/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 181 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), 5 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 95 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 4 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 47 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), 579 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 188 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 107 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 1 ms]
    (42) CpxRNTS
    (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 416 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 181 ms]
    (48) CpxRNTS
    (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 212 ms]
    (52) CpxRNTS
    (53) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (54) CpxRNTS
    (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 192 ms]
    (58) CpxRNTS
    (59) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (60) CpxRNTS
    (61) FinalProof [FINISHED, 0 ms]
    (62) BOUNDS(1, n^2)
(63) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(64) TRS for Loop Detection
    (65) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (66) BEST
        (67) proven lower bound
            (68) LowerBoundPropagationProof [FINISHED, 0 ms]
            (69) BOUNDS(n^1, INF)
        (70) TRS for Loop Detection


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

(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:

   merge(x, nil) -> x
   merge(nil, y) -> y
   merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
   merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))

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(nil) -> nil
   encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
   encArg(u) -> u
   encArg(v) -> v
   encArg(cons_merge(x_1, x_2)) -> merge(encArg(x_1), encArg(x_2))
   encode_merge(x_1, x_2) -> merge(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))
   encode_u -> u
   encode_v -> v


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

(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:

   merge(x, nil) -> x
   merge(nil, y) -> y
   merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
   merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))

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

   encArg(nil) -> nil
   encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
   encArg(u) -> u
   encArg(v) -> v
   encArg(cons_merge(x_1, x_2)) -> merge(encArg(x_1), encArg(x_2))
   encode_merge(x_1, x_2) -> merge(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))
   encode_u -> u
   encode_v -> v

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:

   merge(x, nil) -> x
   merge(nil, y) -> y
   merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
   merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))

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

   encArg(nil) -> nil
   encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
   encArg(u) -> u
   encArg(v) -> v
   encArg(cons_merge(x_1, x_2)) -> merge(encArg(x_1), encArg(x_2))
   encode_merge(x_1, x_2) -> merge(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))
   encode_u -> u
   encode_v -> v

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:

   merge(x, nil) -> x [1]
   merge(nil, y) -> y [1]
   merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v))) [1]
   merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v)) [1]
   encArg(nil) -> nil [0]
   encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2)) [0]
   encArg(u) -> u [0]
   encArg(v) -> v [0]
   encArg(cons_merge(x_1, x_2)) -> merge(encArg(x_1), encArg(x_2)) [0]
   encode_merge(x_1, x_2) -> merge(encArg(x_1), encArg(x_2)) [0]
   encode_nil -> nil [0]
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2)) [0]
   encode_u -> u [0]
   encode_v -> v [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:

   merge(x, nil) -> x [1]
   merge(nil, y) -> y [1]
   merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v))) [1]
   merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v)) [1]
   encArg(nil) -> nil [0]
   encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2)) [0]
   encArg(u) -> u [0]
   encArg(v) -> v [0]
   encArg(cons_merge(x_1, x_2)) -> merge(encArg(x_1), encArg(x_2)) [0]
   encode_merge(x_1, x_2) -> merge(encArg(x_1), encArg(x_2)) [0]
   encode_nil -> nil [0]
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2)) [0]
   encode_u -> u [0]
   encode_v -> v [0]

The TRS has the following type information:
merge :: nil:++:u:v:cons_merge -> nil:++:u:v:cons_merge -> nil:++:u:v:cons_merge
nil :: nil:++:u:v:cons_merge
++ :: nil:++:u:v:cons_merge -> nil:++:u:v:cons_merge -> nil:++:u:v:cons_merge
u :: nil:++:u:v:cons_merge
v :: nil:++:u:v:cons_merge
encArg :: nil:++:u:v:cons_merge -> nil:++:u:v:cons_merge
cons_merge :: nil:++:u:v:cons_merge -> nil:++:u:v:cons_merge -> nil:++:u:v:cons_merge
encode_merge :: nil:++:u:v:cons_merge -> nil:++:u:v:cons_merge -> nil:++:u:v:cons_merge
encode_nil :: nil:++:u:v:cons_merge
encode_++ :: nil:++:u:v:cons_merge -> nil:++:u:v:cons_merge -> nil:++:u:v:cons_merge
encode_u :: nil:++:u:v:cons_merge
encode_v :: nil:++:u:v:cons_merge

