), g(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
S tuples:
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(x0, x0) -> c10(F(i(x0), i(g(x0))), G(g(x0)), G(x0))
   F(z0, z0) -> c10(F(i(z0), g(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
K tuples:none
Defined Rule Symbols:   encArg_1, f_2, g_1

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

Compound Symbols:   c_1, c2_3, c3_2, c9, c12_1, c1_1, c10_3, c4_1


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

(79) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
We considered the (Usable) Rules:
   g(z0) -> c_g(z0)
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   encArg(i(z0)) -> i(encArg(z0))
   g(z0) -> i(z0)
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
And the Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(x0, x0) -> c10(F(i(x0), i(g(x0))), G(g(x0)), G(x0))
   F(z0, z0) -> c10(F(i(z0), g(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_F(x_1, x_2)) = [1]
   POL(ENCODE_G(x_1)) = x_1
   POL(F(x_1, x_2)) = [1] + x_2
   POL(G(x_1)) = 0
   POL(a) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1)) = x_1
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c9) = 0
   POL(c_g(x_1)) = x_1
   POL(cons_f(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_g(x_1)) = [1] + x_1
   POL(encArg(x_1)) = 0
   POL(f(x_1, x_2)) = x_1 + x_2
   POL(g(x_1)) = x_1
   POL(i(x_1)) = x_1

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(x0, x0) -> c10(F(i(x0), i(g(x0))), G(g(x0)), G(x0))
   F(z0, z0) -> c10(F(i(z0), g(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
S tuples:
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(x0, x0) -> c10(F(i(x0), i(g(x0))), G(g(x0)), G(x0))
   F(z0, z0) -> c10(F(i(z0), g(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
K tuples:
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
Defined Rule Symbols:   encArg_1, f_2, g_1

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

Compound Symbols:   c_1, c2_3, c3_2, c9, c12_1, c1_1, c10_3, c4_1


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

(81) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(x0, x0) -> c10(F(i(x0), i(g(x0))), G(g(x0)), G(x0)) by 
   F(z0, z0) -> c10(F(i(z0), i(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(z0, z0) -> c10(F(i(z0), g(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(z0, z0) -> c10(F(i(z0), i(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)))
S tuples:
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(z0, z0) -> c10(F(i(z0), g(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), i(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)))
K tuples:
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
Defined Rule Symbols:   encArg_1, f_2, g_1

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

Compound Symbols:   c_1, c2_3, c3_2, c9, c12_1, c1_1, c10_3, c4_1, c10_1


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

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(z0, z0) -> c10(F(i(z0), g(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c10(G(g(z0)), G(z0))
S tuples:
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(z0, z0) -> c10(F(i(z0), g(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c10(G(g(z0)), G(z0))
K tuples:
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
Defined Rule Symbols:   encArg_1, f_2, g_1

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

Compound Symbols:   c_1, c2_3, c3_2, c9, c12_1, c1_1, c10_3, c4_1, c10_1, c10_2


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

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(z0, z0) -> c10(F(i(z0), g(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
S tuples:
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(z0, z0) -> c10(F(i(z0), g(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
K tuples:
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
Defined Rule Symbols:   encArg_1, f_2, g_1

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

Compound Symbols:   c_1, c2_3, c3_2, c9, c12_1, c1_1, c10_3, c4_1, c10_1, c5_1


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

(87) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
We considered the (Usable) Rules:
   g(z0) -> c_g(z0)
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   encArg(i(z0)) -> i(encArg(z0))
   g(z0) -> i(z0)
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
And the Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(z0, z0) -> c10(F(i(z0), g(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_F(x_1, x_2)) = [1]
   POL(ENCODE_G(x_1)) = x_1
   POL(F(x_1, x_2)) = [1] + x_2
   POL(G(x_1)) = 0
   POL(a) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1)) = x_1
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c9) = 0
   POL(c_g(x_1)) = x_1
   POL(cons_f(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_g(x_1)) = [1] + x_1
   POL(encArg(x_1)) = 0
   POL(f(x_1, x_2)) = x_1 + x_2
   POL(g(x_1)) = x_1
   POL(i(x_1)) = x_1

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(z0, z0) -> c10(F(i(z0), g(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
S tuples:
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(z0, z0) -> c10(F(i(z0), g(c_g(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
K tuples:
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
Defined Rule Symbols:   encArg_1, f_2, g_1

