/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 240 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), 83 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (14) CpxRNTS
    (15) InliningProof [UPPER BOUND(ID), 87 ms]
    (16) CpxRNTS
    (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRNTS
    (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 208 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 67 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 26 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 78 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 31 ms]
    (38) CpxRNTS
    (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 126 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 21 ms]
    (44) CpxRNTS
    (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 439 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 180 ms]
    (50) CpxRNTS
    (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (52) CpxRNTS
    (53) IntTrsBoundProof [UPPER BOUND(ID), 296 ms]
    (54) CpxRNTS
    (55) IntTrsBoundProof [UPPER BOUND(ID), 278 ms]
    (56) CpxRNTS
    (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (58) CpxRNTS
    (59) IntTrsBoundProof [UPPER BOUND(ID), 149 ms]
    (60) CpxRNTS
    (61) IntTrsBoundProof [UPPER BOUND(ID), 4 ms]
    (62) CpxRNTS
    (63) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (64) CpxRNTS
    (65) IntTrsBoundProof [UPPER BOUND(ID), 241 ms]
    (66) CpxRNTS
    (67) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (68) CpxRNTS
    (69) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (70) CpxRNTS
    (71) IntTrsBoundProof [UPPER BOUND(ID), 191 ms]
    (72) CpxRNTS
    (73) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
    (74) CpxRNTS
    (75) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (76) CpxRNTS
    (77) IntTrsBoundProof [UPPER BOUND(ID), 229 ms]
    (78) CpxRNTS
    (79) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (80) CpxRNTS
    (81) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (82) CpxRNTS
    (83) IntTrsBoundProof [UPPER BOUND(ID), 199 ms]
    (84) CpxRNTS
    (85) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
    (86) CpxRNTS
    (87) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (88) CpxRNTS
    (89) IntTrsBoundProof [UPPER BOUND(ID), 228 ms]
    (90) CpxRNTS
    (91) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (92) CpxRNTS
    (93) FinalProof [FINISHED, 0 ms]
    (94) BOUNDS(1, n^2)
(95) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(96) TRS for Loop Detection
    (97) DecreasingLoopProof [LOWER BOUND(ID), 4 ms]
    (98) BEST
        (99) proven lower bound
            (100) LowerBoundPropagationProof [FINISHED, 0 ms]
            (101) BOUNDS(n^1, INF)
        (102) 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:

   admit(x, nil) -> nil
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
   cond(true, y) -> y

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(w) -> w
   encArg(=(x_1, x_2)) -> =(encArg(x_1), encArg(x_2))
   encArg(sum(x_1, x_2, x_3)) -> sum(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(carry(x_1, x_2, x_3)) -> carry(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(true) -> true
   encArg(cons_admit(x_1, x_2)) -> admit(encArg(x_1), encArg(x_2))
   encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2))
   encode_admit(x_1, x_2) -> admit(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_w -> w
   encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2))
   encode_=(x_1, x_2) -> =(encArg(x_1), encArg(x_2))
   encode_sum(x_1, x_2, x_3) -> sum(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_carry(x_1, x_2, x_3) -> carry(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_true -> true


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

(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:

   admit(x, nil) -> nil
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
   cond(true, y) -> y

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

   encArg(nil) -> nil
   encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(w) -> w
   encArg(=(x_1, x_2)) -> =(encArg(x_1), encArg(x_2))
   encArg(sum(x_1, x_2, x_3)) -> sum(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(carry(x_1, x_2, x_3)) -> carry(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(true) -> true
   encArg(cons_admit(x_1, x_2)) -> admit(encArg(x_1), encArg(x_2))
   encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2))
   encode_admit(x_1, x_2) -> admit(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_w -> w
   encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2))
   encode_=(x_1, x_2) -> =(encArg(x_1), encArg(x_2))
   encode_sum(x_1, x_2, x_3) -> sum(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_carry(x_1, x_2, x_3) -> carry(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_true -> true

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:

   admit(x, nil) -> nil
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
   cond(true, y) -> y

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

   encArg(nil) -> nil
   encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(w) -> w
   encArg(=(x_1, x_2)) -> =(encArg(x_1), encArg(x_2))
   encArg(sum(x_1, x_2, x_3)) -> sum(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(carry(x_1, x_2, x_3)) -> carry(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(true) -> true
   encArg(cons_admit(x_1, x_2)) -> admit(encArg(x_1), encArg(x_2))
   encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2))
   encode_admit(x_1, x_2) -> admit(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_w -> w
   encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2))
   encode_=(x_1, x_2) -> =(encArg(x_1), encArg(x_2))
   encode_sum(x_1, x_2, x_3) -> sum(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_carry(x_1, x_2, x_3) -> carry(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_true -> true

