/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 63 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxRelTRS
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 3 ms]
    (10) typed CpxTrs
    (11) OrderProof [LOWER BOUND(ID), 0 ms]
    (12) typed CpxTrs
    (13) RewriteLemmaProof [LOWER BOUND(ID), 2033 ms]
    (14) BOUNDS(1, INF)
(15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(16) CpxTRS
(17) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(18) CpxRelTRS
    (19) RcToIrcProof [BOTH BOUNDS(ID, ID), 6 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) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
            (28) CpxRNTS
        (29) CompletionProof [UPPER BOUND(ID), 0 ms]
        (30) CpxTypedWeightedCompleteTrs
            (31) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
            (32) CpxTypedWeightedCompleteTrs
            (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
            (34) CpxRNTS
            (35) InliningProof [UPPER BOUND(ID), 191 ms]
            (36) CpxRNTS
            (37) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
            (38) CpxRNTS
            (39) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
            (40) CpxRNTS
            (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
            (42) CpxRNTS
            (43) IntTrsBoundProof [UPPER BOUND(ID), 89 ms]
            (44) CpxRNTS
            (45) IntTrsBoundProof [UPPER BOUND(ID), 33 ms]
            (46) CpxRNTS
            (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
            (48) CpxRNTS
            (49) IntTrsBoundProof [UPPER BOUND(ID), 300 ms]
            (50) CpxRNTS
            (51) IntTrsBoundProof [UPPER BOUND(ID), 124 ms]
            (52) CpxRNTS
    (53) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
    (54) CdtProblem
        (55) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
        (56) CdtProblem
        (57) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (58) CdtProblem
        (59) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
        (60) CdtProblem
        (61) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
        (62) CdtProblem
        (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (64) CdtProblem
        (65) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (66) CdtProblem
        (67) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (68) CdtProblem
        (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (70) CdtProblem
        (71) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (72) CdtProblem
        (73) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 19 ms]
        (74) CdtProblem
        (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (76) CdtProblem
        (77) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (78) CdtProblem
        (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (80) CdtProblem
        (81) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (82) CdtProblem
        (83) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (84) CdtProblem
        (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (86) CdtProblem
        (87) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (88) CdtProblem
        (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (90) CdtProblem
        (91) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (92) CdtProblem
        (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (94) CdtProblem
        (95) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
        (96) 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(a) -> f(a)
   a -> b

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(b) -> b
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_a) -> a
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_b -> b


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

(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(a) -> f(a)
   a -> b

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

   encArg(b) -> b
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_a) -> a
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_b -> b

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(a) -> f(a)
   a -> b

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

   encArg(b) -> b
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_a) -> a
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_b -> b

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

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

(6)
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(a) -> f(a)
   a -> b

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

   encArg(b) -> b
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_a) -> a
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_b -> b

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

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

(8)
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(a) -> f(a)
   a -> b

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

   encArg(b) -> b
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_a) -> a
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_b -> b

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

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

(10)
Obligation:
TRS:
Rules:
f(a) -> f(a)
a -> b
encArg(b) -> b
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_a) -> a
encode_f(x_1) -> f(encArg(x_1))
encode_a -> a
encode_b -> b

Types:
f :: b:cons_f:cons_a -> b:cons_f:cons_a
a :: b:cons_f:cons_a
b :: b:cons_f:cons_a
encArg :: b:cons_f:cons_a -> b:cons_f:cons_a
cons_f :: b:cons_f:cons_a -> b:cons_f:cons_a
cons_a :: b:cons_f:cons_a
encode_f :: b:cons_f:cons_a -> b:cons_f:cons_a
encode_a :: b:cons_f:cons_a
encode_b :: b:cons_f:cons_a
hole_b:cons_f:cons_a1_0 :: b:cons_f:cons_a
gen_b:cons_f:cons_a2_0 :: Nat -> b:cons_f:cons_a

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

(11) 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

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

(12)
Obligation:
TRS:
Rules:
f(a) -> f(a)
a -> b
encArg(b) -> b
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_a) -> a
encode_f(x_1) -> f(encArg(x_1))
encode_a -> a
encode_b -> b

Types:
f :: b:cons_f:cons_a -> b:cons_f:cons_a
a :: b:cons_f:cons_a
b :: b:cons_f:cons_a
encArg :: b:cons_f:cons_a -> b:cons_f:cons_a
cons_f :: b:cons_f:cons_a -> b:cons_f:cons_a
cons_a :: b:cons_f:cons_a
encode_f :: b:cons_f:cons_a -> b:cons_f:cons_a
encode_a :: b:cons_f:cons_a
encode_b :: b:cons_f:cons_a
hole_b:cons_f:cons_a1_0 :: b:cons_f:cons_a
gen_b:cons_f:cons_a2_0 :: Nat -> b:cons_f:cons_a


Generator Equations:
gen_b:cons_f:cons_a2_0(0) <=> b
gen_b:cons_f:cons_a2_0(+(x, 1)) <=> cons_f(gen_b:cons_f:cons_a2_0(x))


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

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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_b:cons_f:cons_a2_0(+(1, n7_0))) -> *3_0, rt in Omega(0)

Induction Base:
encArg(gen_b:cons_f:cons_a2_0(+(1, 0)))

Induction Step:
encArg(gen_b:cons_f:cons_a2_0(+(1, +(n7_0, 1)))) ->_R^Omega(0)
f(encArg(gen_b:cons_f:cons_a2_0(+(1, n7_0)))) ->_IH
f(*3_0)

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

(14)
BOUNDS(1, INF)

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

(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(a) -> f(a)
   a -> b
   encArg(b) -> b
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_a) -> a
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_b -> b

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:

   a -> b
   f(c_a) -> f(a)

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

   encArg(b) -> b
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_a) -> a
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_b -> b
   a -> c_a

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:

   a -> b
   f(c_a) -> f(a)

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

   encArg(b) -> b
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_a) -> a
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_b -> b
   a -> c_a