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:

   merge_2
   encArg_1
   encode_merge_2
   encode_nil
   encode_++_2
   encode_u
   encode_v

Due to the following rules being added:

   encArg(v0) -> null_encArg [0]
   encode_merge(v0, v1) -> null_encode_merge [0]
   encode_nil -> null_encode_nil [0]
   encode_++(v0, v1) -> null_encode_++ [0]
   encode_u -> null_encode_u [0]
   encode_v -> null_encode_v [0]
   merge(v0, v1) -> null_merge [0]

And the following fresh constants:    null_encArg, null_encode_merge, null_encode_nil, null_encode_++, null_encode_u, null_encode_v, null_merge

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

(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:

   merge(x, nil) -> x [1]
   merge(nil, y) -> y [1]
   merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v))) [1]
   merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v)) [1]
   encArg(nil) -> nil [0]
   encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2)) [0]
   encArg(u) -> u [0]
   encArg(v) -> v [0]
   encArg(cons_merge(x_1, x_2)) -> merge(encArg(x_1), encArg(x_2)) [0]
   encode_merge(x_1, x_2) -> merge(encArg(x_1), encArg(x_2)) [0]
   encode_nil -> nil [0]
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2)) [0]
   encode_u -> u [0]
   encode_v -> v [0]
   encArg(v0) -> null_encArg [0]
   encode_merge(v0, v1) -> null_encode_merge [0]
   encode_nil -> null_encode_nil [0]
   encode_++(v0, v1) -> null_encode_++ [0]
   encode_u -> null_encode_u [0]
   encode_v -> null_encode_v [0]
   merge(v0, v1) -> null_merge [0]

The TRS has the following type information:
merge :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge -> nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge -> nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
nil :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
++ :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge -> nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge -> nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
u :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
v :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
encArg :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge -> nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
cons_merge :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge -> nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge -> nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
encode_merge :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge -> nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge -> nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
encode_nil :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
encode_++ :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge -> nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge -> nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
encode_u :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
encode_v :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
null_encArg :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
null_encode_merge :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
null_encode_nil :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
null_encode_++ :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
null_encode_u :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
null_encode_v :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
null_merge :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge

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:

   merge(x, nil) -> x [1]
   merge(nil, y) -> y [1]
   merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v))) [1]
   merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v)) [1]
   encArg(nil) -> nil [0]
   encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2)) [0]
   encArg(u) -> u [0]
   encArg(v) -> v [0]
   encArg(cons_merge(x_1, x_2)) -> merge(encArg(x_1), encArg(x_2)) [0]
   encode_merge(x_1, x_2) -> merge(encArg(x_1), encArg(x_2)) [0]
   encode_nil -> nil [0]
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2)) [0]
   encode_u -> u [0]
   encode_v -> v [0]
   encArg(v0) -> null_encArg [0]
   encode_merge(v0, v1) -> null_encode_merge [0]
   encode_nil -> null_encode_nil [0]
   encode_++(v0, v1) -> null_encode_++ [0]
   encode_u -> null_encode_u [0]
   encode_v -> null_encode_v [0]
   merge(v0, v1) -> null_merge [0]

The TRS has the following type information:
merge :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge -> nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge -> nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
nil :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
++ :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge -> nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge -> nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
u :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
v :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
encArg :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge -> nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
cons_merge :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge -> nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge -> nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
encode_merge :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge -> nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge -> nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
encode_nil :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
encode_++ :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge -> nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge -> nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
encode_u :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
encode_v :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
null_encArg :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
null_encode_merge :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
null_encode_nil :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
null_encode_++ :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
null_encode_u :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
null_encode_v :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge
null_merge :: nil:++:u:v:cons_merge:null_encArg:null_encode_merge:null_encode_nil:null_encode_++:null_encode_u:null_encode_v:null_merge