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:

   admit(x, nil) -> nil [1]
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z))))) [1]
   cond(true, y) -> y [1]
   encArg(nil) -> nil [0]
   encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2)) [0]
   encArg(w) -> w [0]
   encArg(=(x_1, x_2)) -> =(encArg(x_1), encArg(x_2)) [0]
   encArg(sum(x_1, x_2, x_3)) -> sum(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(carry(x_1, x_2, x_3)) -> carry(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(true) -> true [0]
   encArg(cons_admit(x_1, x_2)) -> admit(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2)) [0]
   encode_admit(x_1, x_2) -> admit(encArg(x_1), encArg(x_2)) [0]
   encode_nil -> nil [0]
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2)) [0]
   encode_w -> w [0]
   encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2)) [0]
   encode_=(x_1, x_2) -> =(encArg(x_1), encArg(x_2)) [0]
   encode_sum(x_1, x_2, x_3) -> sum(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_carry(x_1, x_2, x_3) -> carry(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_true -> true [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:

   admit(x, nil) -> nil [1]
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z))))) [1]
   cond(true, y) -> y [1]
   encArg(nil) -> nil [0]
   encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2)) [0]
   encArg(w) -> w [0]
   encArg(=(x_1, x_2)) -> =(encArg(x_1), encArg(x_2)) [0]
   encArg(sum(x_1, x_2, x_3)) -> sum(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(carry(x_1, x_2, x_3)) -> carry(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(true) -> true [0]
   encArg(cons_admit(x_1, x_2)) -> admit(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2)) [0]
   encode_admit(x_1, x_2) -> admit(encArg(x_1), encArg(x_2)) [0]
   encode_nil -> nil [0]
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2)) [0]
   encode_w -> w [0]
   encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2)) [0]
   encode_=(x_1, x_2) -> =(encArg(x_1), encArg(x_2)) [0]
   encode_sum(x_1, x_2, x_3) -> sum(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_carry(x_1, x_2, x_3) -> carry(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_true -> true [0]

The TRS has the following type information:
admit :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond
nil :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond
. :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond
w :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond
cond :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond
= :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond
sum :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond
carry :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond
true :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond
encArg :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond
cons_admit :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond
cons_cond :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond
encode_admit :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond
encode_nil :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond
encode_. :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond
encode_w :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond
encode_cond :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond
encode_= :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond
encode_sum :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond
encode_carry :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond
encode_true :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond

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:

   admit_2
   cond_2
   encArg_1
   encode_admit_2
   encode_nil
   encode_._2
   encode_w
   encode_cond_2
   encode_=_2
   encode_sum_3
   encode_carry_3
   encode_true

Due to the following rules being added:

   encArg(v0) -> null_encArg [0]
   encode_admit(v0, v1) -> null_encode_admit [0]
   encode_nil -> null_encode_nil [0]
   encode_.(v0, v1) -> null_encode_. [0]
   encode_w -> null_encode_w [0]
   encode_cond(v0, v1) -> null_encode_cond [0]
   encode_=(v0, v1) -> null_encode_= [0]
   encode_sum(v0, v1, v2) -> null_encode_sum [0]
   encode_carry(v0, v1, v2) -> null_encode_carry [0]
   encode_true -> null_encode_true [0]
   admit(v0, v1) -> null_admit [0]
   cond(v0, v1) -> null_cond [0]

And the following fresh constants:    null_encArg, null_encode_admit, null_encode_nil, null_encode_., null_encode_w, null_encode_cond, null_encode_=, null_encode_sum, null_encode_carry, null_encode_true, null_admit, null_cond

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

(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:

   admit(x, nil) -> nil [1]
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z))))) [1]
   cond(true, y) -> y [1]
   encArg(nil) -> nil [0]
   encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2)) [0]
   encArg(w) -> w [0]
   encArg(=(x_1, x_2)) -> =(encArg(x_1), encArg(x_2)) [0]
   encArg(sum(x_1, x_2, x_3)) -> sum(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(carry(x_1, x_2, x_3)) -> carry(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(true) -> true [0]
   encArg(cons_admit(x_1, x_2)) -> admit(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2)) [0]
   encode_admit(x_1, x_2) -> admit(encArg(x_1), encArg(x_2)) [0]
   encode_nil -> nil [0]
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2)) [0]
   encode_w -> w [0]
   encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2)) [0]
   encode_=(x_1, x_2) -> =(encArg(x_1), encArg(x_2)) [0]
   encode_sum(x_1, x_2, x_3) -> sum(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_carry(x_1, x_2, x_3) -> carry(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_true -> true [0]
   encArg(v0) -> null_encArg [0]
   encode_admit(v0, v1) -> null_encode_admit [0]
   encode_nil -> null_encode_nil [0]
   encode_.(v0, v1) -> null_encode_. [0]
   encode_w -> null_encode_w [0]
   encode_cond(v0, v1) -> null_encode_cond [0]
   encode_=(v0, v1) -> null_encode_= [0]
   encode_sum(v0, v1, v2) -> null_encode_sum [0]
   encode_carry(v0, v1, v2) -> null_encode_carry [0]
   encode_true -> null_encode_true [0]
   admit(v0, v1) -> null_admit [0]
   cond(v0, v1) -> null_cond [0]