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:

   a -> b [1]
   f(c_a) -> f(a) [1]
   encArg(b) -> b [0]
   encArg(cons_f(x_1)) -> f(encArg(x_1)) [0]
   encArg(cons_a) -> a [0]
   encode_f(x_1) -> f(encArg(x_1)) [0]
   encode_a -> a [0]
   encode_b -> b [0]
   a -> c_a [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:

   a -> b [1]
   f(c_a) -> f(a) [1]
   encArg(b) -> b [0]
   encArg(cons_f(x_1)) -> f(encArg(x_1)) [0]
   encArg(cons_a) -> a [0]
   encode_f(x_1) -> f(encArg(x_1)) [0]
   encode_a -> a [0]
   encode_b -> b [0]
   a -> c_a [0]

The TRS has the following type information:
a :: b:c_a:cons_f:cons_a
b :: b:c_a:cons_f:cons_a
f :: b:c_a:cons_f:cons_a -> b:c_a:cons_f:cons_a
c_a :: b:c_a:cons_f:cons_a
encArg :: b:c_a:cons_f:cons_a -> b:c_a:cons_f:cons_a
cons_f :: b:c_a:cons_f:cons_a -> b:c_a:cons_f:cons_a
cons_a :: b:c_a:cons_f:cons_a
encode_f :: b:c_a:cons_f:cons_a -> b:c_a:cons_f:cons_a
encode_a :: b:c_a:cons_f:cons_a
encode_b :: b:c_a:cons_f:cons_a

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

(25) 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) -> null_encode_f [0]
   encode_a -> null_encode_a [0]
   encode_b -> null_encode_b [0]
   a -> null_a [0]
   f(v0) -> null_f [0]

And the following fresh constants:    null_encArg, null_encode_f, null_encode_a, null_encode_b, null_a, null_f

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

(26)
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:

   a -> b [1]
   f(c_a) -> f(a) [1]
   encArg(b) -> b [0]
   encArg(cons_f(x_1)) -> f(encArg(x_1)) [0]
   encArg(cons_a) -> a [0]
   encode_f(x_1) -> f(encArg(x_1)) [0]
   encode_a -> a [0]
   encode_b -> b [0]
   a -> c_a [0]
   encArg(v0) -> null_encArg [0]
   encode_f(v0) -> null_encode_f [0]
   encode_a -> null_encode_a [0]
   encode_b -> null_encode_b [0]
   a -> null_a [0]
   f(v0) -> null_f [0]

The TRS has the following type information:
a :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
b :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
f :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f -> b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
c_a :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
encArg :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f -> b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
cons_f :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f -> b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
cons_a :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
encode_f :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f -> b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
encode_a :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
encode_b :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
null_encArg :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
null_encode_f :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
null_encode_a :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
null_encode_b :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
null_a :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
null_f :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f

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

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

   b => 0
   c_a => 1
   cons_a => 2
   null_encArg => 0
   null_encode_f => 0
   null_encode_a => 0
   null_encode_b => 0
   null_a => 0
   null_f => 0

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

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 0 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   a -{ 0 }-> 0 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> a :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_a -{ 0 }-> a :|: 
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_f(z) -{ 0 }-> f(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   f(z) -{ 1 }-> f(a) :|: z = 1
   f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.

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

(29) 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_1
   encArg_1
   encode_f_1
   encode_a
   encode_b
   a

Due to the following rules being added:

   encArg(v0) -> null_encArg [0]
   encode_f(v0) -> null_encode_f [0]
   encode_a -> null_encode_a [0]
   encode_b -> null_encode_b [0]
   a -> null_a [0]
   f(v0) -> null_f [0]

And the following fresh constants:    null_encArg, null_encode_f, null_encode_a, null_encode_b, null_a, null_f

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

(30)
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:

   a -> b [1]
   f(c_a) -> f(a) [1]
   encArg(b) -> b [0]
   encArg(cons_f(x_1)) -> f(encArg(x_1)) [0]
   encArg(cons_a) -> a [0]
   encode_f(x_1) -> f(encArg(x_1)) [0]
   encode_a -> a [0]
   encode_b -> b [0]
   a -> c_a [0]
   encArg(v0) -> null_encArg [0]
   encode_f(v0) -> null_encode_f [0]
   encode_a -> null_encode_a [0]
   encode_b -> null_encode_b [0]
   a -> null_a [0]
   f(v0) -> null_f [0]

The TRS has the following type information:
a :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
b :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
f :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f -> b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
c_a :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
encArg :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f -> b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
cons_f :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f -> b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
cons_a :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
encode_f :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f -> b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
encode_a :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
encode_b :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
null_encArg :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
null_encode_f :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
null_encode_a :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
null_encode_b :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
null_a :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
null_f :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f

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

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

(32)
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:

   a -> b [1]
   f(c_a) -> f(b) [2]
   f(c_a) -> f(c_a) [1]
   f(c_a) -> f(null_a) [1]
   encArg(b) -> b [0]
   encArg(cons_f(b)) -> f(b) [0]
   encArg(cons_f(cons_f(x_1'))) -> f(f(encArg(x_1'))) [0]
   encArg(cons_f(cons_a)) -> f(a) [0]
   encArg(cons_f(x_1)) -> f(null_encArg) [0]
   encArg(cons_a) -> a [0]
   encode_f(b) -> f(b) [0]
   encode_f(cons_f(x_1'')) -> f(f(encArg(x_1''))) [0]
   encode_f(cons_a) -> f(a) [0]
   encode_f(x_1) -> f(null_encArg) [0]
   encode_a -> a [0]
   encode_b -> b [0]
   a -> c_a [0]
   encArg(v0) -> null_encArg [0]
   encode_f(v0) -> null_encode_f [0]
   encode_a -> null_encode_a [0]
   encode_b -> null_encode_b [0]
   a -> null_a [0]
   f(v0) -> null_f [0]

The TRS has the following type information:
a :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
b :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
f :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f -> b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
c_a :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
encArg :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f -> b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
cons_f :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f -> b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
cons_a :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
encode_f :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f -> b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
encode_a :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
encode_b :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
null_encArg :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
null_encode_f :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
null_encode_a :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
null_encode_b :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
null_a :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f
null_f :: b:c_a:cons_f:cons_a:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_a:null_f

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

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

   b => 0
   c_a => 1
   cons_a => 2
   null_encArg => 0
   null_encode_f => 0
   null_encode_a => 0
   null_encode_b => 0
   null_a => 0
   null_f => 0

