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


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


WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 157 ms]
(4) CpxRelTRS
(5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (14) CpxRNTS
    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 359 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 97 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 76 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 129 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 780 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 401 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), 4 ms]
    (48) CpxRNTS
    (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 248 ms]
    (52) CpxRNTS
    (53) IntTrsBoundProof [UPPER BOUND(ID), 108 ms]
    (54) CpxRNTS
    (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 201 ms]
    (58) CpxRNTS
    (59) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (60) CpxRNTS
    (61) FinalProof [FINISHED, 0 ms]
    (62) BOUNDS(1, n^2)
(63) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(64) CpxRelTRS
    (65) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) typed CpxTrs
    (67) OrderProof [LOWER BOUND(ID), 0 ms]
    (68) typed CpxTrs
    (69) RewriteLemmaProof [LOWER BOUND(ID), 303 ms]
    (70) BEST
        (71) proven lower bound
            (72) LowerBoundPropagationProof [FINISHED, 0 ms]
            (73) BOUNDS(n^1, INF)
        (74) typed CpxTrs
            (75) RewriteLemmaProof [LOWER BOUND(ID), 48 ms]
            (76) BOUNDS(1, INF)


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   rev(ls) -> r1(ls, empty)
   r1(empty, a) -> a
   r1(cons(x, k), a) -> r1(k, cons(x, a))

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

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(empty) -> empty
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_rev(x_1)) -> rev(encArg(x_1))
   encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2))
   encode_rev(x_1) -> rev(encArg(x_1))
   encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2))
   encode_empty -> empty
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))


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

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


The TRS R consists of the following rules:

   rev(ls) -> r1(ls, empty)
   r1(empty, a) -> a
   r1(cons(x, k), a) -> r1(k, cons(x, a))

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

   encArg(empty) -> empty
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_rev(x_1)) -> rev(encArg(x_1))
   encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2))
   encode_rev(x_1) -> rev(encArg(x_1))
   encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2))
   encode_empty -> empty
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))

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

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   rev(ls) -> r1(ls, empty)
   r1(empty, a) -> a
   r1(cons(x, k), a) -> r1(k, cons(x, a))

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

   encArg(empty) -> empty
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_rev(x_1)) -> rev(encArg(x_1))
   encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2))
   encode_rev(x_1) -> rev(encArg(x_1))
   encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2))
   encode_empty -> empty
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))

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

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

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   rev(ls) -> r1(ls, empty) [1]
   r1(empty, a) -> a [1]
   r1(cons(x, k), a) -> r1(k, cons(x, a)) [1]
   encArg(empty) -> empty [0]
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_rev(x_1)) -> rev(encArg(x_1)) [0]
   encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2)) [0]
   encode_rev(x_1) -> rev(encArg(x_1)) [0]
   encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2)) [0]
   encode_empty -> empty [0]
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) [0]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   rev(ls) -> r1(ls, empty) [1]
   r1(empty, a) -> a [1]
   r1(cons(x, k), a) -> r1(k, cons(x, a)) [1]
   encArg(empty) -> empty [0]
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_rev(x_1)) -> rev(encArg(x_1)) [0]
   encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2)) [0]
   encode_rev(x_1) -> rev(encArg(x_1)) [0]
   encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2)) [0]
   encode_empty -> empty [0]
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) [0]

The TRS has the following type information:
rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
empty :: empty:cons:cons_rev:cons_r1
cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encArg :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_empty :: empty:cons:cons_rev:cons_r1
encode_cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1

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

(9) 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:

   rev_1
   r1_2
   encArg_1
   encode_rev_1
   encode_r1_2
   encode_empty
   encode_cons_2

Due to the following rules being added:

   encArg(v0) -> empty [0]
   encode_rev(v0) -> empty [0]
   encode_r1(v0, v1) -> empty [0]
   encode_empty -> empty [0]
   encode_cons(v0, v1) -> empty [0]
   r1(v0, v1) -> empty [0]

And the following fresh constants: none

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

(10)
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:

   rev(ls) -> r1(ls, empty) [1]
   r1(empty, a) -> a [1]
   r1(cons(x, k), a) -> r1(k, cons(x, a)) [1]
   encArg(empty) -> empty [0]
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_rev(x_1)) -> rev(encArg(x_1)) [0]
   encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2)) [0]
   encode_rev(x_1) -> rev(encArg(x_1)) [0]
   encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2)) [0]
   encode_empty -> empty [0]
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) [0]
   encArg(v0) -> empty [0]
   encode_rev(v0) -> empty [0]
   encode_r1(v0, v1) -> empty [0]
   encode_empty -> empty [0]
   encode_cons(v0, v1) -> empty [0]
   r1(v0, v1) -> empty [0]

The TRS has the following type information:
rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
empty :: empty:cons:cons_rev:cons_r1
cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encArg :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_empty :: empty:cons:cons_rev:cons_r1
encode_cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1

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

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

(12)
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:

   rev(ls) -> r1(ls, empty) [1]
   r1(empty, a) -> a [1]
   r1(cons(x, k), a) -> r1(k, cons(x, a)) [1]
   encArg(empty) -> empty [0]
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_rev(empty)) -> rev(empty) [0]
   encArg(cons_rev(cons(x_1', x_2'))) -> rev(cons(encArg(x_1'), encArg(x_2'))) [0]
   encArg(cons_rev(cons_rev(x_1''))) -> rev(rev(encArg(x_1''))) [0]
   encArg(cons_rev(cons_r1(x_11, x_2''))) -> rev(r1(encArg(x_11), encArg(x_2''))) [0]
   encArg(cons_rev(x_1)) -> rev(empty) [0]
   encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2)) [0]
   encode_rev(empty) -> rev(empty) [0]
   encode_rev(cons(x_120, x_213)) -> rev(cons(encArg(x_120), encArg(x_213))) [0]
   encode_rev(cons_rev(x_121)) -> rev(rev(encArg(x_121))) [0]
   encode_rev(cons_r1(x_122, x_214)) -> rev(r1(encArg(x_122), encArg(x_214))) [0]
   encode_rev(x_1) -> rev(empty) [0]
   encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2)) [0]
   encode_empty -> empty [0]
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) [0]
   encArg(v0) -> empty [0]
   encode_rev(v0) -> empty [0]
   encode_r1(v0, v1) -> empty [0]
   encode_empty -> empty [0]
   encode_cons(v0, v1) -> empty [0]
   r1(v0, v1) -> empty [0]

The TRS has the following type information:
rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
empty :: empty:cons:cons_rev:cons_r1
cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encArg :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_empty :: empty:cons:cons_rev:cons_r1
encode_cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1