The TRS has the following type information:
admit :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
nil :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
. :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
w :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
cond :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
= :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
sum :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
carry :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
true :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
encArg :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
cons_admit :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
cons_cond :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
encode_admit :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
encode_nil :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
encode_. :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
encode_w :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
encode_cond :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
encode_= :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
encode_sum :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
encode_carry :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
encode_true :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_encArg :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_encode_admit :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_encode_nil :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_encode_. :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_encode_w :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_encode_cond :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_encode_= :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_encode_sum :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_encode_carry :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_encode_true :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_admit :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_cond :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond

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:

   admit(x, nil) -> nil [1]
   admit(x, .(u, .(v, .(w, nil)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, nil)))) [2]
   admit(x, .(u, .(v, .(w, .(u', .(v', .(w, z'))))))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, cond(=(sum(carry(x, u, v), u', v'), w), .(u', .(v', .(w, admit(carry(carry(x, u, v), u', v'), z'))))))))) [2]
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, null_admit)))) [1]
   cond(true, y) -> y [1]
   encArg(nil) -> nil [0]
   encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2)) [0]
   encArg(w) -> w [0]
   encArg(=(x_1, x_2)) -> =(encArg(x_1), encArg(x_2)) [0]
   encArg(sum(x_1, x_2, x_3)) -> sum(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(carry(x_1, x_2, x_3)) -> carry(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(true) -> true [0]
   encArg(cons_admit(x_1, x_2)) -> admit(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2)) [0]
   encode_admit(x_1, x_2) -> admit(encArg(x_1), encArg(x_2)) [0]
   encode_nil -> nil [0]
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2)) [0]
   encode_w -> w [0]
   encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2)) [0]
   encode_=(x_1, x_2) -> =(encArg(x_1), encArg(x_2)) [0]
   encode_sum(x_1, x_2, x_3) -> sum(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_carry(x_1, x_2, x_3) -> carry(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_true -> true [0]
   encArg(v0) -> null_encArg [0]
   encode_admit(v0, v1) -> null_encode_admit [0]
   encode_nil -> null_encode_nil [0]
   encode_.(v0, v1) -> null_encode_. [0]
   encode_w -> null_encode_w [0]
   encode_cond(v0, v1) -> null_encode_cond [0]
   encode_=(v0, v1) -> null_encode_= [0]
   encode_sum(v0, v1, v2) -> null_encode_sum [0]
   encode_carry(v0, v1, v2) -> null_encode_carry [0]
   encode_true -> null_encode_true [0]
   admit(v0, v1) -> null_admit [0]
   cond(v0, v1) -> null_cond [0]

The TRS has the following type information:
admit :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
nil :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
. :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
w :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
cond :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
= :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
sum :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
carry :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
true :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
encArg :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
cons_admit :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
cons_cond :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
encode_admit :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
encode_nil :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
encode_. :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
encode_w :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
encode_cond :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
encode_= :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
encode_sum :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
encode_carry :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond -> nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
encode_true :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_encArg :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_encode_admit :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_encode_nil :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_encode_. :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_encode_w :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_encode_cond :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_encode_= :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_encode_sum :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_encode_carry :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_encode_true :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_admit :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond
null_cond :: nil:w:.:sum:=:carry:true:cons_admit:cons_cond:null_encArg:null_encode_admit:null_encode_nil:null_encode_.:null_encode_w:null_encode_cond:null_encode_=:null_encode_sum:null_encode_carry:null_encode_true:null_admit:null_cond

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
   w => 2
   true => 1
   null_encArg => 0
   null_encode_admit => 0
   null_encode_nil => 0
   null_encode_. => 0
   null_encode_w => 0
   null_encode_cond => 0
   null_encode_= => 0
   null_encode_sum => 0
   null_encode_carry => 0
   null_encode_true => 0
   null_admit => 0
   null_cond => 0