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

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

   a -{ 0 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   a -{ 0 }-> 0 :|: 
   encArg(z) -{ 0 }-> f(f(encArg(x_1'))) :|: z = 1 + (1 + x_1'), x_1' >= 0
   encArg(z) -{ 0 }-> f(a) :|: z = 1 + 2
   encArg(z) -{ 0 }-> f(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> f(0) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> a :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_a -{ 0 }-> a :|: 
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_f(z) -{ 0 }-> f(f(encArg(x_1''))) :|: z = 1 + x_1'', x_1'' >= 0
   encode_f(z) -{ 0 }-> f(a) :|: z = 2
   encode_f(z) -{ 0 }-> f(0) :|: z = 0
   encode_f(z) -{ 0 }-> f(0) :|: x_1 >= 0, z = x_1
   encode_f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   f(z) -{ 1 }-> f(1) :|: z = 1
   f(z) -{ 2 }-> f(0) :|: z = 1
   f(z) -{ 1 }-> f(0) :|: z = 1
   f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

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

   a -{ 0 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   a -{ 0 }-> 0 :|: 

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

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

   a -{ 0 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   a -{ 0 }-> 0 :|: 
   encArg(z) -{ 0 }-> f(f(encArg(x_1'))) :|: z = 1 + (1 + x_1'), x_1' >= 0
   encArg(z) -{ 0 }-> f(1) :|: z = 1 + 2
   encArg(z) -{ 0 }-> f(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> f(0) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 1 }-> f(0) :|: z = 1 + 2
   encArg(z) -{ 0 }-> f(0) :|: z = 1 + 2
   encArg(z) -{ 0 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 1 }-> 0 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 2
   encode_a -{ 0 }-> 1 :|: 
   encode_a -{ 0 }-> 0 :|: 
   encode_a -{ 1 }-> 0 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_f(z) -{ 0 }-> f(f(encArg(x_1''))) :|: z = 1 + x_1'', x_1'' >= 0
   encode_f(z) -{ 0 }-> f(1) :|: z = 2
   encode_f(z) -{ 0 }-> f(0) :|: z = 0
   encode_f(z) -{ 0 }-> f(0) :|: x_1 >= 0, z = x_1
   encode_f(z) -{ 1 }-> f(0) :|: z = 2
   encode_f(z) -{ 0 }-> f(0) :|: z = 2
   encode_f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   f(z) -{ 1 }-> f(1) :|: z = 1
   f(z) -{ 2 }-> f(0) :|: z = 1
   f(z) -{ 1 }-> f(0) :|: z = 1
   f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

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

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

   a -{ 0 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   a -{ 0 }-> 0 :|: 
   encArg(z) -{ 0 }-> f(f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(1) :|: z = 1 + 2
   encArg(z) -{ 0 }-> f(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> f(0) :|: z - 1 >= 0
   encArg(z) -{ 1 }-> f(0) :|: z = 1 + 2
   encArg(z) -{ 0 }-> f(0) :|: z = 1 + 2
   encArg(z) -{ 0 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 1 }-> 0 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 2
   encode_a -{ 0 }-> 1 :|: 
   encode_a -{ 0 }-> 0 :|: 
   encode_a -{ 1 }-> 0 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_f(z) -{ 0 }-> f(f(encArg(z - 1))) :|: z - 1 >= 0
   encode_f(z) -{ 0 }-> f(1) :|: z = 2
   encode_f(z) -{ 0 }-> f(0) :|: z = 0
   encode_f(z) -{ 0 }-> f(0) :|: z >= 0
   encode_f(z) -{ 1 }-> f(0) :|: z = 2
   encode_f(z) -{ 0 }-> f(0) :|: z = 2
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   f(z) -{ 1 }-> f(1) :|: z = 1
   f(z) -{ 2 }-> f(0) :|: z = 1
   f(z) -{ 1 }-> f(0) :|: z = 1
   f(z) -{ 0 }-> 0 :|: z >= 0


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

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

   { encode_a }
   { f }
   { encode_b }
   { a }
   { encArg }
   { encode_f }

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

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

   a -{ 0 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   a -{ 0 }-> 0 :|: 
   encArg(z) -{ 0 }-> f(f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(1) :|: z = 1 + 2
   encArg(z) -{ 0 }-> f(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> f(0) :|: z - 1 >= 0
   encArg(z) -{ 1 }-> f(0) :|: z = 1 + 2
   encArg(z) -{ 0 }-> f(0) :|: z = 1 + 2
   encArg(z) -{ 0 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 1 }-> 0 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 2
   encode_a -{ 0 }-> 1 :|: 
   encode_a -{ 0 }-> 0 :|: 
   encode_a -{ 1 }-> 0 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_f(z) -{ 0 }-> f(f(encArg(z - 1))) :|: z - 1 >= 0
   encode_f(z) -{ 0 }-> f(1) :|: z = 2
   encode_f(z) -{ 0 }-> f(0) :|: z = 0
   encode_f(z) -{ 0 }-> f(0) :|: z >= 0
   encode_f(z) -{ 1 }-> f(0) :|: z = 2
   encode_f(z) -{ 0 }-> f(0) :|: z = 2
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   f(z) -{ 1 }-> f(1) :|: z = 1
   f(z) -{ 2 }-> f(0) :|: z = 1
   f(z) -{ 1 }-> f(0) :|: z = 1
   f(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encode_a}, {f}, {encode_b}, {a}, {encArg}, {encode_f}

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

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

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

   a -{ 0 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   a -{ 0 }-> 0 :|: 
   encArg(z) -{ 0 }-> f(f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(1) :|: z = 1 + 2
   encArg(z) -{ 0 }-> f(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> f(0) :|: z - 1 >= 0
   encArg(z) -{ 1 }-> f(0) :|: z = 1 + 2
   encArg(z) -{ 0 }-> f(0) :|: z = 1 + 2
   encArg(z) -{ 0 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 1 }-> 0 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 2
   encode_a -{ 0 }-> 1 :|: 
   encode_a -{ 0 }-> 0 :|: 
   encode_a -{ 1 }-> 0 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_f(z) -{ 0 }-> f(f(encArg(z - 1))) :|: z - 1 >= 0
   encode_f(z) -{ 0 }-> f(1) :|: z = 2
   encode_f(z) -{ 0 }-> f(0) :|: z = 0
   encode_f(z) -{ 0 }-> f(0) :|: z >= 0
   encode_f(z) -{ 1 }-> f(0) :|: z = 2
   encode_f(z) -{ 0 }-> f(0) :|: z = 2
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   f(z) -{ 1 }-> f(1) :|: z = 1
   f(z) -{ 2 }-> f(0) :|: z = 1
   f(z) -{ 1 }-> f(0) :|: z = 1
   f(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encode_a}, {f}, {encode_b}, {a}, {encArg}, {encode_f}