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

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

   empty => 0

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

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> rev(rev(encArg(x_1''))) :|: z = 1 + (1 + x_1''), x_1'' >= 0
   encArg(z) -{ 0 }-> rev(r1(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 0 }-> rev(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> rev(0) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> rev(1 + encArg(x_1') + encArg(x_2')) :|: z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> r1(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) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_cons(z, z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 0 }-> r1(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_r1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_rev(z) -{ 0 }-> rev(rev(encArg(x_121))) :|: z = 1 + x_121, x_121 >= 0
   encode_rev(z) -{ 0 }-> rev(r1(encArg(x_122), encArg(x_214))) :|: z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z = 0
   encode_rev(z) -{ 0 }-> rev(0) :|: x_1 >= 0, z = x_1
   encode_rev(z) -{ 0 }-> rev(1 + encArg(x_120) + encArg(x_213)) :|: x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   r1(z, z') -{ 1 }-> a :|: z' = a, a >= 0, z = 0
   r1(z, z') -{ 1 }-> r1(k, 1 + x + a) :|: z' = a, a >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   rev(z) -{ 1 }-> r1(ls, 0) :|: ls >= 0, z = ls


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

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

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> rev(rev(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> rev(r1(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 0 }-> rev(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> rev(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> rev(1 + encArg(x_1') + encArg(x_2')) :|: z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> r1(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 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 0 }-> r1(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 0 }-> rev(rev(encArg(z - 1))) :|: z - 1 >= 0
   encode_rev(z) -{ 0 }-> rev(r1(encArg(x_122), encArg(x_214))) :|: z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z = 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z >= 0
   encode_rev(z) -{ 0 }-> rev(1 + encArg(x_120) + encArg(x_213)) :|: x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 1 }-> r1(k, 1 + x + z') :|: z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 1 }-> r1(z, 0) :|: z >= 0


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

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

   { r1 }
   { encode_empty }
   { rev }
   { encArg }
   { encode_r1 }
   { encode_rev }
   { encode_cons }

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

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> rev(rev(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> rev(r1(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 0 }-> rev(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> rev(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> rev(1 + encArg(x_1') + encArg(x_2')) :|: z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> r1(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 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 0 }-> r1(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 0 }-> rev(rev(encArg(z - 1))) :|: z - 1 >= 0
   encode_rev(z) -{ 0 }-> rev(r1(encArg(x_122), encArg(x_214))) :|: z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z = 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z >= 0
   encode_rev(z) -{ 0 }-> rev(1 + encArg(x_120) + encArg(x_213)) :|: x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 1 }-> r1(k, 1 + x + z') :|: z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 1 }-> r1(z, 0) :|: z >= 0

Function symbols to be analyzed: {r1}, {encode_empty}, {rev}, {encArg}, {encode_r1}, {encode_rev}, {encode_cons}

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

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

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> rev(rev(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> rev(r1(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 0 }-> rev(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> rev(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> rev(1 + encArg(x_1') + encArg(x_2')) :|: z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> r1(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 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 0 }-> r1(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 0 }-> rev(rev(encArg(z - 1))) :|: z - 1 >= 0
   encode_rev(z) -{ 0 }-> rev(r1(encArg(x_122), encArg(x_214))) :|: z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z = 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z >= 0
   encode_rev(z) -{ 0 }-> rev(1 + encArg(x_120) + encArg(x_213)) :|: x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 1 }-> r1(k, 1 + x + z') :|: z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 1 }-> r1(z, 0) :|: z >= 0

Function symbols to be analyzed: {r1}, {encode_empty}, {rev}, {encArg}, {encode_r1}, {encode_rev}, {encode_cons}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: r1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z + z'

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

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> rev(rev(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> rev(r1(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 0 }-> rev(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> rev(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> rev(1 + encArg(x_1') + encArg(x_2')) :|: z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> r1(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 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 0 }-> r1(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 0 }-> rev(rev(encArg(z - 1))) :|: z - 1 >= 0
   encode_rev(z) -{ 0 }-> rev(r1(encArg(x_122), encArg(x_214))) :|: z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z = 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z >= 0
   encode_rev(z) -{ 0 }-> rev(1 + encArg(x_120) + encArg(x_213)) :|: x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 1 }-> r1(k, 1 + x + z') :|: z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 1 }-> r1(z, 0) :|: z >= 0

Function symbols to be analyzed: {r1}, {encode_empty}, {rev}, {encArg}, {encode_r1}, {encode_rev}, {encode_cons}
Previous analysis results are:
  r1: runtime: ?, size: O(n^1) [z + z']

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: r1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z

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

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> rev(rev(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> rev(r1(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 0 }-> rev(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> rev(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> rev(1 + encArg(x_1') + encArg(x_2')) :|: z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> r1(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 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 0 }-> r1(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 0 }-> rev(rev(encArg(z - 1))) :|: z - 1 >= 0
   encode_rev(z) -{ 0 }-> rev(r1(encArg(x_122), encArg(x_214))) :|: z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z = 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z >= 0
   encode_rev(z) -{ 0 }-> rev(1 + encArg(x_120) + encArg(x_213)) :|: x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 1 }-> r1(k, 1 + x + z') :|: z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 1 }-> r1(z, 0) :|: z >= 0

Function symbols to be analyzed: {encode_empty}, {rev}, {encArg}, {encode_r1}, {encode_rev}, {encode_cons}
Previous analysis results are:
  r1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

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

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

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> rev(rev(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> rev(r1(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 0 }-> rev(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> rev(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> rev(1 + encArg(x_1') + encArg(x_2')) :|: z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> r1(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 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 0 }-> r1(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 0 }-> rev(rev(encArg(z - 1))) :|: z - 1 >= 0
   encode_rev(z) -{ 0 }-> rev(r1(encArg(x_122), encArg(x_214))) :|: z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z = 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z >= 0
   encode_rev(z) -{ 0 }-> rev(1 + encArg(x_120) + encArg(x_213)) :|: x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 2 + k }-> s' :|: s' >= 0, s' <= k + (1 + x + z'), z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 2 + z }-> s :|: s >= 0, s <= z + 0, z >= 0

Function symbols to be analyzed: {encode_empty}, {rev}, {encArg}, {encode_r1}, {encode_rev}, {encode_cons}
Previous analysis results are:
  r1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> rev(rev(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> rev(r1(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 0 }-> rev(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> rev(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> rev(1 + encArg(x_1') + encArg(x_2')) :|: z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> r1(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 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 0 }-> r1(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 0 }-> rev(rev(encArg(z - 1))) :|: z - 1 >= 0
   encode_rev(z) -{ 0 }-> rev(r1(encArg(x_122), encArg(x_214))) :|: z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z = 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z >= 0
   encode_rev(z) -{ 0 }-> rev(1 + encArg(x_120) + encArg(x_213)) :|: x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 2 + k }-> s' :|: s' >= 0, s' <= k + (1 + x + z'), z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 2 + z }-> s :|: s >= 0, s <= z + 0, z >= 0

Function symbols to be analyzed: {encode_empty}, {rev}, {encArg}, {encode_r1}, {encode_rev}, {encode_cons}
Previous analysis results are:
  r1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encode_empty: runtime: ?, size: O(1) [0]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> rev(rev(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> rev(r1(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 0 }-> rev(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> rev(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> rev(1 + encArg(x_1') + encArg(x_2')) :|: z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> r1(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 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 0 }-> r1(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 0 }-> rev(rev(encArg(z - 1))) :|: z - 1 >= 0
   encode_rev(z) -{ 0 }-> rev(r1(encArg(x_122), encArg(x_214))) :|: z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z = 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z >= 0
   encode_rev(z) -{ 0 }-> rev(1 + encArg(x_120) + encArg(x_213)) :|: x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 2 + k }-> s' :|: s' >= 0, s' <= k + (1 + x + z'), z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 2 + z }-> s :|: s >= 0, s <= z + 0, z >= 0