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + x + u + v) + 2, 1 + u + (1 + v + (1 + 2 + cond(1 + (1 + (1 + x + u + v) + u' + v') + 2, 1 + u' + (1 + v' + (1 + 2 + admit(1 + (1 + x + u + v) + u' + v', z'))))))) :|: v >= 0, x >= 0, z' >= 0, u' >= 0, v' >= 0, z'' = x, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 2 }-> cond(1 + (1 + x + u + v) + 2, 1 + u + (1 + v + (1 + 2 + 0))) :|: v >= 0, x >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), z'' = x, u >= 0
   admit(z'', z1) -{ 1 }-> cond(1 + (1 + x + u + v) + 2, 1 + u + (1 + v + (1 + 2 + 0))) :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), x >= 0, z'' = x, u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, x >= 0, z'' = x
   admit(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> y :|: z1 = y, y >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   encArg(z'') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 2 :|: z'' = 2
   encArg(z'') -{ 0 }-> 1 :|: z'' = 1
   encArg(z'') -{ 0 }-> 0 :|: z'' = 0
   encArg(z'') -{ 0 }-> 0 :|: z'' = v0, v0 >= 0
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   encode_.(z'', z1) -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z'' = x_1, z1 = x_2, x_2 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   encode_=(z'', z1) -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z'' = x_1, z1 = x_2, x_2 >= 0
   encode_admit(z'', z1) -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z'' = x_1, z1 = x_2, x_2 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' = v0, z2 = v2, v0 >= 0, z1 = v1, v1 >= 0, v2 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: x_1 >= 0, z'' = x_1, z1 = x_2, z2 = x_3, x_3 >= 0, x_2 >= 0
   encode_cond(z'', z1) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z'' = x_1, z1 = x_2, x_2 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' = v0, z2 = v2, v0 >= 0, z1 = v1, v1 >= 0, v2 >= 0
   encode_sum(z'', z1, z2) -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: x_1 >= 0, z'' = x_1, z1 = x_2, z2 = x_3, x_3 >= 0, x_2 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 


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

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

   cond(z'', z1) -{ 1 }-> y :|: z1 = y, y >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + x + u + v) + 2, 1 + u + (1 + v + (1 + 2 + cond(1 + (1 + (1 + x + u + v) + u' + v') + 2, 1 + u' + (1 + v' + (1 + 2 + admit(1 + (1 + x + u + v) + u' + v', z'))))))) :|: v >= 0, x >= 0, z' >= 0, u' >= 0, v' >= 0, z'' = x, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, x >= 0, z'' = x
   admit(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, x >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), z'' = x, u >= 0, 1 + (1 + x + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), x >= 0, z'' = x, u >= 0, 1 + (1 + x + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> y :|: z1 = y, y >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   encArg(z'') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 2 :|: z'' = 2
   encArg(z'') -{ 0 }-> 1 :|: z'' = 1
   encArg(z'') -{ 0 }-> 0 :|: z'' = 0
   encArg(z'') -{ 0 }-> 0 :|: z'' = v0, v0 >= 0
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   encode_.(z'', z1) -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z'' = x_1, z1 = x_2, x_2 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   encode_=(z'', z1) -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z'' = x_1, z1 = x_2, x_2 >= 0
   encode_admit(z'', z1) -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z'' = x_1, z1 = x_2, x_2 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' = v0, z2 = v2, v0 >= 0, z1 = v1, v1 >= 0, v2 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: x_1 >= 0, z'' = x_1, z1 = x_2, z2 = x_3, x_3 >= 0, x_2 >= 0
   encode_cond(z'', z1) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z'' = x_1, z1 = x_2, x_2 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' = v0, z2 = v2, v0 >= 0, z1 = v1, v1 >= 0, v2 >= 0
   encode_sum(z'', z1, z2) -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: x_1 >= 0, z'' = x_1, z1 = x_2, z2 = x_3, x_3 >= 0, x_2 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 


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

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 2, 1 + u + (1 + v + (1 + 2 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 2, 1 + u' + (1 + v' + (1 + 2 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> admit(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> cond(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 


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

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

   { cond }
   { encode_nil }
   { encode_true }
   { encode_w }
   { admit }
   { encArg }
   { encode_admit }
   { encode_= }
   { encode_carry }
   { encode_. }
   { encode_sum }
   { encode_cond }

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 2, 1 + u + (1 + v + (1 + 2 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 2, 1 + u' + (1 + v' + (1 + 2 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> admit(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> cond(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {cond}, {encode_nil}, {encode_true}, {encode_w}, {admit}, {encArg}, {encode_admit}, {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}

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

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 2, 1 + u + (1 + v + (1 + 2 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 2, 1 + u' + (1 + v' + (1 + 2 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> admit(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> cond(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {cond}, {encode_nil}, {encode_true}, {encode_w}, {admit}, {encArg}, {encode_admit}, {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: cond
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z1

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 2, 1 + u + (1 + v + (1 + 2 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 2, 1 + u' + (1 + v' + (1 + 2 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> admit(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> cond(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {cond}, {encode_nil}, {encode_true}, {encode_w}, {admit}, {encArg}, {encode_admit}, {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: ?, size: O(n^1) [z1]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 2, 1 + u + (1 + v + (1 + 2 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 2, 1 + u' + (1 + v' + (1 + 2 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> admit(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> cond(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_nil}, {encode_true}, {encode_w}, {admit}, {encArg}, {encode_admit}, {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]

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

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 2, 1 + u + (1 + v + (1 + 2 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 2, 1 + u' + (1 + v' + (1 + 2 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> admit(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> cond(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_nil}, {encode_true}, {encode_w}, {admit}, {encArg}, {encode_admit}, {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]