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

Compound Symbols:   c_1, c2_3, c3_2, c9, c12_1, c1_1, c10_3, c4_1, c10_1, c5_1


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

(89) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(z0, z0) -> c10(F(i(z0), g(c_g(z0))), G(g(z0)), G(z0)) by 
   F(x0, x0) -> c10(F(i(x0), c_g(c_g(x0))), G(g(x0)), G(x0))
   F(x0, x0) -> c10(F(i(x0), i(c_g(x0))), G(g(x0)), G(x0))
   F(x0, x0) -> c10(G(g(x0)), G(x0))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
   F(x0, x0) -> c10(F(i(x0), c_g(c_g(x0))), G(g(x0)), G(x0))
   F(x0, x0) -> c10(F(i(x0), i(c_g(x0))), G(g(x0)), G(x0))
   F(x0, x0) -> c10(G(g(x0)), G(x0))
S tuples:
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(F(i(x0), c_g(c_g(x0))), G(g(x0)), G(x0))
   F(x0, x0) -> c10(F(i(x0), i(c_g(x0))), G(g(x0)), G(x0))
   F(x0, x0) -> c10(G(g(x0)), G(x0))
K tuples:
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
Defined Rule Symbols:   encArg_1, f_2, g_1

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

Compound Symbols:   c_1, c2_3, c3_2, c9, c12_1, c1_1, c10_3, c4_1, c10_1, c5_1, c10_2


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

(91) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
   F(x0, x0) -> c10(G(g(x0)), G(x0))
S tuples:
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)), G(x0))
K tuples:
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
Defined Rule Symbols:   encArg_1, f_2, g_1

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

Compound Symbols:   c_1, c2_3, c3_2, c9, c12_1, c1_1, c10_3, c4_1, c10_1, c5_1, c10_2


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

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
   F(x0, x0) -> c6(G(g(x0)))
   F(x0, x0) -> c6(G(x0))
S tuples:
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c6(G(g(x0)))
   F(x0, x0) -> c6(G(x0))
K tuples:
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
Defined Rule Symbols:   encArg_1, f_2, g_1

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

Compound Symbols:   c_1, c2_3, c3_2, c9, c12_1, c1_1, c10_3, c4_1, c10_1, c5_1, c6_1


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

(95) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(x0, x0) -> c6(G(g(x0)))
   F(x0, x0) -> c6(G(x0))
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
   F(x0, x0) -> c6(G(g(x0)))
   F(x0, x0) -> c6(G(x0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_F(x_1, x_2)) = [1]
   POL(ENCODE_G(x_1)) = 0
   POL(F(x_1, x_2)) = [1]
   POL(G(x_1)) = 0
   POL(a) = [3]
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1)) = x_1
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c9) = 0
   POL(c_g(x_1)) = [3] + x_1
   POL(cons_f(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_g(x_1)) = x_1
   POL(encArg(x_1)) = [3]x_1
   POL(f(x_1, x_2)) = [3]
   POL(g(x_1)) = 0
   POL(i(x_1)) = x_1

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
   F(x0, x0) -> c6(G(g(x0)))
   F(x0, x0) -> c6(G(x0))
S tuples:
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
K tuples:
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
   F(x0, x0) -> c6(G(g(x0)))
   F(x0, x0) -> c6(G(x0))
Defined Rule Symbols:   encArg_1, f_2, g_1

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

Compound Symbols:   c_1, c2_3, c3_2, c9, c12_1, c1_1, c10_3, c4_1, c10_1, c5_1, c6_1


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

(97) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(z0, z0) -> c10(F(i(z0), g(i(z0))), G(g(z0)), G(z0)) by 
   F(x0, x0) -> c10(F(i(x0), c_g(i(x0))), G(g(x0)), G(x0))
   F(x0, x0) -> c10(F(i(x0), i(i(x0))), G(g(x0)), G(x0))
   F(x0, x0) -> c10(G(g(x0)), G(x0))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
   F(x0, x0) -> c6(G(g(x0)))
   F(x0, x0) -> c6(G(x0))
   F(x0, x0) -> c10(F(i(x0), c_g(i(x0))), G(g(x0)), G(x0))
   F(x0, x0) -> c10(G(g(x0)), G(x0))
S tuples:
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(F(i(x0), c_g(i(x0))), G(g(x0)), G(x0))
   F(x0, x0) -> c10(G(g(x0)), G(x0))
K tuples:
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
   F(x0, x0) -> c6(G(g(x0)))
   F(x0, x0) -> c6(G(x0))
Defined Rule Symbols:   encArg_1, f_2, g_1