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:

   nil => 0
   u => 1
   v => 2
   null_encArg => 0
   null_encode_merge => 0
   null_encode_nil => 0
   null_encode_++ => 0
   null_encode_u => 0
   null_encode_v => 0
   null_merge => 0

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

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

   encArg(z) -{ 0 }-> merge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   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_++(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_++(z, z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_merge(z, z') -{ 0 }-> merge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_merge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   merge(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   merge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   merge(z, z') -{ 1 }-> 1 + x + merge(y, 1 + 1 + 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 1 }-> 1 + 1 + merge(1 + x + y, 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2


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

(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 }-> merge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   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_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> merge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 1 }-> 1 + x + merge(y, 1 + 1 + 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 1 }-> 1 + 1 + merge(1 + x + y, 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2


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

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

   { encode_v }
   { encode_nil }
   { merge }
   { encode_u }
   { encArg }
   { encode_++ }
   { encode_merge }

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

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

   encArg(z) -{ 0 }-> merge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   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_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> merge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 1 }-> 1 + x + merge(y, 1 + 1 + 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 1 }-> 1 + 1 + merge(1 + x + y, 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2

Function symbols to be analyzed: {encode_v}, {encode_nil}, {merge}, {encode_u}, {encArg}, {encode_++}, {encode_merge}

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

(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 }-> merge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   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_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> merge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 1 }-> 1 + x + merge(y, 1 + 1 + 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 1 }-> 1 + 1 + merge(1 + x + y, 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2

Function symbols to be analyzed: {encode_v}, {encode_nil}, {merge}, {encode_u}, {encArg}, {encode_++}, {encode_merge}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_v
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

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

   encArg(z) -{ 0 }-> merge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   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_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> merge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 1 }-> 1 + x + merge(y, 1 + 1 + 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 1 }-> 1 + 1 + merge(1 + x + y, 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2

Function symbols to be analyzed: {encode_v}, {encode_nil}, {merge}, {encode_u}, {encArg}, {encode_++}, {encode_merge}
Previous analysis results are:
  encode_v: runtime: ?, size: O(1) [2]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> merge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   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_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> merge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 1 }-> 1 + x + merge(y, 1 + 1 + 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 1 }-> 1 + 1 + merge(1 + x + y, 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2

Function symbols to be analyzed: {encode_nil}, {merge}, {encode_u}, {encArg}, {encode_++}, {encode_merge}
Previous analysis results are:
  encode_v: runtime: O(1) [0], size: O(1) [2]

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

(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 }-> merge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   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_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> merge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 1 }-> 1 + x + merge(y, 1 + 1 + 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 1 }-> 1 + 1 + merge(1 + x + y, 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2

Function symbols to be analyzed: {encode_nil}, {merge}, {encode_u}, {encArg}, {encode_++}, {encode_merge}
Previous analysis results are:
  encode_v: runtime: O(1) [0], size: O(1) [2]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_nil
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 }-> merge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   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_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> merge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 1 }-> 1 + x + merge(y, 1 + 1 + 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 1 }-> 1 + 1 + merge(1 + x + y, 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2

Function symbols to be analyzed: {encode_nil}, {merge}, {encode_u}, {encArg}, {encode_++}, {encode_merge}
Previous analysis results are:
  encode_v: runtime: O(1) [0], size: O(1) [2]
  encode_nil: runtime: ?, size: O(1) [0]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: encode_nil
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 }-> merge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   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_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> merge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 1 }-> 1 + x + merge(y, 1 + 1 + 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 1 }-> 1 + 1 + merge(1 + x + y, 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2

Function symbols to be analyzed: {merge}, {encode_u}, {encArg}, {encode_++}, {encode_merge}
Previous analysis results are:
  encode_v: runtime: O(1) [0], size: O(1) [2]
  encode_nil: 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 }-> merge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   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_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> merge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 1 }-> 1 + x + merge(y, 1 + 1 + 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 1 }-> 1 + 1 + merge(1 + x + y, 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2

Function symbols to be analyzed: {merge}, {encode_u}, {encArg}, {encode_++}, {encode_merge}
Previous analysis results are:
  encode_v: runtime: O(1) [0], size: O(1) [2]
  encode_nil: runtime: O(1) [0], size: O(1) [0]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> merge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   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_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> merge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 1 }-> 1 + x + merge(y, 1 + 1 + 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 1 }-> 1 + 1 + merge(1 + x + y, 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2