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

(43) 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: 1

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

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

   a -{ 0 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   a -{ 0 }-> 0 :|: 
   encArg(z) -{ 0 }-> f(f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(1) :|: z = 1 + 2
   encArg(z) -{ 0 }-> f(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> f(0) :|: z - 1 >= 0
   encArg(z) -{ 1 }-> f(0) :|: z = 1 + 2
   encArg(z) -{ 0 }-> f(0) :|: z = 1 + 2
   encArg(z) -{ 0 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 1 }-> 0 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 2
   encode_a -{ 0 }-> 1 :|: 
   encode_a -{ 0 }-> 0 :|: 
   encode_a -{ 1 }-> 0 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_f(z) -{ 0 }-> f(f(encArg(z - 1))) :|: z - 1 >= 0
   encode_f(z) -{ 0 }-> f(1) :|: z = 2
   encode_f(z) -{ 0 }-> f(0) :|: z = 0
   encode_f(z) -{ 0 }-> f(0) :|: z >= 0
   encode_f(z) -{ 1 }-> f(0) :|: z = 2
   encode_f(z) -{ 0 }-> f(0) :|: z = 2
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   f(z) -{ 1 }-> f(1) :|: z = 1
   f(z) -{ 2 }-> f(0) :|: z = 1
   f(z) -{ 1 }-> f(0) :|: z = 1
   f(z) -{ 0 }-> 0 :|: z >= 0

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

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

(45) 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: 1

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

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

   a -{ 0 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   a -{ 0 }-> 0 :|: 
   encArg(z) -{ 0 }-> f(f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(1) :|: z = 1 + 2
   encArg(z) -{ 0 }-> f(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> f(0) :|: z - 1 >= 0
   encArg(z) -{ 1 }-> f(0) :|: z = 1 + 2
   encArg(z) -{ 0 }-> f(0) :|: z = 1 + 2
   encArg(z) -{ 0 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 1 }-> 0 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 2
   encode_a -{ 0 }-> 1 :|: 
   encode_a -{ 0 }-> 0 :|: 
   encode_a -{ 1 }-> 0 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_f(z) -{ 0 }-> f(f(encArg(z - 1))) :|: z - 1 >= 0
   encode_f(z) -{ 0 }-> f(1) :|: z = 2
   encode_f(z) -{ 0 }-> f(0) :|: z = 0
   encode_f(z) -{ 0 }-> f(0) :|: z >= 0
   encode_f(z) -{ 1 }-> f(0) :|: z = 2
   encode_f(z) -{ 0 }-> f(0) :|: z = 2
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   f(z) -{ 1 }-> f(1) :|: z = 1
   f(z) -{ 2 }-> f(0) :|: z = 1
   f(z) -{ 1 }-> f(0) :|: z = 1
   f(z) -{ 0 }-> 0 :|: z >= 0

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

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

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

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

   a -{ 0 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   a -{ 0 }-> 0 :|: 
   encArg(z) -{ 0 }-> f(f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(1) :|: z = 1 + 2
   encArg(z) -{ 0 }-> f(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> f(0) :|: z - 1 >= 0
   encArg(z) -{ 1 }-> f(0) :|: z = 1 + 2
   encArg(z) -{ 0 }-> f(0) :|: z = 1 + 2
   encArg(z) -{ 0 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 1 }-> 0 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 2
   encode_a -{ 0 }-> 1 :|: 
   encode_a -{ 0 }-> 0 :|: 
   encode_a -{ 1 }-> 0 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_f(z) -{ 0 }-> f(f(encArg(z - 1))) :|: z - 1 >= 0
   encode_f(z) -{ 0 }-> f(1) :|: z = 2
   encode_f(z) -{ 0 }-> f(0) :|: z = 0
   encode_f(z) -{ 0 }-> f(0) :|: z >= 0
   encode_f(z) -{ 1 }-> f(0) :|: z = 2
   encode_f(z) -{ 0 }-> f(0) :|: z = 2
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   f(z) -{ 1 }-> f(1) :|: z = 1
   f(z) -{ 2 }-> f(0) :|: z = 1
   f(z) -{ 1 }-> f(0) :|: z = 1
   f(z) -{ 0 }-> 0 :|: z >= 0

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

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a -{ 0 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   a -{ 0 }-> 0 :|: 
   encArg(z) -{ 0 }-> f(f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(1) :|: z = 1 + 2
   encArg(z) -{ 0 }-> f(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> f(0) :|: z - 1 >= 0
   encArg(z) -{ 1 }-> f(0) :|: z = 1 + 2
   encArg(z) -{ 0 }-> f(0) :|: z = 1 + 2
   encArg(z) -{ 0 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 1 }-> 0 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 2
   encode_a -{ 0 }-> 1 :|: 
   encode_a -{ 0 }-> 0 :|: 
   encode_a -{ 1 }-> 0 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_f(z) -{ 0 }-> f(f(encArg(z - 1))) :|: z - 1 >= 0
   encode_f(z) -{ 0 }-> f(1) :|: z = 2
   encode_f(z) -{ 0 }-> f(0) :|: z = 0
   encode_f(z) -{ 0 }-> f(0) :|: z >= 0
   encode_f(z) -{ 1 }-> f(0) :|: z = 2
   encode_f(z) -{ 0 }-> f(0) :|: z = 2
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   f(z) -{ 1 }-> f(1) :|: z = 1
   f(z) -{ 2 }-> f(0) :|: z = 1
   f(z) -{ 1 }-> f(0) :|: z = 1
   f(z) -{ 0 }-> 0 :|: z >= 0

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

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

(51) 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: ?