Function symbols to be analyzed: {rev}, {encArg}, {encode_r1}, {encode_rev}, {encode_cons}
Previous analysis results are:
  r1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encode_empty: runtime: O(1) [0], size: O(1) [0]

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

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

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

   encArg(z) -{ 0 }-> rev(rev(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> rev(r1(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 0 }-> rev(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> rev(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> rev(1 + encArg(x_1') + encArg(x_2')) :|: z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> r1(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 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 0 }-> r1(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 0 }-> rev(rev(encArg(z - 1))) :|: z - 1 >= 0
   encode_rev(z) -{ 0 }-> rev(r1(encArg(x_122), encArg(x_214))) :|: z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z = 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z >= 0
   encode_rev(z) -{ 0 }-> rev(1 + encArg(x_120) + encArg(x_213)) :|: x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 2 + k }-> s' :|: s' >= 0, s' <= k + (1 + x + z'), z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 2 + z }-> s :|: s >= 0, s <= z + 0, z >= 0

Function symbols to be analyzed: {rev}, {encArg}, {encode_r1}, {encode_rev}, {encode_cons}
Previous analysis results are:
  r1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encode_empty: runtime: O(1) [0], size: O(1) [0]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> rev(rev(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> rev(r1(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 0 }-> rev(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> rev(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> rev(1 + encArg(x_1') + encArg(x_2')) :|: z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> r1(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 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 0 }-> r1(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 0 }-> rev(rev(encArg(z - 1))) :|: z - 1 >= 0
   encode_rev(z) -{ 0 }-> rev(r1(encArg(x_122), encArg(x_214))) :|: z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z = 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z >= 0
   encode_rev(z) -{ 0 }-> rev(1 + encArg(x_120) + encArg(x_213)) :|: x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 2 + k }-> s' :|: s' >= 0, s' <= k + (1 + x + z'), z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 2 + z }-> s :|: s >= 0, s <= z + 0, z >= 0

Function symbols to be analyzed: {rev}, {encArg}, {encode_r1}, {encode_rev}, {encode_cons}
Previous analysis results are:
  r1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encode_empty: runtime: O(1) [0], size: O(1) [0]
  rev: runtime: ?, size: O(n^1) [z]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: rev
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z

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

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

   encArg(z) -{ 0 }-> rev(rev(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> rev(r1(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 0 }-> rev(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> rev(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> rev(1 + encArg(x_1') + encArg(x_2')) :|: z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> r1(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 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 0 }-> r1(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 0 }-> rev(rev(encArg(z - 1))) :|: z - 1 >= 0
   encode_rev(z) -{ 0 }-> rev(r1(encArg(x_122), encArg(x_214))) :|: z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z = 0
   encode_rev(z) -{ 0 }-> rev(0) :|: z >= 0
   encode_rev(z) -{ 0 }-> rev(1 + encArg(x_120) + encArg(x_213)) :|: x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 2 + k }-> s' :|: s' >= 0, s' <= k + (1 + x + z'), z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 2 + z }-> s :|: s >= 0, s <= z + 0, z >= 0

Function symbols to be analyzed: {encArg}, {encode_r1}, {encode_rev}, {encode_cons}
Previous analysis results are:
  r1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encode_empty: runtime: O(1) [0], size: O(1) [0]
  rev: runtime: O(n^1) [2 + z], size: O(n^1) [z]

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

(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) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   encArg(z) -{ 0 }-> rev(rev(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> rev(r1(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 0 }-> rev(1 + encArg(x_1') + encArg(x_2')) :|: z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> r1(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 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 0 }-> r1(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0, z = 0
   encode_rev(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0, z >= 0
   encode_rev(z) -{ 0 }-> rev(rev(encArg(z - 1))) :|: z - 1 >= 0
   encode_rev(z) -{ 0 }-> rev(r1(encArg(x_122), encArg(x_214))) :|: z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 0 }-> rev(1 + encArg(x_120) + encArg(x_213)) :|: x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 2 + k }-> s' :|: s' >= 0, s' <= k + (1 + x + z'), z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 2 + z }-> s :|: s >= 0, s <= z + 0, z >= 0

Function symbols to be analyzed: {encArg}, {encode_r1}, {encode_rev}, {encode_cons}
Previous analysis results are:
  r1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encode_empty: runtime: O(1) [0], size: O(1) [0]
  rev: runtime: O(n^1) [2 + z], size: O(n^1) [z]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   encArg(z) -{ 0 }-> rev(rev(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> rev(r1(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 0 }-> rev(1 + encArg(x_1') + encArg(x_2')) :|: z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> r1(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 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 0 }-> r1(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0, z = 0
   encode_rev(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0, z >= 0
   encode_rev(z) -{ 0 }-> rev(rev(encArg(z - 1))) :|: z - 1 >= 0
   encode_rev(z) -{ 0 }-> rev(r1(encArg(x_122), encArg(x_214))) :|: z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 0 }-> rev(1 + encArg(x_120) + encArg(x_213)) :|: x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 2 + k }-> s' :|: s' >= 0, s' <= k + (1 + x + z'), z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 2 + z }-> s :|: s >= 0, s <= z + 0, z >= 0

Function symbols to be analyzed: {encArg}, {encode_r1}, {encode_rev}, {encode_cons}
Previous analysis results are:
  r1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encode_empty: runtime: O(1) [0], size: O(1) [0]
  rev: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  encArg: runtime: ?, size: O(n^1) [z]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: encArg
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 2 + 2*z + 2*z^2

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

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

   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   encArg(z) -{ 0 }-> rev(rev(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> rev(r1(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 0 }-> rev(1 + encArg(x_1') + encArg(x_2')) :|: z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> r1(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 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 0 }-> r1(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0, z = 0
   encode_rev(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0, z >= 0
   encode_rev(z) -{ 0 }-> rev(rev(encArg(z - 1))) :|: z - 1 >= 0
   encode_rev(z) -{ 0 }-> rev(r1(encArg(x_122), encArg(x_214))) :|: z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 0 }-> rev(1 + encArg(x_120) + encArg(x_213)) :|: x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 2 + k }-> s' :|: s' >= 0, s' <= k + (1 + x + z'), z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 2 + z }-> s :|: s >= 0, s <= z + 0, z >= 0

Function symbols to be analyzed: {encode_r1}, {encode_rev}, {encode_cons}
Previous analysis results are:
  r1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encode_empty: runtime: O(1) [0], size: O(1) [0]
  rev: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  encArg: runtime: O(n^2) [2 + 2*z + 2*z^2], size: O(n^1) [z]