Function symbols to be analyzed: {merge}, {encode_u}, {encArg}, {encode_++}, {encode_merge}
Previous analysis results are:
  encode_v: runtime: O(1) [0], size: O(1) [2]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  merge: runtime: ?, size: O(n^1) [z + z']

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: merge
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 }-> merge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   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_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> merge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 1 }-> 1 + x + merge(y, 1 + 1 + 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 1 }-> 1 + 1 + merge(1 + x + y, 2) :|: z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2

Function symbols to be analyzed: {encode_u}, {encArg}, {encode_++}, {encode_merge}
Previous analysis results are:
  encode_v: runtime: O(1) [0], size: O(1) [2]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  merge: runtime: O(n^1) [2 + z], size: O(n^1) [z + 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) -{ 0 }-> merge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   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_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> merge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 3 + y }-> 1 + x + s :|: s >= 0, s <= y + (1 + 1 + 2), z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 4 + x + y }-> 1 + 1 + s' :|: s' >= 0, s' <= 1 + x + y + 2, z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2

Function symbols to be analyzed: {encode_u}, {encArg}, {encode_++}, {encode_merge}
Previous analysis results are:
  encode_v: runtime: O(1) [0], size: O(1) [2]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  merge: runtime: O(n^1) [2 + z], size: O(n^1) [z + z']

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> merge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   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_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> merge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 3 + y }-> 1 + x + s :|: s >= 0, s <= y + (1 + 1 + 2), z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 4 + x + y }-> 1 + 1 + s' :|: s' >= 0, s' <= 1 + x + y + 2, z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2

Function symbols to be analyzed: {encode_u}, {encArg}, {encode_++}, {encode_merge}
Previous analysis results are:
  encode_v: runtime: O(1) [0], size: O(1) [2]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  merge: runtime: O(n^1) [2 + z], size: O(n^1) [z + z']
  encode_u: runtime: ?, size: O(1) [1]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> merge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   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_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> merge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 3 + y }-> 1 + x + s :|: s >= 0, s <= y + (1 + 1 + 2), z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 4 + x + y }-> 1 + 1 + s' :|: s' >= 0, s' <= 1 + x + y + 2, z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2

Function symbols to be analyzed: {encArg}, {encode_++}, {encode_merge}
Previous analysis results are:
  encode_v: runtime: O(1) [0], size: O(1) [2]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  merge: runtime: O(n^1) [2 + z], size: O(n^1) [z + z']
  encode_u: runtime: O(1) [0], size: O(1) [1]

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

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

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

   encArg(z) -{ 0 }-> merge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   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_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> merge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 3 + y }-> 1 + x + s :|: s >= 0, s <= y + (1 + 1 + 2), z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 4 + x + y }-> 1 + 1 + s' :|: s' >= 0, s' <= 1 + x + y + 2, z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2

Function symbols to be analyzed: {encArg}, {encode_++}, {encode_merge}
Previous analysis results are:
  encode_v: runtime: O(1) [0], size: O(1) [2]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  merge: runtime: O(n^1) [2 + z], size: O(n^1) [z + z']
  encode_u: runtime: O(1) [0], size: O(1) [1]

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

(45) 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: 2 + 2*z

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

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

   encArg(z) -{ 0 }-> merge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   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_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> merge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 3 + y }-> 1 + x + s :|: s >= 0, s <= y + (1 + 1 + 2), z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 4 + x + y }-> 1 + 1 + s' :|: s' >= 0, s' <= 1 + x + y + 2, z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2

Function symbols to be analyzed: {encArg}, {encode_++}, {encode_merge}
Previous analysis results are:
  encode_v: runtime: O(1) [0], size: O(1) [2]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  merge: runtime: O(n^1) [2 + z], size: O(n^1) [z + z']
  encode_u: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: ?, size: O(n^1) [2 + 2*z]