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

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

   a -{ 0 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   a -{ 0 }-> 0 :|: 
   encArg(z) -{ 0 }-> f(f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(1) :|: z = 1 + 2
   encArg(z) -{ 0 }-> f(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> f(0) :|: z - 1 >= 0
   encArg(z) -{ 1 }-> f(0) :|: z = 1 + 2
   encArg(z) -{ 0 }-> f(0) :|: z = 1 + 2
   encArg(z) -{ 0 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 1 }-> 0 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 2
   encode_a -{ 0 }-> 1 :|: 
   encode_a -{ 0 }-> 0 :|: 
   encode_a -{ 1 }-> 0 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_f(z) -{ 0 }-> f(f(encArg(z - 1))) :|: z - 1 >= 0
   encode_f(z) -{ 0 }-> f(1) :|: z = 2
   encode_f(z) -{ 0 }-> f(0) :|: z = 0
   encode_f(z) -{ 0 }-> f(0) :|: z >= 0
   encode_f(z) -{ 1 }-> f(0) :|: z = 2
   encode_f(z) -{ 0 }-> f(0) :|: z = 2
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   f(z) -{ 1 }-> f(1) :|: z = 1
   f(z) -{ 2 }-> f(0) :|: z = 1
   f(z) -{ 1 }-> f(0) :|: z = 1
   f(z) -{ 0 }-> 0 :|: z >= 0

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

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

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   encode_f(z0) -> f(encArg(z0))
   encode_a -> a
   encode_b -> b
   a -> c_a
   a -> b
   f(c_a) -> f(a)
Tuples:
   ENCARG(b) -> c
   ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c2(A)
   ENCODE_F(z0) -> c3(F(encArg(z0)), ENCARG(z0))
   ENCODE_A -> c4(A)
   ENCODE_B -> c5
   A -> c6
   A -> c7
   F(c_a) -> c8(F(a), A)
S tuples:
   A -> c7
   F(c_a) -> c8(F(a), A)
K tuples:none
Defined Rule Symbols:   a, f_1, encArg_1, encode_f_1, encode_a, encode_b

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, ENCODE_A, ENCODE_B, A, F_1

Compound Symbols:   c, c1_2, c2_1, c3_2, c4_1, c5, c6, c7, c8_2


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

(55) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   ENCODE_A -> c4(A)
Removed 3 trailing nodes:
   ENCARG(b) -> c
   A -> c6
   ENCODE_B -> c5

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   encode_f(z0) -> f(encArg(z0))
   encode_a -> a
   encode_b -> b
   a -> c_a
   a -> b
   f(c_a) -> f(a)
Tuples:
   ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c2(A)
   ENCODE_F(z0) -> c3(F(encArg(z0)), ENCARG(z0))
   A -> c7
   F(c_a) -> c8(F(a), A)
S tuples:
   A -> c7
   F(c_a) -> c8(F(a), A)
K tuples:none
Defined Rule Symbols:   a, f_1, encArg_1, encode_f_1, encode_a, encode_b

Defined Pair Symbols:   ENCARG_1, ENCODE_F_1, A, F_1

Compound Symbols:   c1_2, c2_1, c3_2, c7, c8_2


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

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   encode_f(z0) -> f(encArg(z0))
   encode_a -> a
   encode_b -> b
   a -> c_a
   a -> b
   f(c_a) -> f(a)
Tuples:
   ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c2(A)
   A -> c7
   F(c_a) -> c8(F(a), A)
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCODE_F(z0) -> c(ENCARG(z0))
S tuples:
   A -> c7
   F(c_a) -> c8(F(a), A)
K tuples:none
Defined Rule Symbols:   a, f_1, encArg_1, encode_f_1, encode_a, encode_b

Defined Pair Symbols:   ENCARG_1, A, F_1, ENCODE_F_1

Compound Symbols:   c1_2, c2_1, c7, c8_2, c_1


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

(59) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   ENCODE_F(z0) -> c(ENCARG(z0))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   encode_f(z0) -> f(encArg(z0))
   encode_a -> a
   encode_b -> b
   a -> c_a
   a -> b
   f(c_a) -> f(a)
Tuples:
   ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c2(A)
   A -> c7
   F(c_a) -> c8(F(a), A)
   ENCODE_F(z0) -> c(F(encArg(z0)))
S tuples:
   A -> c7
   F(c_a) -> c8(F(a), A)
K tuples:none
Defined Rule Symbols:   a, f_1, encArg_1, encode_f_1, encode_a, encode_b

Defined Pair Symbols:   ENCARG_1, A, F_1, ENCODE_F_1

Compound Symbols:   c1_2, c2_1, c7, c8_2, c_1


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

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

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   f(c_a) -> f(a)
   a -> c_a
   a -> b
Tuples:
   ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0))
   ENCARG(cons_a) -> c2(A)
   A -> c7
   F(c_a) -> c8(F(a), A)
   ENCODE_F(z0) -> c(F(encArg(z0)))
S tuples:
   A -> c7
   F(c_a) -> c8(F(a), A)
K tuples:none
Defined Rule Symbols:   encArg_1, f_1, a

Defined Pair Symbols:   ENCARG_1, A, F_1, ENCODE_F_1

Compound Symbols:   c1_2, c2_1, c7, c8_2, c_1


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

(63) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_f(z0)) -> c1(F(encArg(z0)), ENCARG(z0)) by 
   ENCARG(cons_f(b)) -> c1(F(b), ENCARG(b))
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_a)) -> c1(F(a), ENCARG(cons_a))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   f(c_a) -> f(a)
   a -> c_a
   a -> b
Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   F(c_a) -> c8(F(a), A)
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCARG(cons_f(b)) -> c1(F(b), ENCARG(b))
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_a)) -> c1(F(a), ENCARG(cons_a))
S tuples:
   A -> c7
   F(c_a) -> c8(F(a), A)
K tuples:none
Defined Rule Symbols:   encArg_1, f_1, a

Defined Pair Symbols:   ENCARG_1, A, F_1, ENCODE_F_1

Compound Symbols:   c2_1, c7, c8_2, c_1, c1_2


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

(65) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   ENCARG(cons_f(b)) -> c1(F(b), ENCARG(b))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   f(c_a) -> f(a)
   a -> c_a
   a -> b
Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   F(c_a) -> c8(F(a), A)
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_a)) -> c1(F(a), ENCARG(cons_a))
S tuples:
   A -> c7
   F(c_a) -> c8(F(a), A)
K tuples:none
Defined Rule Symbols:   encArg_1, f_1, a