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

(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) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   encArg(z) -{ 10 + s10 + s9 + -6*z + 2*z^2 }-> s11 :|: s9 >= 0, s9 <= z - 2, s10 >= 0, s10 <= s9, s11 >= 0, s11 <= s10, z - 2 >= 0
   encArg(z) -{ 7 + s12 + s14 + 2*x_11 + 2*x_11^2 + 2*x_2'' + 2*x_2''^2 }-> s15 :|: s12 >= 0, s12 <= x_11, s13 >= 0, s13 <= x_2'', s14 >= 0, s14 <= s12 + s13, s15 >= 0, s15 <= s14, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 5 + s16 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> s18 :|: s16 >= 0, s16 <= x_1, s17 >= 0, s17 <= x_2, s18 >= 0, s18 <= s16 + s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 7 + s6 + s7 + 2*x_1' + 2*x_1'^2 + 2*x_2' + 2*x_2'^2 }-> s8 :|: s6 >= 0, s6 <= x_1', s7 >= 0, s7 <= x_2', s8 >= 0, s8 <= 1 + s6 + s7, z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 4 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> 1 + s4 + s5 :|: s4 >= 0, s4 <= x_1, s5 >= 0, s5 <= x_2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 4 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> 1 + s32 + s33 :|: s32 >= 0, s32 <= z, s33 >= 0, s33 <= z', z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 5 + s29 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> s31 :|: s29 >= 0, s29 <= z, s30 >= 0, s30 <= z', s31 >= 0, s31 <= s29 + s30, z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0, z = 0
   encode_rev(z) -{ 7 + s19 + s20 + 2*x_120 + 2*x_120^2 + 2*x_213 + 2*x_213^2 }-> s21 :|: s19 >= 0, s19 <= x_120, s20 >= 0, s20 <= x_213, s21 >= 0, s21 <= 1 + s19 + s20, x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 6 + s22 + s23 + -2*z + 2*z^2 }-> s24 :|: s22 >= 0, s22 <= z - 1, s23 >= 0, s23 <= s22, s24 >= 0, s24 <= s23, z - 1 >= 0
   encode_rev(z) -{ 7 + s25 + s27 + 2*x_122 + 2*x_122^2 + 2*x_214 + 2*x_214^2 }-> s28 :|: s25 >= 0, s25 <= x_122, s26 >= 0, s26 <= x_214, s27 >= 0, s27 <= s25 + s26, s28 >= 0, s28 <= s27, z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0, z >= 0
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 2 + k }-> s' :|: s' >= 0, s' <= k + (1 + x + z'), z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 2 + z }-> s :|: s >= 0, s <= z + 0, z >= 0

Function symbols to be analyzed: {encode_r1}, {encode_rev}, {encode_cons}
Previous analysis results are:
  r1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encode_empty: runtime: O(1) [0], size: O(1) [0]
  rev: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  encArg: runtime: O(n^2) [2 + 2*z + 2*z^2], size: O(n^1) [z]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_r1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z + z'

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

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

   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   encArg(z) -{ 10 + s10 + s9 + -6*z + 2*z^2 }-> s11 :|: s9 >= 0, s9 <= z - 2, s10 >= 0, s10 <= s9, s11 >= 0, s11 <= s10, z - 2 >= 0
   encArg(z) -{ 7 + s12 + s14 + 2*x_11 + 2*x_11^2 + 2*x_2'' + 2*x_2''^2 }-> s15 :|: s12 >= 0, s12 <= x_11, s13 >= 0, s13 <= x_2'', s14 >= 0, s14 <= s12 + s13, s15 >= 0, s15 <= s14, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 5 + s16 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> s18 :|: s16 >= 0, s16 <= x_1, s17 >= 0, s17 <= x_2, s18 >= 0, s18 <= s16 + s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 7 + s6 + s7 + 2*x_1' + 2*x_1'^2 + 2*x_2' + 2*x_2'^2 }-> s8 :|: s6 >= 0, s6 <= x_1', s7 >= 0, s7 <= x_2', s8 >= 0, s8 <= 1 + s6 + s7, z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 4 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> 1 + s4 + s5 :|: s4 >= 0, s4 <= x_1, s5 >= 0, s5 <= x_2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 4 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> 1 + s32 + s33 :|: s32 >= 0, s32 <= z, s33 >= 0, s33 <= z', z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 5 + s29 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> s31 :|: s29 >= 0, s29 <= z, s30 >= 0, s30 <= z', s31 >= 0, s31 <= s29 + s30, z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0, z = 0
   encode_rev(z) -{ 7 + s19 + s20 + 2*x_120 + 2*x_120^2 + 2*x_213 + 2*x_213^2 }-> s21 :|: s19 >= 0, s19 <= x_120, s20 >= 0, s20 <= x_213, s21 >= 0, s21 <= 1 + s19 + s20, x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 6 + s22 + s23 + -2*z + 2*z^2 }-> s24 :|: s22 >= 0, s22 <= z - 1, s23 >= 0, s23 <= s22, s24 >= 0, s24 <= s23, z - 1 >= 0
   encode_rev(z) -{ 7 + s25 + s27 + 2*x_122 + 2*x_122^2 + 2*x_214 + 2*x_214^2 }-> s28 :|: s25 >= 0, s25 <= x_122, s26 >= 0, s26 <= x_214, s27 >= 0, s27 <= s25 + s26, s28 >= 0, s28 <= s27, z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0, z >= 0
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 2 + k }-> s' :|: s' >= 0, s' <= k + (1 + x + z'), z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 2 + z }-> s :|: s >= 0, s <= z + 0, z >= 0

Function symbols to be analyzed: {encode_r1}, {encode_rev}, {encode_cons}
Previous analysis results are:
  r1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encode_empty: runtime: O(1) [0], size: O(1) [0]
  rev: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  encArg: runtime: O(n^2) [2 + 2*z + 2*z^2], size: O(n^1) [z]
  encode_r1: runtime: ?, size: O(n^1) [z + z']

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_r1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 5 + 3*z + 2*z^2 + 2*z' + 2*z'^2

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

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

   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   encArg(z) -{ 10 + s10 + s9 + -6*z + 2*z^2 }-> s11 :|: s9 >= 0, s9 <= z - 2, s10 >= 0, s10 <= s9, s11 >= 0, s11 <= s10, z - 2 >= 0
   encArg(z) -{ 7 + s12 + s14 + 2*x_11 + 2*x_11^2 + 2*x_2'' + 2*x_2''^2 }-> s15 :|: s12 >= 0, s12 <= x_11, s13 >= 0, s13 <= x_2'', s14 >= 0, s14 <= s12 + s13, s15 >= 0, s15 <= s14, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 5 + s16 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> s18 :|: s16 >= 0, s16 <= x_1, s17 >= 0, s17 <= x_2, s18 >= 0, s18 <= s16 + s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 7 + s6 + s7 + 2*x_1' + 2*x_1'^2 + 2*x_2' + 2*x_2'^2 }-> s8 :|: s6 >= 0, s6 <= x_1', s7 >= 0, s7 <= x_2', s8 >= 0, s8 <= 1 + s6 + s7, z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 4 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> 1 + s4 + s5 :|: s4 >= 0, s4 <= x_1, s5 >= 0, s5 <= x_2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 4 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> 1 + s32 + s33 :|: s32 >= 0, s32 <= z, s33 >= 0, s33 <= z', z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 5 + s29 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> s31 :|: s29 >= 0, s29 <= z, s30 >= 0, s30 <= z', s31 >= 0, s31 <= s29 + s30, z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0, z = 0
   encode_rev(z) -{ 7 + s19 + s20 + 2*x_120 + 2*x_120^2 + 2*x_213 + 2*x_213^2 }-> s21 :|: s19 >= 0, s19 <= x_120, s20 >= 0, s20 <= x_213, s21 >= 0, s21 <= 1 + s19 + s20, x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 6 + s22 + s23 + -2*z + 2*z^2 }-> s24 :|: s22 >= 0, s22 <= z - 1, s23 >= 0, s23 <= s22, s24 >= 0, s24 <= s23, z - 1 >= 0
   encode_rev(z) -{ 7 + s25 + s27 + 2*x_122 + 2*x_122^2 + 2*x_214 + 2*x_214^2 }-> s28 :|: s25 >= 0, s25 <= x_122, s26 >= 0, s26 <= x_214, s27 >= 0, s27 <= s25 + s26, s28 >= 0, s28 <= s27, z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0, z >= 0
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 2 + k }-> s' :|: s' >= 0, s' <= k + (1 + x + z'), z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 2 + z }-> s :|: s >= 0, s <= z + 0, z >= 0