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

(47) 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*z + 2*z^2

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

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

   encArg(z) -{ 0 }-> merge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   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_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> merge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 3 + y }-> 1 + x + s :|: s >= 0, s <= y + (1 + 1 + 2), z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 4 + x + y }-> 1 + 1 + s' :|: s' >= 0, s' <= 1 + x + y + 2, z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2

Function symbols to be analyzed: {encode_++}, {encode_merge}
Previous analysis results are:
  encode_v: runtime: O(1) [0], size: O(1) [2]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  merge: runtime: O(n^1) [2 + z], size: O(n^1) [z + z']
  encode_u: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2*z + 2*z^2], size: O(n^1) [2 + 2*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 + s2 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> s4 :|: s2 >= 0, s2 <= 2 * x_1 + 2, s3 >= 0, s3 <= 2 * x_2 + 2, s4 >= 0, s4 <= s2 + s3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> 1 + s'' + s1 :|: s'' >= 0, s'' <= 2 * x_1 + 2, s1 >= 0, s1 <= 2 * x_2 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> 1 + s8 + s9 :|: s8 >= 0, s8 <= 2 * z + 2, s9 >= 0, s9 <= 2 * z' + 2, z >= 0, z' >= 0
   encode_merge(z, z') -{ 2 + s5 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> s7 :|: s5 >= 0, s5 <= 2 * z + 2, s6 >= 0, s6 <= 2 * z' + 2, s7 >= 0, s7 <= s5 + s6, z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 3 + y }-> 1 + x + s :|: s >= 0, s <= y + (1 + 1 + 2), z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 4 + x + y }-> 1 + 1 + s' :|: s' >= 0, s' <= 1 + x + y + 2, z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2

Function symbols to be analyzed: {encode_++}, {encode_merge}
Previous analysis results are:
  encode_v: runtime: O(1) [0], size: O(1) [2]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  merge: runtime: O(n^1) [2 + z], size: O(n^1) [z + z']
  encode_u: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2*z + 2*z^2], size: O(n^1) [2 + 2*z]

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_++
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 5 + 2*z + 2*z'

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

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

   encArg(z) -{ 2 + s2 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> s4 :|: s2 >= 0, s2 <= 2 * x_1 + 2, s3 >= 0, s3 <= 2 * x_2 + 2, s4 >= 0, s4 <= s2 + s3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> 1 + s'' + s1 :|: s'' >= 0, s'' <= 2 * x_1 + 2, s1 >= 0, s1 <= 2 * x_2 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> 1 + s8 + s9 :|: s8 >= 0, s8 <= 2 * z + 2, s9 >= 0, s9 <= 2 * z' + 2, z >= 0, z' >= 0
   encode_merge(z, z') -{ 2 + s5 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> s7 :|: s5 >= 0, s5 <= 2 * z + 2, s6 >= 0, s6 <= 2 * z' + 2, s7 >= 0, s7 <= s5 + s6, z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 3 + y }-> 1 + x + s :|: s >= 0, s <= y + (1 + 1 + 2), z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 4 + x + y }-> 1 + 1 + s' :|: s' >= 0, s' <= 1 + x + y + 2, z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2

Function symbols to be analyzed: {encode_++}, {encode_merge}
Previous analysis results are:
  encode_v: runtime: O(1) [0], size: O(1) [2]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  merge: runtime: O(n^1) [2 + z], size: O(n^1) [z + z']
  encode_u: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2*z + 2*z^2], size: O(n^1) [2 + 2*z]
  encode_++: runtime: ?, size: O(n^1) [5 + 2*z + 2*z']

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 2 + s2 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> s4 :|: s2 >= 0, s2 <= 2 * x_1 + 2, s3 >= 0, s3 <= 2 * x_2 + 2, s4 >= 0, s4 <= s2 + s3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> 1 + s'' + s1 :|: s'' >= 0, s'' <= 2 * x_1 + 2, s1 >= 0, s1 <= 2 * x_2 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> 1 + s8 + s9 :|: s8 >= 0, s8 <= 2 * z + 2, s9 >= 0, s9 <= 2 * z' + 2, z >= 0, z' >= 0
   encode_merge(z, z') -{ 2 + s5 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> s7 :|: s5 >= 0, s5 <= 2 * z + 2, s6 >= 0, s6 <= 2 * z' + 2, s7 >= 0, s7 <= s5 + s6, z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 3 + y }-> 1 + x + s :|: s >= 0, s <= y + (1 + 1 + 2), z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 4 + x + y }-> 1 + 1 + s' :|: s' >= 0, s' <= 1 + x + y + 2, z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2