Defined Pair Symbols:   ENCARG_1, A, F_1, ENCODE_F_1

Compound Symbols:   c2_1, c7, c8_2, c_1, c1_2


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

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   f(c_a) -> f(a)
   a -> c_a
   a -> b
Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   F(c_a) -> c8(F(a), A)
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_a)) -> c3(F(a))
   ENCARG(cons_f(cons_a)) -> c3(ENCARG(cons_a))
S tuples:
   A -> c7
   F(c_a) -> c8(F(a), A)
K tuples:none
Defined Rule Symbols:   encArg_1, f_1, a

Defined Pair Symbols:   ENCARG_1, A, F_1, ENCODE_F_1

Compound Symbols:   c2_1, c7, c8_2, c_1, c1_2, c3_1


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

(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(c_a) -> c8(F(a), A) by 
   F(c_a) -> c8(F(c_a), A)
   F(c_a) -> c8(F(b), A)

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   f(c_a) -> f(a)
   a -> c_a
   a -> b
Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_a)) -> c3(F(a))
   ENCARG(cons_f(cons_a)) -> c3(ENCARG(cons_a))
   F(c_a) -> c8(F(c_a), A)
   F(c_a) -> c8(F(b), A)
S tuples:
   A -> c7
   F(c_a) -> c8(F(c_a), A)
   F(c_a) -> c8(F(b), A)
K tuples:none
Defined Rule Symbols:   encArg_1, f_1, a

Defined Pair Symbols:   ENCARG_1, A, ENCODE_F_1, F_1

Compound Symbols:   c2_1, c7, c_1, c1_2, c3_1, c8_2


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

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   f(c_a) -> f(a)
   a -> c_a
   a -> b
Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_a)) -> c3(F(a))
   ENCARG(cons_f(cons_a)) -> c3(ENCARG(cons_a))
   F(c_a) -> c8(F(c_a), A)
   F(c_a) -> c8(A)
S tuples:
   A -> c7
   F(c_a) -> c8(F(c_a), A)
   F(c_a) -> c8(A)
K tuples:none
Defined Rule Symbols:   encArg_1, f_1, a

Defined Pair Symbols:   ENCARG_1, A, ENCODE_F_1, F_1

Compound Symbols:   c2_1, c7, c_1, c1_2, c3_1, c8_2, c8_1


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

(73) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(c_a) -> c8(A)
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_a)) -> c3(F(a))
   ENCARG(cons_f(cons_a)) -> c3(ENCARG(cons_a))
   F(c_a) -> c8(F(c_a), A)
   F(c_a) -> c8(A)
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A) = 0
   POL(ENCARG(x_1)) = [2] + x_1
   POL(ENCODE_F(x_1)) = [1] + x_1
   POL(F(x_1)) = [1]
   POL(a) = [2]
   POL(b) = [3]
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c7) = 0
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c_a) = [3]
   POL(cons_a) = [2]
   POL(cons_f(x_1)) = [2] + x_1
   POL(encArg(x_1)) = [3]
   POL(f(x_1)) = [3] + [3]x_1

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   f(c_a) -> f(a)
   a -> c_a
   a -> b
Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   ENCODE_F(z0) -> c(F(encArg(z0)))
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_a)) -> c3(F(a))
   ENCARG(cons_f(cons_a)) -> c3(ENCARG(cons_a))
   F(c_a) -> c8(F(c_a), A)
   F(c_a) -> c8(A)
S tuples:
   A -> c7
   F(c_a) -> c8(F(c_a), A)
K tuples:
   F(c_a) -> c8(A)
Defined Rule Symbols:   encArg_1, f_1, a

Defined Pair Symbols:   ENCARG_1, A, ENCODE_F_1, F_1

Compound Symbols:   c2_1, c7, c_1, c1_2, c3_1, c8_2, c8_1


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

(75) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_F(z0) -> c(F(encArg(z0))) by 
   ENCODE_F(b) -> c(F(b))
   ENCODE_F(cons_f(z0)) -> c(F(f(encArg(z0))))
   ENCODE_F(cons_a) -> c(F(a))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   f(c_a) -> f(a)
   a -> c_a
   a -> b
Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_a)) -> c3(F(a))
   ENCARG(cons_f(cons_a)) -> c3(ENCARG(cons_a))
   F(c_a) -> c8(F(c_a), A)
   F(c_a) -> c8(A)
   ENCODE_F(b) -> c(F(b))
   ENCODE_F(cons_f(z0)) -> c(F(f(encArg(z0))))
   ENCODE_F(cons_a) -> c(F(a))
S tuples:
   A -> c7
   F(c_a) -> c8(F(c_a), A)
K tuples:
   F(c_a) -> c8(A)
Defined Rule Symbols:   encArg_1, f_1, a

Defined Pair Symbols:   ENCARG_1, A, F_1, ENCODE_F_1

Compound Symbols:   c2_1, c7, c1_2, c3_1, c8_2, c8_1, c_1


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

(77) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   ENCODE_F(b) -> c(F(b))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   f(c_a) -> f(a)
   a -> c_a
   a -> b
Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0)))
   ENCARG(cons_f(cons_a)) -> c3(F(a))
   ENCARG(cons_f(cons_a)) -> c3(ENCARG(cons_a))
   F(c_a) -> c8(F(c_a), A)
   F(c_a) -> c8(A)
   ENCODE_F(cons_f(z0)) -> c(F(f(encArg(z0))))
   ENCODE_F(cons_a) -> c(F(a))
S tuples:
   A -> c7
   F(c_a) -> c8(F(c_a), A)
K tuples:
   F(c_a) -> c8(A)
Defined Rule Symbols:   encArg_1, f_1, a