Function symbols to be analyzed: {encode_rev}, {encode_cons}
Previous analysis results are:
  r1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encode_empty: runtime: O(1) [0], size: O(1) [0]
  rev: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  encArg: runtime: O(n^2) [2 + 2*z + 2*z^2], size: O(n^1) [z]
  encode_r1: runtime: O(n^2) [5 + 3*z + 2*z^2 + 2*z' + 2*z'^2], size: O(n^1) [z + 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) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   encArg(z) -{ 10 + s10 + s9 + -6*z + 2*z^2 }-> s11 :|: s9 >= 0, s9 <= z - 2, s10 >= 0, s10 <= s9, s11 >= 0, s11 <= s10, z - 2 >= 0
   encArg(z) -{ 7 + s12 + s14 + 2*x_11 + 2*x_11^2 + 2*x_2'' + 2*x_2''^2 }-> s15 :|: s12 >= 0, s12 <= x_11, s13 >= 0, s13 <= x_2'', s14 >= 0, s14 <= s12 + s13, s15 >= 0, s15 <= s14, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 5 + s16 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> s18 :|: s16 >= 0, s16 <= x_1, s17 >= 0, s17 <= x_2, s18 >= 0, s18 <= s16 + s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 7 + s6 + s7 + 2*x_1' + 2*x_1'^2 + 2*x_2' + 2*x_2'^2 }-> s8 :|: s6 >= 0, s6 <= x_1', s7 >= 0, s7 <= x_2', s8 >= 0, s8 <= 1 + s6 + s7, z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 4 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> 1 + s4 + s5 :|: s4 >= 0, s4 <= x_1, s5 >= 0, s5 <= x_2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 4 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> 1 + s32 + s33 :|: s32 >= 0, s32 <= z, s33 >= 0, s33 <= z', z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 5 + s29 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> s31 :|: s29 >= 0, s29 <= z, s30 >= 0, s30 <= z', s31 >= 0, s31 <= s29 + s30, z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0, z = 0
   encode_rev(z) -{ 7 + s19 + s20 + 2*x_120 + 2*x_120^2 + 2*x_213 + 2*x_213^2 }-> s21 :|: s19 >= 0, s19 <= x_120, s20 >= 0, s20 <= x_213, s21 >= 0, s21 <= 1 + s19 + s20, x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 6 + s22 + s23 + -2*z + 2*z^2 }-> s24 :|: s22 >= 0, s22 <= z - 1, s23 >= 0, s23 <= s22, s24 >= 0, s24 <= s23, z - 1 >= 0
   encode_rev(z) -{ 7 + s25 + s27 + 2*x_122 + 2*x_122^2 + 2*x_214 + 2*x_214^2 }-> s28 :|: s25 >= 0, s25 <= x_122, s26 >= 0, s26 <= x_214, s27 >= 0, s27 <= s25 + s26, s28 >= 0, s28 <= s27, z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0, z >= 0
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 2 + k }-> s' :|: s' >= 0, s' <= k + (1 + x + z'), z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 2 + z }-> s :|: s >= 0, s <= z + 0, z >= 0

Function symbols to be analyzed: {encode_rev}, {encode_cons}
Previous analysis results are:
  r1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encode_empty: runtime: O(1) [0], size: O(1) [0]
  rev: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  encArg: runtime: O(n^2) [2 + 2*z + 2*z^2], size: O(n^1) [z]
  encode_r1: runtime: O(n^2) [5 + 3*z + 2*z^2 + 2*z' + 2*z'^2], size: O(n^1) [z + z']

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   encArg(z) -{ 10 + s10 + s9 + -6*z + 2*z^2 }-> s11 :|: s9 >= 0, s9 <= z - 2, s10 >= 0, s10 <= s9, s11 >= 0, s11 <= s10, z - 2 >= 0
   encArg(z) -{ 7 + s12 + s14 + 2*x_11 + 2*x_11^2 + 2*x_2'' + 2*x_2''^2 }-> s15 :|: s12 >= 0, s12 <= x_11, s13 >= 0, s13 <= x_2'', s14 >= 0, s14 <= s12 + s13, s15 >= 0, s15 <= s14, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 5 + s16 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> s18 :|: s16 >= 0, s16 <= x_1, s17 >= 0, s17 <= x_2, s18 >= 0, s18 <= s16 + s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 7 + s6 + s7 + 2*x_1' + 2*x_1'^2 + 2*x_2' + 2*x_2'^2 }-> s8 :|: s6 >= 0, s6 <= x_1', s7 >= 0, s7 <= x_2', s8 >= 0, s8 <= 1 + s6 + s7, z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 4 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> 1 + s4 + s5 :|: s4 >= 0, s4 <= x_1, s5 >= 0, s5 <= x_2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 4 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> 1 + s32 + s33 :|: s32 >= 0, s32 <= z, s33 >= 0, s33 <= z', z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 5 + s29 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> s31 :|: s29 >= 0, s29 <= z, s30 >= 0, s30 <= z', s31 >= 0, s31 <= s29 + s30, z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0, z = 0
   encode_rev(z) -{ 7 + s19 + s20 + 2*x_120 + 2*x_120^2 + 2*x_213 + 2*x_213^2 }-> s21 :|: s19 >= 0, s19 <= x_120, s20 >= 0, s20 <= x_213, s21 >= 0, s21 <= 1 + s19 + s20, x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 6 + s22 + s23 + -2*z + 2*z^2 }-> s24 :|: s22 >= 0, s22 <= z - 1, s23 >= 0, s23 <= s22, s24 >= 0, s24 <= s23, z - 1 >= 0
   encode_rev(z) -{ 7 + s25 + s27 + 2*x_122 + 2*x_122^2 + 2*x_214 + 2*x_214^2 }-> s28 :|: s25 >= 0, s25 <= x_122, s26 >= 0, s26 <= x_214, s27 >= 0, s27 <= s25 + s26, s28 >= 0, s28 <= s27, z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0, z >= 0
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 2 + k }-> s' :|: s' >= 0, s' <= k + (1 + x + z'), z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 2 + z }-> s :|: s >= 0, s <= z + 0, z >= 0