Function symbols to be analyzed: {encode_merge}
Previous analysis results are:
  encode_v: runtime: O(1) [0], size: O(1) [2]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  merge: runtime: O(n^1) [2 + z], size: O(n^1) [z + z']
  encode_u: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2*z + 2*z^2], size: O(n^1) [2 + 2*z]
  encode_++: runtime: O(n^2) [2*z + 2*z^2 + 2*z' + 2*z'^2], size: O(n^1) [5 + 2*z + 2*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 + s2 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> s4 :|: s2 >= 0, s2 <= 2 * x_1 + 2, s3 >= 0, s3 <= 2 * x_2 + 2, s4 >= 0, s4 <= s2 + s3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> 1 + s'' + s1 :|: s'' >= 0, s'' <= 2 * x_1 + 2, s1 >= 0, s1 <= 2 * x_2 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> 1 + s8 + s9 :|: s8 >= 0, s8 <= 2 * z + 2, s9 >= 0, s9 <= 2 * z' + 2, z >= 0, z' >= 0
   encode_merge(z, z') -{ 2 + s5 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> s7 :|: s5 >= 0, s5 <= 2 * z + 2, s6 >= 0, s6 <= 2 * z' + 2, s7 >= 0, s7 <= s5 + s6, z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 3 + y }-> 1 + x + s :|: s >= 0, s <= y + (1 + 1 + 2), z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 4 + x + y }-> 1 + 1 + s' :|: s' >= 0, s' <= 1 + x + y + 2, z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2

Function symbols to be analyzed: {encode_merge}
Previous analysis results are:
  encode_v: runtime: O(1) [0], size: O(1) [2]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  merge: runtime: O(n^1) [2 + z], size: O(n^1) [z + z']
  encode_u: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2*z + 2*z^2], size: O(n^1) [2 + 2*z]
  encode_++: runtime: O(n^2) [2*z + 2*z^2 + 2*z' + 2*z'^2], size: O(n^1) [5 + 2*z + 2*z']

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_merge
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 4 + 2*z + 2*z'

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

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

   encArg(z) -{ 2 + s2 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> s4 :|: s2 >= 0, s2 <= 2 * x_1 + 2, s3 >= 0, s3 <= 2 * x_2 + 2, s4 >= 0, s4 <= s2 + s3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> 1 + s'' + s1 :|: s'' >= 0, s'' <= 2 * x_1 + 2, s1 >= 0, s1 <= 2 * x_2 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> 1 + s8 + s9 :|: s8 >= 0, s8 <= 2 * z + 2, s9 >= 0, s9 <= 2 * z' + 2, z >= 0, z' >= 0
   encode_merge(z, z') -{ 2 + s5 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> s7 :|: s5 >= 0, s5 <= 2 * z + 2, s6 >= 0, s6 <= 2 * z' + 2, s7 >= 0, s7 <= s5 + s6, z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 3 + y }-> 1 + x + s :|: s >= 0, s <= y + (1 + 1 + 2), z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 4 + x + y }-> 1 + 1 + s' :|: s' >= 0, s' <= 1 + x + y + 2, z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2

Function symbols to be analyzed: {encode_merge}
Previous analysis results are:
  encode_v: runtime: O(1) [0], size: O(1) [2]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  merge: runtime: O(n^1) [2 + z], size: O(n^1) [z + z']
  encode_u: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2*z + 2*z^2], size: O(n^1) [2 + 2*z]
  encode_++: runtime: O(n^2) [2*z + 2*z^2 + 2*z' + 2*z'^2], size: O(n^1) [5 + 2*z + 2*z']
  encode_merge: runtime: ?, size: O(n^1) [4 + 2*z + 2*z']