Defined Pair Symbols:   ENCARG_1, A, F_1, ENCODE_F_1

Compound Symbols:   c2_1, c7, c1_2, c3_1, c8_2, c8_1, c_1


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

(79) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_f(cons_f(z0))) -> c1(F(f(encArg(z0))), ENCARG(cons_f(z0))) by 
   ENCARG(cons_f(cons_f(b))) -> c1(F(f(b)), ENCARG(cons_f(b)))
   ENCARG(cons_f(cons_f(cons_f(z0)))) -> c1(F(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_f(cons_f(cons_a))) -> c1(F(f(a)), ENCARG(cons_f(cons_a)))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   f(c_a) -> f(a)
   a -> c_a
   a -> b
Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   ENCARG(cons_f(cons_a)) -> c3(F(a))
   ENCARG(cons_f(cons_a)) -> c3(ENCARG(cons_a))
   F(c_a) -> c8(F(c_a), A)
   F(c_a) -> c8(A)
   ENCODE_F(cons_f(z0)) -> c(F(f(encArg(z0))))
   ENCODE_F(cons_a) -> c(F(a))
   ENCARG(cons_f(cons_f(b))) -> c1(F(f(b)), ENCARG(cons_f(b)))
   ENCARG(cons_f(cons_f(cons_f(z0)))) -> c1(F(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_f(cons_f(cons_a))) -> c1(F(f(a)), ENCARG(cons_f(cons_a)))
S tuples:
   A -> c7
   F(c_a) -> c8(F(c_a), A)
K tuples:
   F(c_a) -> c8(A)
Defined Rule Symbols:   encArg_1, f_1, a

Defined Pair Symbols:   ENCARG_1, A, F_1, ENCODE_F_1

Compound Symbols:   c2_1, c7, c3_1, c8_2, c8_1, c_1, c1_2


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

(81) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   ENCARG(cons_f(cons_f(b))) -> c1(F(f(b)), ENCARG(cons_f(b)))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   f(c_a) -> f(a)
   a -> c_a
   a -> b
Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   ENCARG(cons_f(cons_a)) -> c3(F(a))
   ENCARG(cons_f(cons_a)) -> c3(ENCARG(cons_a))
   F(c_a) -> c8(F(c_a), A)
   F(c_a) -> c8(A)
   ENCODE_F(cons_f(z0)) -> c(F(f(encArg(z0))))
   ENCODE_F(cons_a) -> c(F(a))
   ENCARG(cons_f(cons_f(cons_f(z0)))) -> c1(F(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_f(cons_f(cons_a))) -> c1(F(f(a)), ENCARG(cons_f(cons_a)))
S tuples:
   A -> c7
   F(c_a) -> c8(F(c_a), A)
K tuples:
   F(c_a) -> c8(A)
Defined Rule Symbols:   encArg_1, f_1, a

Defined Pair Symbols:   ENCARG_1, A, F_1, ENCODE_F_1

Compound Symbols:   c2_1, c7, c3_1, c8_2, c8_1, c_1, c1_2


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

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   f(c_a) -> f(a)
   a -> c_a
   a -> b
Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   ENCARG(cons_f(cons_a)) -> c3(F(a))
   ENCARG(cons_f(cons_a)) -> c3(ENCARG(cons_a))
   F(c_a) -> c8(F(c_a), A)
   F(c_a) -> c8(A)
   ENCODE_F(cons_f(z0)) -> c(F(f(encArg(z0))))
   ENCODE_F(cons_a) -> c(F(a))
   ENCARG(cons_f(cons_f(cons_f(z0)))) -> c1(F(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_f(cons_f(cons_a))) -> c4(F(f(a)))
   ENCARG(cons_f(cons_f(cons_a))) -> c4(ENCARG(cons_f(cons_a)))
S tuples:
   A -> c7
   F(c_a) -> c8(F(c_a), A)
K tuples:
   F(c_a) -> c8(A)
Defined Rule Symbols:   encArg_1, f_1, a

Defined Pair Symbols:   ENCARG_1, A, F_1, ENCODE_F_1

Compound Symbols:   c2_1, c7, c3_1, c8_2, c8_1, c_1, c1_2, c4_1


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

(85) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_f(cons_a)) -> c3(F(a)) by 
   ENCARG(cons_f(cons_a)) -> c3(F(c_a))
   ENCARG(cons_f(cons_a)) -> c3(F(b))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   f(c_a) -> f(a)
   a -> c_a
   a -> b
Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   ENCARG(cons_f(cons_a)) -> c3(ENCARG(cons_a))
   F(c_a) -> c8(F(c_a), A)
   F(c_a) -> c8(A)
   ENCODE_F(cons_f(z0)) -> c(F(f(encArg(z0))))
   ENCODE_F(cons_a) -> c(F(a))
   ENCARG(cons_f(cons_f(cons_f(z0)))) -> c1(F(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_f(cons_f(cons_a))) -> c4(F(f(a)))
   ENCARG(cons_f(cons_f(cons_a))) -> c4(ENCARG(cons_f(cons_a)))
   ENCARG(cons_f(cons_a)) -> c3(F(c_a))
   ENCARG(cons_f(cons_a)) -> c3(F(b))
S tuples:
   A -> c7
   F(c_a) -> c8(F(c_a), A)
K tuples:
   F(c_a) -> c8(A)
Defined Rule Symbols:   encArg_1, f_1, a

Defined Pair Symbols:   ENCARG_1, A, F_1, ENCODE_F_1

Compound Symbols:   c2_1, c7, c3_1, c8_2, c8_1, c_1, c1_2, c4_1


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

(87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   ENCARG(cons_f(cons_a)) -> c3(F(b))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   f(c_a) -> f(a)
   a -> c_a
   a -> b
Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   ENCARG(cons_f(cons_a)) -> c3(ENCARG(cons_a))
   F(c_a) -> c8(F(c_a), A)
   F(c_a) -> c8(A)
   ENCODE_F(cons_f(z0)) -> c(F(f(encArg(z0))))
   ENCODE_F(cons_a) -> c(F(a))
   ENCARG(cons_f(cons_f(cons_f(z0)))) -> c1(F(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_f(cons_f(cons_a))) -> c4(F(f(a)))
   ENCARG(cons_f(cons_f(cons_a))) -> c4(ENCARG(cons_f(cons_a)))
   ENCARG(cons_f(cons_a)) -> c3(F(c_a))
S tuples:
   A -> c7
   F(c_a) -> c8(F(c_a), A)
K tuples:
   F(c_a) -> c8(A)
Defined Rule Symbols:   encArg_1, f_1, a