Function symbols to be analyzed: {encode_rev}, {encode_cons}
Previous analysis results are:
  r1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encode_empty: runtime: O(1) [0], size: O(1) [0]
  rev: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  encArg: runtime: O(n^2) [2 + 2*z + 2*z^2], size: O(n^1) [z]
  encode_r1: runtime: O(n^2) [5 + 3*z + 2*z^2 + 2*z' + 2*z'^2], size: O(n^1) [z + z']
  encode_rev: runtime: ?, size: O(n^1) [z]

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_rev
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 24 + 13*z + 10*z^2

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

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

   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   encArg(z) -{ 10 + s10 + s9 + -6*z + 2*z^2 }-> s11 :|: s9 >= 0, s9 <= z - 2, s10 >= 0, s10 <= s9, s11 >= 0, s11 <= s10, z - 2 >= 0
   encArg(z) -{ 7 + s12 + s14 + 2*x_11 + 2*x_11^2 + 2*x_2'' + 2*x_2''^2 }-> s15 :|: s12 >= 0, s12 <= x_11, s13 >= 0, s13 <= x_2'', s14 >= 0, s14 <= s12 + s13, s15 >= 0, s15 <= s14, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 5 + s16 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> s18 :|: s16 >= 0, s16 <= x_1, s17 >= 0, s17 <= x_2, s18 >= 0, s18 <= s16 + s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 7 + s6 + s7 + 2*x_1' + 2*x_1'^2 + 2*x_2' + 2*x_2'^2 }-> s8 :|: s6 >= 0, s6 <= x_1', s7 >= 0, s7 <= x_2', s8 >= 0, s8 <= 1 + s6 + s7, z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 4 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> 1 + s4 + s5 :|: s4 >= 0, s4 <= x_1, s5 >= 0, s5 <= x_2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 4 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> 1 + s32 + s33 :|: s32 >= 0, s32 <= z, s33 >= 0, s33 <= z', z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 5 + s29 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> s31 :|: s29 >= 0, s29 <= z, s30 >= 0, s30 <= z', s31 >= 0, s31 <= s29 + s30, z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0, z = 0
   encode_rev(z) -{ 7 + s19 + s20 + 2*x_120 + 2*x_120^2 + 2*x_213 + 2*x_213^2 }-> s21 :|: s19 >= 0, s19 <= x_120, s20 >= 0, s20 <= x_213, s21 >= 0, s21 <= 1 + s19 + s20, x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 6 + s22 + s23 + -2*z + 2*z^2 }-> s24 :|: s22 >= 0, s22 <= z - 1, s23 >= 0, s23 <= s22, s24 >= 0, s24 <= s23, z - 1 >= 0
   encode_rev(z) -{ 7 + s25 + s27 + 2*x_122 + 2*x_122^2 + 2*x_214 + 2*x_214^2 }-> s28 :|: s25 >= 0, s25 <= x_122, s26 >= 0, s26 <= x_214, s27 >= 0, s27 <= s25 + s26, s28 >= 0, s28 <= s27, z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0, z >= 0
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 2 + k }-> s' :|: s' >= 0, s' <= k + (1 + x + z'), z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 2 + z }-> s :|: s >= 0, s <= z + 0, z >= 0

Function symbols to be analyzed: {encode_cons}
Previous analysis results are:
  r1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encode_empty: runtime: O(1) [0], size: O(1) [0]
  rev: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  encArg: runtime: O(n^2) [2 + 2*z + 2*z^2], size: O(n^1) [z]
  encode_r1: runtime: O(n^2) [5 + 3*z + 2*z^2 + 2*z' + 2*z'^2], size: O(n^1) [z + z']
  encode_rev: runtime: O(n^2) [24 + 13*z + 10*z^2], size: O(n^1) [z]

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

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

(56)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   encArg(z) -{ 10 + s10 + s9 + -6*z + 2*z^2 }-> s11 :|: s9 >= 0, s9 <= z - 2, s10 >= 0, s10 <= s9, s11 >= 0, s11 <= s10, z - 2 >= 0
   encArg(z) -{ 7 + s12 + s14 + 2*x_11 + 2*x_11^2 + 2*x_2'' + 2*x_2''^2 }-> s15 :|: s12 >= 0, s12 <= x_11, s13 >= 0, s13 <= x_2'', s14 >= 0, s14 <= s12 + s13, s15 >= 0, s15 <= s14, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 5 + s16 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> s18 :|: s16 >= 0, s16 <= x_1, s17 >= 0, s17 <= x_2, s18 >= 0, s18 <= s16 + s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 7 + s6 + s7 + 2*x_1' + 2*x_1'^2 + 2*x_2' + 2*x_2'^2 }-> s8 :|: s6 >= 0, s6 <= x_1', s7 >= 0, s7 <= x_2', s8 >= 0, s8 <= 1 + s6 + s7, z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 4 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> 1 + s4 + s5 :|: s4 >= 0, s4 <= x_1, s5 >= 0, s5 <= x_2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 4 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> 1 + s32 + s33 :|: s32 >= 0, s32 <= z, s33 >= 0, s33 <= z', z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 5 + s29 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> s31 :|: s29 >= 0, s29 <= z, s30 >= 0, s30 <= z', s31 >= 0, s31 <= s29 + s30, z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0, z = 0
   encode_rev(z) -{ 7 + s19 + s20 + 2*x_120 + 2*x_120^2 + 2*x_213 + 2*x_213^2 }-> s21 :|: s19 >= 0, s19 <= x_120, s20 >= 0, s20 <= x_213, s21 >= 0, s21 <= 1 + s19 + s20, x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 6 + s22 + s23 + -2*z + 2*z^2 }-> s24 :|: s22 >= 0, s22 <= z - 1, s23 >= 0, s23 <= s22, s24 >= 0, s24 <= s23, z - 1 >= 0
   encode_rev(z) -{ 7 + s25 + s27 + 2*x_122 + 2*x_122^2 + 2*x_214 + 2*x_214^2 }-> s28 :|: s25 >= 0, s25 <= x_122, s26 >= 0, s26 <= x_214, s27 >= 0, s27 <= s25 + s26, s28 >= 0, s28 <= s27, z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0, z >= 0
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 2 + k }-> s' :|: s' >= 0, s' <= k + (1 + x + z'), z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 2 + z }-> s :|: s >= 0, s <= z + 0, z >= 0

Function symbols to be analyzed: {encode_cons}
Previous analysis results are:
  r1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encode_empty: runtime: O(1) [0], size: O(1) [0]
  rev: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  encArg: runtime: O(n^2) [2 + 2*z + 2*z^2], size: O(n^1) [z]
  encode_r1: runtime: O(n^2) [5 + 3*z + 2*z^2 + 2*z' + 2*z'^2], size: O(n^1) [z + z']
  encode_rev: runtime: O(n^2) [24 + 13*z + 10*z^2], size: O(n^1) [z]