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

Compound Symbols:   c_1, c2_3, c3_2, c9, c12_1, c1_1, c4_1, c10_3, c10_1, c5_1, c6_1, c10_2


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

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
   F(x0, x0) -> c6(G(g(x0)))
   F(x0, x0) -> c6(G(x0))
   F(x0, x0) -> c10(G(g(x0)), G(x0))
S tuples:
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)), G(x0))
K tuples:
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
   F(x0, x0) -> c6(G(g(x0)))
   F(x0, x0) -> c6(G(x0))
Defined Rule Symbols:   encArg_1, f_2, g_1

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

Compound Symbols:   c_1, c2_3, c3_2, c9, c12_1, c1_1, c4_1, c10_3, c10_1, c5_1, c6_1, c10_2


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

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
   F(x0, x0) -> c6(G(g(x0)))
   F(x0, x0) -> c6(G(x0))
   F(x0, x0) -> c7(G(g(x0)))
   F(x0, x0) -> c7(G(x0))
S tuples:
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c7(G(g(x0)))
   F(x0, x0) -> c7(G(x0))
K tuples:
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
   F(x0, x0) -> c6(G(g(x0)))
   F(x0, x0) -> c6(G(x0))
Defined Rule Symbols:   encArg_1, f_2, g_1

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

Compound Symbols:   c_1, c2_3, c3_2, c9, c12_1, c1_1, c4_1, c10_3, c10_1, c5_1, c6_1, c7_1


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

(103) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(x0, x0) -> c7(G(g(x0)))
   F(x0, x0) -> c7(G(x0))
We considered the (Usable) Rules:
   g(z0) -> c_g(z0)
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   encArg(i(z0)) -> i(encArg(z0))
   g(z0) -> i(z0)
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
And the Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
   F(x0, x0) -> c6(G(g(x0)))
   F(x0, x0) -> c6(G(x0))
   F(x0, x0) -> c7(G(g(x0)))
   F(x0, x0) -> c7(G(x0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_F(x_1, x_2)) = [1]
   POL(ENCODE_G(x_1)) = x_1
   POL(F(x_1, x_2)) = [1] + x_1 + x_2
   POL(G(x_1)) = 0
   POL(a) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1)) = x_1
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9) = 0
   POL(c_g(x_1)) = x_1
   POL(cons_f(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_g(x_1)) = [1] + x_1
   POL(encArg(x_1)) = 0
   POL(f(x_1, x_2)) = x_1 + x_2
   POL(g(x_1)) = x_1
   POL(i(x_1)) = x_1

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
   F(x0, x0) -> c6(G(g(x0)))
   F(x0, x0) -> c6(G(x0))
   F(x0, x0) -> c7(G(g(x0)))
   F(x0, x0) -> c7(G(x0))
S tuples:
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
K tuples:
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
   F(x0, x0) -> c6(G(g(x0)))
   F(x0, x0) -> c6(G(x0))
   F(x0, x0) -> c7(G(g(x0)))
   F(x0, x0) -> c7(G(x0))
Defined Rule Symbols:   encArg_1, f_2, g_1

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

Compound Symbols:   c_1, c2_3, c3_2, c9, c12_1, c1_1, c4_1, c10_3, c10_1, c5_1, c6_1, c7_1


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

(105) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(x0, x0) -> c4(G(g(x0))) by 
   F(z0, z0) -> c4(G(c_g(z0)))
   F(z0, z0) -> c4(G(i(z0)))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   ENCARG(i(z0)) -> c(ENCARG(z0))
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(x0, x0) -> c4(G(x0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
   F(x0, x0) -> c6(G(g(x0)))
   F(x0, x0) -> c6(G(x0))
   F(x0, x0) -> c7(G(g(x0)))
   F(x0, x0) -> c7(G(x0))
   F(z0, z0) -> c4(G(c_g(z0)))
   F(z0, z0) -> c4(G(i(z0)))
S tuples:
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
K tuples:
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
   F(x0, x0) -> c6(G(g(x0)))
   F(x0, x0) -> c6(G(x0))
   F(x0, x0) -> c7(G(g(x0)))
   F(x0, x0) -> c7(G(x0))
Defined Rule Symbols:   encArg_1, f_2, g_1

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

Compound Symbols:   c_1, c2_3, c3_2, c9, c12_1, c1_1, c4_1, c10_3, c10_1, c5_1, c6_1, c7_1