Defined Pair Symbols:   ENCARG_1, A, F_1, ENCODE_F_1

Compound Symbols:   c2_1, c7, c3_1, c8_2, c8_1, c_1, c1_2, c4_1


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

(89) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_F(cons_f(z0)) -> c(F(f(encArg(z0)))) by 
   ENCODE_F(cons_f(b)) -> c(F(f(b)))
   ENCODE_F(cons_f(cons_f(z0))) -> c(F(f(f(encArg(z0)))))
   ENCODE_F(cons_f(cons_a)) -> c(F(f(a)))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   f(c_a) -> f(a)
   a -> c_a
   a -> b
Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   ENCARG(cons_f(cons_a)) -> c3(ENCARG(cons_a))
   F(c_a) -> c8(F(c_a), A)
   F(c_a) -> c8(A)
   ENCODE_F(cons_a) -> c(F(a))
   ENCARG(cons_f(cons_f(cons_f(z0)))) -> c1(F(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_f(cons_f(cons_a))) -> c4(F(f(a)))
   ENCARG(cons_f(cons_f(cons_a))) -> c4(ENCARG(cons_f(cons_a)))
   ENCARG(cons_f(cons_a)) -> c3(F(c_a))
   ENCODE_F(cons_f(b)) -> c(F(f(b)))
   ENCODE_F(cons_f(cons_f(z0))) -> c(F(f(f(encArg(z0)))))
   ENCODE_F(cons_f(cons_a)) -> c(F(f(a)))
S tuples:
   A -> c7
   F(c_a) -> c8(F(c_a), A)
K tuples:
   F(c_a) -> c8(A)
Defined Rule Symbols:   encArg_1, f_1, a

Defined Pair Symbols:   ENCARG_1, A, F_1, ENCODE_F_1

Compound Symbols:   c2_1, c7, c3_1, c8_2, c8_1, c_1, c1_2, c4_1


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

(91) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   ENCODE_F(cons_f(b)) -> c(F(f(b)))

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   f(c_a) -> f(a)
   a -> c_a
   a -> b
Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   ENCARG(cons_f(cons_a)) -> c3(ENCARG(cons_a))
   F(c_a) -> c8(F(c_a), A)
   F(c_a) -> c8(A)
   ENCODE_F(cons_a) -> c(F(a))
   ENCARG(cons_f(cons_f(cons_f(z0)))) -> c1(F(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_f(cons_f(cons_a))) -> c4(F(f(a)))
   ENCARG(cons_f(cons_f(cons_a))) -> c4(ENCARG(cons_f(cons_a)))
   ENCARG(cons_f(cons_a)) -> c3(F(c_a))
   ENCODE_F(cons_f(cons_f(z0))) -> c(F(f(f(encArg(z0)))))
   ENCODE_F(cons_f(cons_a)) -> c(F(f(a)))
S tuples:
   A -> c7
   F(c_a) -> c8(F(c_a), A)
K tuples:
   F(c_a) -> c8(A)
Defined Rule Symbols:   encArg_1, f_1, a

Defined Pair Symbols:   ENCARG_1, A, F_1, ENCODE_F_1

Compound Symbols:   c2_1, c7, c3_1, c8_2, c8_1, c_1, c1_2, c4_1


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

(93) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_F(cons_a) -> c(F(a)) by 
   ENCODE_F(cons_a) -> c(F(c_a))
   ENCODE_F(cons_a) -> c(F(b))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   f(c_a) -> f(a)
   a -> c_a
   a -> b
Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   ENCARG(cons_f(cons_a)) -> c3(ENCARG(cons_a))
   F(c_a) -> c8(F(c_a), A)
   F(c_a) -> c8(A)
   ENCARG(cons_f(cons_f(cons_f(z0)))) -> c1(F(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_f(cons_f(cons_a))) -> c4(F(f(a)))
   ENCARG(cons_f(cons_f(cons_a))) -> c4(ENCARG(cons_f(cons_a)))
   ENCARG(cons_f(cons_a)) -> c3(F(c_a))
   ENCODE_F(cons_f(cons_f(z0))) -> c(F(f(f(encArg(z0)))))
   ENCODE_F(cons_f(cons_a)) -> c(F(f(a)))
   ENCODE_F(cons_a) -> c(F(c_a))
   ENCODE_F(cons_a) -> c(F(b))
S tuples:
   A -> c7
   F(c_a) -> c8(F(c_a), A)
K tuples:
   F(c_a) -> c8(A)
Defined Rule Symbols:   encArg_1, f_1, a

Defined Pair Symbols:   ENCARG_1, A, F_1, ENCODE_F_1

Compound Symbols:   c2_1, c7, c3_1, c8_2, c8_1, c1_2, c4_1, c_1


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

(95) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   ENCODE_F(cons_a) -> c(F(c_a))
Removed 1 trailing nodes:
   ENCODE_F(cons_a) -> c(F(b))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(b) -> b
   encArg(cons_f(z0)) -> f(encArg(z0))
   encArg(cons_a) -> a
   f(c_a) -> f(a)
   a -> c_a
   a -> b
Tuples:
   ENCARG(cons_a) -> c2(A)
   A -> c7
   ENCARG(cons_f(cons_a)) -> c3(ENCARG(cons_a))
   F(c_a) -> c8(F(c_a), A)
   F(c_a) -> c8(A)
   ENCARG(cons_f(cons_f(cons_f(z0)))) -> c1(F(f(f(encArg(z0)))), ENCARG(cons_f(cons_f(z0))))
   ENCARG(cons_f(cons_f(cons_a))) -> c4(F(f(a)))
   ENCARG(cons_f(cons_f(cons_a))) -> c4(ENCARG(cons_f(cons_a)))
   ENCARG(cons_f(cons_a)) -> c3(F(c_a))
   ENCODE_F(cons_f(cons_f(z0))) -> c(F(f(f(encArg(z0)))))
   ENCODE_F(cons_f(cons_a)) -> c(F(f(a)))
S tuples:
   A -> c7
   F(c_a) -> c8(F(c_a), A)
K tuples:
   F(c_a) -> c8(A)
Defined Rule Symbols:   encArg_1, f_1, a

Defined Pair Symbols:   ENCARG_1, A, F_1, ENCODE_F_1

Compound Symbols:   c2_1, c7, c3_1, c8_2, c8_1, c1_2, c4_1, c_1