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   encArg(z) -{ 10 + s10 + s9 + -6*z + 2*z^2 }-> s11 :|: s9 >= 0, s9 <= z - 2, s10 >= 0, s10 <= s9, s11 >= 0, s11 <= s10, z - 2 >= 0
   encArg(z) -{ 7 + s12 + s14 + 2*x_11 + 2*x_11^2 + 2*x_2'' + 2*x_2''^2 }-> s15 :|: s12 >= 0, s12 <= x_11, s13 >= 0, s13 <= x_2'', s14 >= 0, s14 <= s12 + s13, s15 >= 0, s15 <= s14, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 5 + s16 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> s18 :|: s16 >= 0, s16 <= x_1, s17 >= 0, s17 <= x_2, s18 >= 0, s18 <= s16 + s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 7 + s6 + s7 + 2*x_1' + 2*x_1'^2 + 2*x_2' + 2*x_2'^2 }-> s8 :|: s6 >= 0, s6 <= x_1', s7 >= 0, s7 <= x_2', s8 >= 0, s8 <= 1 + s6 + s7, z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 4 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> 1 + s4 + s5 :|: s4 >= 0, s4 <= x_1, s5 >= 0, s5 <= x_2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 4 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> 1 + s32 + s33 :|: s32 >= 0, s32 <= z, s33 >= 0, s33 <= z', z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 5 + s29 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> s31 :|: s29 >= 0, s29 <= z, s30 >= 0, s30 <= z', s31 >= 0, s31 <= s29 + s30, z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0, z = 0
   encode_rev(z) -{ 7 + s19 + s20 + 2*x_120 + 2*x_120^2 + 2*x_213 + 2*x_213^2 }-> s21 :|: s19 >= 0, s19 <= x_120, s20 >= 0, s20 <= x_213, s21 >= 0, s21 <= 1 + s19 + s20, x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 6 + s22 + s23 + -2*z + 2*z^2 }-> s24 :|: s22 >= 0, s22 <= z - 1, s23 >= 0, s23 <= s22, s24 >= 0, s24 <= s23, z - 1 >= 0
   encode_rev(z) -{ 7 + s25 + s27 + 2*x_122 + 2*x_122^2 + 2*x_214 + 2*x_214^2 }-> s28 :|: s25 >= 0, s25 <= x_122, s26 >= 0, s26 <= x_214, s27 >= 0, s27 <= s25 + s26, s28 >= 0, s28 <= s27, z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0, z >= 0
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 2 + k }-> s' :|: s' >= 0, s' <= k + (1 + x + z'), z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 2 + z }-> s :|: s >= 0, s <= z + 0, z >= 0

Function symbols to be analyzed: {encode_cons}
Previous analysis results are:
  r1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encode_empty: runtime: O(1) [0], size: O(1) [0]
  rev: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  encArg: runtime: O(n^2) [2 + 2*z + 2*z^2], size: O(n^1) [z]
  encode_r1: runtime: O(n^2) [5 + 3*z + 2*z^2 + 2*z' + 2*z'^2], size: O(n^1) [z + z']
  encode_rev: runtime: O(n^2) [24 + 13*z + 10*z^2], size: O(n^1) [z]
  encode_cons: runtime: ?, size: O(n^1) [1 + z + z']

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

(59) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_cons
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 4 + 2*z + 2*z^2 + 2*z' + 2*z'^2

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

(60)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   encArg(z) -{ 10 + s10 + s9 + -6*z + 2*z^2 }-> s11 :|: s9 >= 0, s9 <= z - 2, s10 >= 0, s10 <= s9, s11 >= 0, s11 <= s10, z - 2 >= 0
   encArg(z) -{ 7 + s12 + s14 + 2*x_11 + 2*x_11^2 + 2*x_2'' + 2*x_2''^2 }-> s15 :|: s12 >= 0, s12 <= x_11, s13 >= 0, s13 <= x_2'', s14 >= 0, s14 <= s12 + s13, s15 >= 0, s15 <= s14, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0
   encArg(z) -{ 5 + s16 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> s18 :|: s16 >= 0, s16 <= x_1, s17 >= 0, s17 <= x_2, s18 >= 0, s18 <= s16 + s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 7 + s6 + s7 + 2*x_1' + 2*x_1'^2 + 2*x_2' + 2*x_2'^2 }-> s8 :|: s6 >= 0, s6 <= x_1', s7 >= 0, s7 <= x_2', s8 >= 0, s8 <= 1 + s6 + s7, z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 4 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> 1 + s4 + s5 :|: s4 >= 0, s4 <= x_1, s5 >= 0, s5 <= x_2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_cons(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_cons(z, z') -{ 4 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> 1 + s32 + s33 :|: s32 >= 0, s32 <= z, s33 >= 0, s33 <= z', z >= 0, z' >= 0
   encode_empty -{ 0 }-> 0 :|: 
   encode_r1(z, z') -{ 5 + s29 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> s31 :|: s29 >= 0, s29 <= z, s30 >= 0, s30 <= z', s31 >= 0, s31 <= s29 + s30, z >= 0, z' >= 0
   encode_r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_rev(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0, z = 0
   encode_rev(z) -{ 7 + s19 + s20 + 2*x_120 + 2*x_120^2 + 2*x_213 + 2*x_213^2 }-> s21 :|: s19 >= 0, s19 <= x_120, s20 >= 0, s20 <= x_213, s21 >= 0, s21 <= 1 + s19 + s20, x_120 >= 0, x_213 >= 0, z = 1 + x_120 + x_213
   encode_rev(z) -{ 6 + s22 + s23 + -2*z + 2*z^2 }-> s24 :|: s22 >= 0, s22 <= z - 1, s23 >= 0, s23 <= s22, s24 >= 0, s24 <= s23, z - 1 >= 0
   encode_rev(z) -{ 7 + s25 + s27 + 2*x_122 + 2*x_122^2 + 2*x_214 + 2*x_214^2 }-> s28 :|: s25 >= 0, s25 <= x_122, s26 >= 0, s26 <= x_214, s27 >= 0, s27 <= s25 + s26, s28 >= 0, s28 <= s27, z = 1 + x_122 + x_214, x_214 >= 0, x_122 >= 0
   encode_rev(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0, z >= 0
   encode_rev(z) -{ 0 }-> 0 :|: z >= 0
   r1(z, z') -{ 2 + k }-> s' :|: s' >= 0, s' <= k + (1 + x + z'), z' >= 0, x >= 0, k >= 0, z = 1 + x + k
   r1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   r1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   rev(z) -{ 2 + z }-> s :|: s >= 0, s <= z + 0, z >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  r1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encode_empty: runtime: O(1) [0], size: O(1) [0]
  rev: runtime: O(n^1) [2 + z], size: O(n^1) [z]
  encArg: runtime: O(n^2) [2 + 2*z + 2*z^2], size: O(n^1) [z]
  encode_r1: runtime: O(n^2) [5 + 3*z + 2*z^2 + 2*z' + 2*z'^2], size: O(n^1) [z + z']
  encode_rev: runtime: O(n^2) [24 + 13*z + 10*z^2], size: O(n^1) [z]
  encode_cons: runtime: O(n^2) [4 + 2*z + 2*z^2 + 2*z' + 2*z'^2], size: O(n^1) [1 + z + z']

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

(61) FinalProof (FINISHED)
Computed overall runtime complexity
----------------------------------------

(62)
BOUNDS(1, n^2)

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

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

(64)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   rev(ls) -> r1(ls, empty)
   r1(empty, a) -> a
   r1(cons(x, k), a) -> r1(k, cons(x, a))