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

(107) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ENCARG(i(z0)) -> c(ENCARG(z0)) by 
   ENCARG(i(i(y0))) -> c(ENCARG(i(y0)))
   ENCARG(i(cons_f(y0, y1))) -> c(ENCARG(cons_f(y0, y1)))
   ENCARG(i(cons_g(y0))) -> c(ENCARG(cons_g(y0)))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(x0, x0) -> c4(G(x0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
   F(x0, x0) -> c6(G(g(x0)))
   F(x0, x0) -> c6(G(x0))
   F(x0, x0) -> c7(G(g(x0)))
   F(x0, x0) -> c7(G(x0))
   F(z0, z0) -> c4(G(c_g(z0)))
   F(z0, z0) -> c4(G(i(z0)))
   ENCARG(i(i(y0))) -> c(ENCARG(i(y0)))
   ENCARG(i(cons_f(y0, y1))) -> c(ENCARG(cons_f(y0, y1)))
   ENCARG(i(cons_g(y0))) -> c(ENCARG(cons_g(y0)))
S tuples:
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
K tuples:
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
   F(x0, x0) -> c6(G(g(x0)))
   F(x0, x0) -> c6(G(x0))
   F(x0, x0) -> c7(G(g(x0)))
   F(x0, x0) -> c7(G(x0))
Defined Rule Symbols:   encArg_1, f_2, g_1

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

Compound Symbols:   c2_3, c3_2, c9, c12_1, c1_1, c4_1, c10_3, c10_1, c5_1, c6_1, c7_1, c_1


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

(109) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ENCARG(i(i(y0))) -> c(ENCARG(i(y0))) by 
   ENCARG(i(i(i(y0)))) -> c(ENCARG(i(i(y0))))
   ENCARG(i(i(cons_f(y0, y1)))) -> c(ENCARG(i(cons_f(y0, y1))))
   ENCARG(i(i(cons_g(y0)))) -> c(ENCARG(i(cons_g(y0))))

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(i(z0)) -> i(encArg(z0))
   encArg(a) -> a
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encArg(cons_g(z0)) -> g(encArg(z0))
   f(z0, z0) -> f(i(z0), g(g(z0)))
   f(z0, z1) -> z0
   f(z0, i(z0)) -> f(z0, z0)
   f(i(z0), i(c_g(z0))) -> a
   g(z0) -> c_g(z0)
   g(z0) -> i(z0)
Tuples:
   ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0))
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)))
   ENCODE_G(z0) -> c1(G(encArg(z0)))
   F(x0, x0) -> c4(G(x0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
   F(x0, x0) -> c6(G(g(x0)))
   F(x0, x0) -> c6(G(x0))
   F(x0, x0) -> c7(G(g(x0)))
   F(x0, x0) -> c7(G(x0))
   F(z0, z0) -> c4(G(c_g(z0)))
   F(z0, z0) -> c4(G(i(z0)))
   ENCARG(i(cons_f(y0, y1))) -> c(ENCARG(cons_f(y0, y1)))
   ENCARG(i(cons_g(y0))) -> c(ENCARG(cons_g(y0)))
   ENCARG(i(i(i(y0)))) -> c(ENCARG(i(i(y0))))
   ENCARG(i(i(cons_f(y0, y1)))) -> c(ENCARG(i(cons_f(y0, y1))))
   ENCARG(i(i(cons_g(y0)))) -> c(ENCARG(i(cons_g(y0))))
S tuples:
   G(z0) -> c9
   F(z0, i(z0)) -> c12(F(z0, z0))
   F(z0, z0) -> c10(F(i(z0), i(i(z0))), G(g(z0)), G(z0))
K tuples:
   F(x0, x0) -> c4(G(g(x0)))
   F(x0, x0) -> c4(G(x0))
   F(x0, x0) -> c10(G(g(x0)))
   F(z0, z0) -> c5(G(g(z0)))
   F(z0, z0) -> c5(G(z0))
   F(x0, x0) -> c6(G(g(x0)))
   F(x0, x0) -> c6(G(x0))
   F(x0, x0) -> c7(G(g(x0)))
   F(x0, x0) -> c7(G(x0))
Defined Rule Symbols:   encArg_1, f_2, g_1

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

Compound Symbols:   c2_3, c3_2, c9, c12_1, c1_1, c4_1, c10_3, c10_1, c5_1, c6_1, c7_1, c_1