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

(29) 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

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 2, 1 + u + (1 + v + (1 + 2 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 2, 1 + u' + (1 + v' + (1 + 2 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> admit(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> cond(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_nil}, {encode_true}, {encode_w}, {admit}, {encArg}, {encode_admit}, {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: ?, size: O(1) [0]

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

(31) 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

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 2, 1 + u + (1 + v + (1 + 2 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 2, 1 + u' + (1 + v' + (1 + 2 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> admit(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> cond(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_true}, {encode_w}, {admit}, {encArg}, {encode_admit}, {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]

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

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 2, 1 + u + (1 + v + (1 + 2 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 2, 1 + u' + (1 + v' + (1 + 2 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> admit(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> cond(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_true}, {encode_w}, {admit}, {encArg}, {encode_admit}, {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 2, 1 + u + (1 + v + (1 + 2 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 2, 1 + u' + (1 + v' + (1 + 2 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> admit(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> cond(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_true}, {encode_w}, {admit}, {encArg}, {encode_admit}, {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: ?, size: O(1) [1]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 2, 1 + u + (1 + v + (1 + 2 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 2, 1 + u' + (1 + v' + (1 + 2 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> admit(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> cond(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_w}, {admit}, {encArg}, {encode_admit}, {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]

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

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 2, 1 + u + (1 + v + (1 + 2 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 2, 1 + u' + (1 + v' + (1 + 2 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> admit(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> cond(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_w}, {admit}, {encArg}, {encode_admit}, {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 2, 1 + u + (1 + v + (1 + 2 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 2, 1 + u' + (1 + v' + (1 + 2 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> admit(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> cond(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_w}, {admit}, {encArg}, {encode_admit}, {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: ?, size: O(1) [2]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 2, 1 + u + (1 + v + (1 + 2 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 2, 1 + u' + (1 + v' + (1 + 2 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> admit(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> cond(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {admit}, {encArg}, {encode_admit}, {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]

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

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 2, 1 + u + (1 + v + (1 + 2 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 2, 1 + u' + (1 + v' + (1 + 2 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> admit(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> cond(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {admit}, {encArg}, {encode_admit}, {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: admit
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z1

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 2, 1 + u + (1 + v + (1 + 2 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 2, 1 + u' + (1 + v' + (1 + 2 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> admit(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> cond(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {admit}, {encArg}, {encode_admit}, {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: ?, size: O(n^1) [z1]

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: admit
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 4 + 4*z1

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 2, 1 + u + (1 + v + (1 + 2 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 2, 1 + u' + (1 + v' + (1 + 2 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> admit(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> cond(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encArg}, {encode_admit}, {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]

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

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

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

   admit(z'', z1) -{ 8 + 4*z' }-> s'' :|: s >= 0, s <= z', s' >= 0, s' <= 1 + u' + (1 + v' + (1 + 2 + s)), s'' >= 0, s'' <= 1 + u + (1 + v + (1 + 2 + s')), v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> admit(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> cond(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encArg}, {encode_admit}, {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]

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

(53) 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''

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

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

   admit(z'', z1) -{ 8 + 4*z' }-> s'' :|: s >= 0, s <= z', s' >= 0, s' <= 1 + u' + (1 + v' + (1 + 2 + s)), s'' >= 0, s'' <= 1 + u + (1 + v + (1 + 2 + s')), v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> admit(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> cond(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encArg}, {encode_admit}, {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]
  encArg: runtime: ?, size: O(n^1) [2 + 2*z'']

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

(55) 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: 4*z'' + 8*z''^2

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

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

   admit(z'', z1) -{ 8 + 4*z' }-> s'' :|: s >= 0, s <= z', s' >= 0, s' <= 1 + u' + (1 + v' + (1 + 2 + s)), s'' >= 0, s'' <= 1 + u + (1 + v + (1 + 2 + s')), v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> admit(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) + encArg(x_3) :|: z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 1 + encArg(z'') + encArg(z1) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> admit(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> cond(encArg(z''), encArg(z1)) :|: z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 0 }-> 1 + encArg(z'') + encArg(z1) + encArg(z2) :|: z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_admit}, {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [4*z'' + 8*z''^2], size: O(n^1) [2 + 2*z'']