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

(59) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 2 + s2 + 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> s4 :|: s2 >= 0, s2 <= 2 * x_1 + 2, s3 >= 0, s3 <= 2 * x_2 + 2, s4 >= 0, s4 <= s2 + s3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 2*x_1 + 2*x_1^2 + 2*x_2 + 2*x_2^2 }-> 1 + s'' + s1 :|: s'' >= 0, s'' <= 2 * x_1 + 2, s1 >= 0, s1 <= 2 * x_2 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_++(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_++(z, z') -{ 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> 1 + s8 + s9 :|: s8 >= 0, s8 <= 2 * z + 2, s9 >= 0, s9 <= 2 * z' + 2, z >= 0, z' >= 0
   encode_merge(z, z') -{ 2 + s5 + 2*z + 2*z^2 + 2*z' + 2*z'^2 }-> s7 :|: s5 >= 0, s5 <= 2 * z + 2, s6 >= 0, s6 <= 2 * z' + 2, s7 >= 0, s7 <= s5 + s6, z >= 0, z' >= 0
   encode_merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_u -{ 0 }-> 1 :|: 
   encode_u -{ 0 }-> 0 :|: 
   encode_v -{ 0 }-> 2 :|: 
   encode_v -{ 0 }-> 0 :|: 
   merge(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 3 + y }-> 1 + x + s :|: s >= 0, s <= y + (1 + 1 + 2), z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2
   merge(z, z') -{ 4 + x + y }-> 1 + 1 + s' :|: s' >= 0, s' <= 1 + x + y + 2, z = 1 + x + y, x >= 0, y >= 0, z' = 1 + 1 + 2

Function symbols to be analyzed: 
Previous analysis results are:
  encode_v: runtime: O(1) [0], size: O(1) [2]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  merge: runtime: O(n^1) [2 + z], size: O(n^1) [z + z']
  encode_u: runtime: O(1) [0], size: O(1) [1]
  encArg: runtime: O(n^2) [2*z + 2*z^2], size: O(n^1) [2 + 2*z]
  encode_++: runtime: O(n^2) [2*z + 2*z^2 + 2*z' + 2*z'^2], size: O(n^1) [5 + 2*z + 2*z']
  encode_merge: runtime: O(n^2) [4 + 4*z + 2*z^2 + 2*z' + 2*z'^2], size: O(n^1) [4 + 2*z + 2*z']

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

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

(62)
BOUNDS(1, n^2)

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

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

(64)
Obligation:
Analyzing the following TRS for decreasing loops:

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:

   merge(x, nil) -> x
   merge(nil, y) -> y
   merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
   merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))

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

   encArg(nil) -> nil
   encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
   encArg(u) -> u
   encArg(v) -> v
   encArg(cons_merge(x_1, x_2)) -> merge(encArg(x_1), encArg(x_2))
   encode_merge(x_1, x_2) -> merge(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))
   encode_u -> u
   encode_v -> v

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

(65) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

merge(++(x, y), ++(u, v)) ->^+ ++(x, merge(y, ++(u, v)))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1].

The pumping substitution is [y / ++(x, y)].

The result substitution is [ ].




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

(66)
Complex Obligation (BEST)

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

(67)
Obligation:
Proved the lower bound n^1 for the following 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:

   merge(x, nil) -> x
   merge(nil, y) -> y
   merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
   merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))

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

   encArg(nil) -> nil
   encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
   encArg(u) -> u
   encArg(v) -> v
   encArg(cons_merge(x_1, x_2)) -> merge(encArg(x_1), encArg(x_2))
   encode_merge(x_1, x_2) -> merge(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))
   encode_u -> u
   encode_v -> v

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

(68) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(69)
BOUNDS(n^1, INF)

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

(70)
Obligation:
Analyzing the following TRS for decreasing loops:

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:

   merge(x, nil) -> x
   merge(nil, y) -> y
   merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
   merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))

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

   encArg(nil) -> nil
   encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
   encArg(u) -> u
   encArg(v) -> v
   encArg(cons_merge(x_1, x_2)) -> merge(encArg(x_1), encArg(x_2))
   encode_merge(x_1, x_2) -> merge(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))
   encode_u -> u
   encode_v -> v

Rewrite Strategy: INNERMOST