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

   encArg(empty) -> empty
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_rev(x_1)) -> rev(encArg(x_1))
   encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2))
   encode_rev(x_1) -> rev(encArg(x_1))
   encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2))
   encode_empty -> empty
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))

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

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

(66)
Obligation:
Innermost TRS:
Rules:
rev(ls) -> r1(ls, empty)
r1(empty, a) -> a
r1(cons(x, k), a) -> r1(k, cons(x, a))
encArg(empty) -> empty
encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
encArg(cons_rev(x_1)) -> rev(encArg(x_1))
encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2))
encode_rev(x_1) -> rev(encArg(x_1))
encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2))
encode_empty -> empty
encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))

Types:
rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
empty :: empty:cons:cons_rev:cons_r1
cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encArg :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_empty :: empty:cons:cons_rev:cons_r1
encode_cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
hole_empty:cons:cons_rev:cons_r11_0 :: empty:cons:cons_rev:cons_r1
gen_empty:cons:cons_rev:cons_r12_0 :: Nat -> empty:cons:cons_rev:cons_r1

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

(67) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
r1, encArg

They will be analysed ascendingly in the following order:
r1 < encArg

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

(68)
Obligation:
Innermost TRS:
Rules:
rev(ls) -> r1(ls, empty)
r1(empty, a) -> a
r1(cons(x, k), a) -> r1(k, cons(x, a))
encArg(empty) -> empty
encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
encArg(cons_rev(x_1)) -> rev(encArg(x_1))
encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2))
encode_rev(x_1) -> rev(encArg(x_1))
encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2))
encode_empty -> empty
encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))

Types:
rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
empty :: empty:cons:cons_rev:cons_r1
cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encArg :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_empty :: empty:cons:cons_rev:cons_r1
encode_cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
hole_empty:cons:cons_rev:cons_r11_0 :: empty:cons:cons_rev:cons_r1
gen_empty:cons:cons_rev:cons_r12_0 :: Nat -> empty:cons:cons_rev:cons_r1


Generator Equations:
gen_empty:cons:cons_rev:cons_r12_0(0) <=> empty
gen_empty:cons:cons_rev:cons_r12_0(+(x, 1)) <=> cons(empty, gen_empty:cons:cons_rev:cons_r12_0(x))


The following defined symbols remain to be analysed:
r1, encArg

They will be analysed ascendingly in the following order:
r1 < encArg

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

(69) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
r1(gen_empty:cons:cons_rev:cons_r12_0(n4_0), gen_empty:cons:cons_rev:cons_r12_0(b)) -> gen_empty:cons:cons_rev:cons_r12_0(+(n4_0, b)), rt in Omega(1 + n4_0)

Induction Base:
r1(gen_empty:cons:cons_rev:cons_r12_0(0), gen_empty:cons:cons_rev:cons_r12_0(b)) ->_R^Omega(1)
gen_empty:cons:cons_rev:cons_r12_0(b)

Induction Step:
r1(gen_empty:cons:cons_rev:cons_r12_0(+(n4_0, 1)), gen_empty:cons:cons_rev:cons_r12_0(b)) ->_R^Omega(1)
r1(gen_empty:cons:cons_rev:cons_r12_0(n4_0), cons(empty, gen_empty:cons:cons_rev:cons_r12_0(b))) ->_IH
gen_empty:cons:cons_rev:cons_r12_0(+(+(b, 1), c5_0))

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

(70)
Complex Obligation (BEST)

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

(71)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
rev(ls) -> r1(ls, empty)
r1(empty, a) -> a
r1(cons(x, k), a) -> r1(k, cons(x, a))
encArg(empty) -> empty
encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
encArg(cons_rev(x_1)) -> rev(encArg(x_1))
encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2))
encode_rev(x_1) -> rev(encArg(x_1))
encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2))
encode_empty -> empty
encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))

Types:
rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
empty :: empty:cons:cons_rev:cons_r1
cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encArg :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_empty :: empty:cons:cons_rev:cons_r1
encode_cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
hole_empty:cons:cons_rev:cons_r11_0 :: empty:cons:cons_rev:cons_r1
gen_empty:cons:cons_rev:cons_r12_0 :: Nat -> empty:cons:cons_rev:cons_r1


Generator Equations:
gen_empty:cons:cons_rev:cons_r12_0(0) <=> empty
gen_empty:cons:cons_rev:cons_r12_0(+(x, 1)) <=> cons(empty, gen_empty:cons:cons_rev:cons_r12_0(x))


The following defined symbols remain to be analysed:
r1, encArg

They will be analysed ascendingly in the following order:
r1 < encArg

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(72) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(73)
BOUNDS(n^1, INF)

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(74)
Obligation:
Innermost TRS:
Rules:
rev(ls) -> r1(ls, empty)
r1(empty, a) -> a
r1(cons(x, k), a) -> r1(k, cons(x, a))
encArg(empty) -> empty
encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
encArg(cons_rev(x_1)) -> rev(encArg(x_1))
encArg(cons_r1(x_1, x_2)) -> r1(encArg(x_1), encArg(x_2))
encode_rev(x_1) -> rev(encArg(x_1))
encode_r1(x_1, x_2) -> r1(encArg(x_1), encArg(x_2))
encode_empty -> empty
encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))

Types:
rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
empty :: empty:cons:cons_rev:cons_r1
cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encArg :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
cons_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_rev :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_r1 :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
encode_empty :: empty:cons:cons_rev:cons_r1
encode_cons :: empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1 -> empty:cons:cons_rev:cons_r1
hole_empty:cons:cons_rev:cons_r11_0 :: empty:cons:cons_rev:cons_r1
gen_empty:cons:cons_rev:cons_r12_0 :: Nat -> empty:cons:cons_rev:cons_r1


Lemmas:
r1(gen_empty:cons:cons_rev:cons_r12_0(n4_0), gen_empty:cons:cons_rev:cons_r12_0(b)) -> gen_empty:cons:cons_rev:cons_r12_0(+(n4_0, b)), rt in Omega(1 + n4_0)


Generator Equations:
gen_empty:cons:cons_rev:cons_r12_0(0) <=> empty
gen_empty:cons:cons_rev:cons_r12_0(+(x, 1)) <=> cons(empty, gen_empty:cons:cons_rev:cons_r12_0(x))


The following defined symbols remain to be analysed:
encArg
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(75) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_empty:cons:cons_rev:cons_r12_0(n646_0)) -> gen_empty:cons:cons_rev:cons_r12_0(n646_0), rt in Omega(0)

Induction Base:
encArg(gen_empty:cons:cons_rev:cons_r12_0(0)) ->_R^Omega(0)
empty

Induction Step:
encArg(gen_empty:cons:cons_rev:cons_r12_0(+(n646_0, 1))) ->_R^Omega(0)
cons(encArg(empty), encArg(gen_empty:cons:cons_rev:cons_r12_0(n646_0))) ->_R^Omega(0)
cons(empty, encArg(gen_empty:cons:cons_rev:cons_r12_0(n646_0))) ->_IH
cons(empty, gen_empty:cons:cons_rev:cons_r12_0(c647_0))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(76)
BOUNDS(1, INF)