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

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

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

   admit(z'', z1) -{ 8 + 4*z' }-> s'' :|: s >= 0, s <= z', s' >= 0, s' <= 1 + u' + (1 + v' + (1 + 2 + s)), s'' >= 0, s'' <= 1 + u + (1 + v + (1 + 2 + s')), v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 1 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s24 :|: s22 >= 0, s22 <= 2 * x_1 + 2, s23 >= 0, s23 <= 2 * x_2 + 2, s24 >= 0, s24 <= s23, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4 + 4*s7 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s8 :|: s6 >= 0, s6 <= 2 * x_1 + 2, s7 >= 0, s7 <= 2 * x_2 + 2, s8 >= 0, s8 <= s7, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= 2 * x_1 + 2, s2 >= 0, s2 <= 2 * x_2 + 2, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 + 4*x_3 + 8*x_3^2 }-> 1 + s3 + s4 + s5 :|: s3 >= 0, s3 <= 2 * x_1 + 2, s4 >= 0, s4 <= 2 * x_2 + 2, s5 >= 0, s5 <= 2 * x_3 + 2, z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s12 + s13 :|: s12 >= 0, s12 <= 2 * z'' + 2, s13 >= 0, s13 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s14 + s15 :|: s14 >= 0, s14 <= 2 * z'' + 2, s15 >= 0, s15 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 4 + 4*s10 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s11 :|: s9 >= 0, s9 <= 2 * z'' + 2, s10 >= 0, s10 <= 2 * z1 + 2, s11 >= 0, s11 <= s10, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s19 + s20 + s21 :|: s19 >= 0, s19 <= 2 * z'' + 2, s20 >= 0, s20 <= 2 * z1 + 2, s21 >= 0, s21 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 1 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s27 :|: s25 >= 0, s25 <= 2 * z'' + 2, s26 >= 0, s26 <= 2 * z1 + 2, s27 >= 0, s27 <= s26, z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s16 + s17 + s18 :|: s16 >= 0, s16 <= 2 * z'' + 2, s17 >= 0, s17 <= 2 * z1 + 2, s18 >= 0, s18 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_admit}, {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [4*z'' + 8*z''^2], size: O(n^1) [2 + 2*z'']

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

(59) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_admit
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + 2*z1

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

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

   admit(z'', z1) -{ 8 + 4*z' }-> s'' :|: s >= 0, s <= z', s' >= 0, s' <= 1 + u' + (1 + v' + (1 + 2 + s)), s'' >= 0, s'' <= 1 + u + (1 + v + (1 + 2 + s')), v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 1 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s24 :|: s22 >= 0, s22 <= 2 * x_1 + 2, s23 >= 0, s23 <= 2 * x_2 + 2, s24 >= 0, s24 <= s23, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4 + 4*s7 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s8 :|: s6 >= 0, s6 <= 2 * x_1 + 2, s7 >= 0, s7 <= 2 * x_2 + 2, s8 >= 0, s8 <= s7, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= 2 * x_1 + 2, s2 >= 0, s2 <= 2 * x_2 + 2, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 + 4*x_3 + 8*x_3^2 }-> 1 + s3 + s4 + s5 :|: s3 >= 0, s3 <= 2 * x_1 + 2, s4 >= 0, s4 <= 2 * x_2 + 2, s5 >= 0, s5 <= 2 * x_3 + 2, z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s12 + s13 :|: s12 >= 0, s12 <= 2 * z'' + 2, s13 >= 0, s13 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s14 + s15 :|: s14 >= 0, s14 <= 2 * z'' + 2, s15 >= 0, s15 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 4 + 4*s10 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s11 :|: s9 >= 0, s9 <= 2 * z'' + 2, s10 >= 0, s10 <= 2 * z1 + 2, s11 >= 0, s11 <= s10, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s19 + s20 + s21 :|: s19 >= 0, s19 <= 2 * z'' + 2, s20 >= 0, s20 <= 2 * z1 + 2, s21 >= 0, s21 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 1 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s27 :|: s25 >= 0, s25 <= 2 * z'' + 2, s26 >= 0, s26 <= 2 * z1 + 2, s27 >= 0, s27 <= s26, z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s16 + s17 + s18 :|: s16 >= 0, s16 <= 2 * z'' + 2, s17 >= 0, s17 <= 2 * z1 + 2, s18 >= 0, s18 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_admit}, {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [4*z'' + 8*z''^2], size: O(n^1) [2 + 2*z'']
  encode_admit: runtime: ?, size: O(n^1) [2 + 2*z1]

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

(61) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_admit
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 12 + 4*z'' + 8*z''^2 + 12*z1 + 8*z1^2

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

(62)
Obligation:
Complexity RNTS consisting of the following rules:

   admit(z'', z1) -{ 8 + 4*z' }-> s'' :|: s >= 0, s <= z', s' >= 0, s' <= 1 + u' + (1 + v' + (1 + 2 + s)), s'' >= 0, s'' <= 1 + u + (1 + v + (1 + 2 + s')), v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 1 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s24 :|: s22 >= 0, s22 <= 2 * x_1 + 2, s23 >= 0, s23 <= 2 * x_2 + 2, s24 >= 0, s24 <= s23, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4 + 4*s7 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s8 :|: s6 >= 0, s6 <= 2 * x_1 + 2, s7 >= 0, s7 <= 2 * x_2 + 2, s8 >= 0, s8 <= s7, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= 2 * x_1 + 2, s2 >= 0, s2 <= 2 * x_2 + 2, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 + 4*x_3 + 8*x_3^2 }-> 1 + s3 + s4 + s5 :|: s3 >= 0, s3 <= 2 * x_1 + 2, s4 >= 0, s4 <= 2 * x_2 + 2, s5 >= 0, s5 <= 2 * x_3 + 2, z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s12 + s13 :|: s12 >= 0, s12 <= 2 * z'' + 2, s13 >= 0, s13 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s14 + s15 :|: s14 >= 0, s14 <= 2 * z'' + 2, s15 >= 0, s15 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 4 + 4*s10 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s11 :|: s9 >= 0, s9 <= 2 * z'' + 2, s10 >= 0, s10 <= 2 * z1 + 2, s11 >= 0, s11 <= s10, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s19 + s20 + s21 :|: s19 >= 0, s19 <= 2 * z'' + 2, s20 >= 0, s20 <= 2 * z1 + 2, s21 >= 0, s21 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 1 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s27 :|: s25 >= 0, s25 <= 2 * z'' + 2, s26 >= 0, s26 <= 2 * z1 + 2, s27 >= 0, s27 <= s26, z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s16 + s17 + s18 :|: s16 >= 0, s16 <= 2 * z'' + 2, s17 >= 0, s17 <= 2 * z1 + 2, s18 >= 0, s18 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [4*z'' + 8*z''^2], size: O(n^1) [2 + 2*z'']
  encode_admit: runtime: O(n^2) [12 + 4*z'' + 8*z''^2 + 12*z1 + 8*z1^2], size: O(n^1) [2 + 2*z1]

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

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

(64)
Obligation:
Complexity RNTS consisting of the following rules:

   admit(z'', z1) -{ 8 + 4*z' }-> s'' :|: s >= 0, s <= z', s' >= 0, s' <= 1 + u' + (1 + v' + (1 + 2 + s)), s'' >= 0, s'' <= 1 + u + (1 + v + (1 + 2 + s')), v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 1 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s24 :|: s22 >= 0, s22 <= 2 * x_1 + 2, s23 >= 0, s23 <= 2 * x_2 + 2, s24 >= 0, s24 <= s23, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4 + 4*s7 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s8 :|: s6 >= 0, s6 <= 2 * x_1 + 2, s7 >= 0, s7 <= 2 * x_2 + 2, s8 >= 0, s8 <= s7, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= 2 * x_1 + 2, s2 >= 0, s2 <= 2 * x_2 + 2, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 + 4*x_3 + 8*x_3^2 }-> 1 + s3 + s4 + s5 :|: s3 >= 0, s3 <= 2 * x_1 + 2, s4 >= 0, s4 <= 2 * x_2 + 2, s5 >= 0, s5 <= 2 * x_3 + 2, z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s12 + s13 :|: s12 >= 0, s12 <= 2 * z'' + 2, s13 >= 0, s13 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s14 + s15 :|: s14 >= 0, s14 <= 2 * z'' + 2, s15 >= 0, s15 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 4 + 4*s10 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s11 :|: s9 >= 0, s9 <= 2 * z'' + 2, s10 >= 0, s10 <= 2 * z1 + 2, s11 >= 0, s11 <= s10, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s19 + s20 + s21 :|: s19 >= 0, s19 <= 2 * z'' + 2, s20 >= 0, s20 <= 2 * z1 + 2, s21 >= 0, s21 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 1 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s27 :|: s25 >= 0, s25 <= 2 * z'' + 2, s26 >= 0, s26 <= 2 * z1 + 2, s27 >= 0, s27 <= s26, z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s16 + s17 + s18 :|: s16 >= 0, s16 <= 2 * z'' + 2, s17 >= 0, s17 <= 2 * z1 + 2, s18 >= 0, s18 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [4*z'' + 8*z''^2], size: O(n^1) [2 + 2*z'']
  encode_admit: runtime: O(n^2) [12 + 4*z'' + 8*z''^2 + 12*z1 + 8*z1^2], size: O(n^1) [2 + 2*z1]

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

(65) 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*z1

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

(66)
Obligation:
Complexity RNTS consisting of the following rules:

   admit(z'', z1) -{ 8 + 4*z' }-> s'' :|: s >= 0, s <= z', s' >= 0, s' <= 1 + u' + (1 + v' + (1 + 2 + s)), s'' >= 0, s'' <= 1 + u + (1 + v + (1 + 2 + s')), v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 1 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s24 :|: s22 >= 0, s22 <= 2 * x_1 + 2, s23 >= 0, s23 <= 2 * x_2 + 2, s24 >= 0, s24 <= s23, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4 + 4*s7 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s8 :|: s6 >= 0, s6 <= 2 * x_1 + 2, s7 >= 0, s7 <= 2 * x_2 + 2, s8 >= 0, s8 <= s7, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= 2 * x_1 + 2, s2 >= 0, s2 <= 2 * x_2 + 2, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 + 4*x_3 + 8*x_3^2 }-> 1 + s3 + s4 + s5 :|: s3 >= 0, s3 <= 2 * x_1 + 2, s4 >= 0, s4 <= 2 * x_2 + 2, s5 >= 0, s5 <= 2 * x_3 + 2, z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s12 + s13 :|: s12 >= 0, s12 <= 2 * z'' + 2, s13 >= 0, s13 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s14 + s15 :|: s14 >= 0, s14 <= 2 * z'' + 2, s15 >= 0, s15 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 4 + 4*s10 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s11 :|: s9 >= 0, s9 <= 2 * z'' + 2, s10 >= 0, s10 <= 2 * z1 + 2, s11 >= 0, s11 <= s10, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s19 + s20 + s21 :|: s19 >= 0, s19 <= 2 * z'' + 2, s20 >= 0, s20 <= 2 * z1 + 2, s21 >= 0, s21 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 1 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s27 :|: s25 >= 0, s25 <= 2 * z'' + 2, s26 >= 0, s26 <= 2 * z1 + 2, s27 >= 0, s27 <= s26, z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s16 + s17 + s18 :|: s16 >= 0, s16 <= 2 * z'' + 2, s17 >= 0, s17 <= 2 * z1 + 2, s18 >= 0, s18 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_=}, {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [4*z'' + 8*z''^2], size: O(n^1) [2 + 2*z'']
  encode_admit: runtime: O(n^2) [12 + 4*z'' + 8*z''^2 + 12*z1 + 8*z1^2], size: O(n^1) [2 + 2*z1]
  encode_=: runtime: ?, size: O(n^1) [5 + 2*z'' + 2*z1]

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

(67) 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: 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2

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

(68)
Obligation:
Complexity RNTS consisting of the following rules:

   admit(z'', z1) -{ 8 + 4*z' }-> s'' :|: s >= 0, s <= z', s' >= 0, s' <= 1 + u' + (1 + v' + (1 + 2 + s)), s'' >= 0, s'' <= 1 + u + (1 + v + (1 + 2 + s')), v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 1 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s24 :|: s22 >= 0, s22 <= 2 * x_1 + 2, s23 >= 0, s23 <= 2 * x_2 + 2, s24 >= 0, s24 <= s23, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4 + 4*s7 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s8 :|: s6 >= 0, s6 <= 2 * x_1 + 2, s7 >= 0, s7 <= 2 * x_2 + 2, s8 >= 0, s8 <= s7, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= 2 * x_1 + 2, s2 >= 0, s2 <= 2 * x_2 + 2, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 + 4*x_3 + 8*x_3^2 }-> 1 + s3 + s4 + s5 :|: s3 >= 0, s3 <= 2 * x_1 + 2, s4 >= 0, s4 <= 2 * x_2 + 2, s5 >= 0, s5 <= 2 * x_3 + 2, z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s12 + s13 :|: s12 >= 0, s12 <= 2 * z'' + 2, s13 >= 0, s13 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s14 + s15 :|: s14 >= 0, s14 <= 2 * z'' + 2, s15 >= 0, s15 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 4 + 4*s10 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s11 :|: s9 >= 0, s9 <= 2 * z'' + 2, s10 >= 0, s10 <= 2 * z1 + 2, s11 >= 0, s11 <= s10, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s19 + s20 + s21 :|: s19 >= 0, s19 <= 2 * z'' + 2, s20 >= 0, s20 <= 2 * z1 + 2, s21 >= 0, s21 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 1 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s27 :|: s25 >= 0, s25 <= 2 * z'' + 2, s26 >= 0, s26 <= 2 * z1 + 2, s27 >= 0, s27 <= s26, z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s16 + s17 + s18 :|: s16 >= 0, s16 <= 2 * z'' + 2, s17 >= 0, s17 <= 2 * z1 + 2, s18 >= 0, s18 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [4*z'' + 8*z''^2], size: O(n^1) [2 + 2*z'']
  encode_admit: runtime: O(n^2) [12 + 4*z'' + 8*z''^2 + 12*z1 + 8*z1^2], size: O(n^1) [2 + 2*z1]
  encode_=: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2], size: O(n^1) [5 + 2*z'' + 2*z1]

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

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

(70)
Obligation:
Complexity RNTS consisting of the following rules:

   admit(z'', z1) -{ 8 + 4*z' }-> s'' :|: s >= 0, s <= z', s' >= 0, s' <= 1 + u' + (1 + v' + (1 + 2 + s)), s'' >= 0, s'' <= 1 + u + (1 + v + (1 + 2 + s')), v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 1 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s24 :|: s22 >= 0, s22 <= 2 * x_1 + 2, s23 >= 0, s23 <= 2 * x_2 + 2, s24 >= 0, s24 <= s23, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4 + 4*s7 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s8 :|: s6 >= 0, s6 <= 2 * x_1 + 2, s7 >= 0, s7 <= 2 * x_2 + 2, s8 >= 0, s8 <= s7, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= 2 * x_1 + 2, s2 >= 0, s2 <= 2 * x_2 + 2, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 + 4*x_3 + 8*x_3^2 }-> 1 + s3 + s4 + s5 :|: s3 >= 0, s3 <= 2 * x_1 + 2, s4 >= 0, s4 <= 2 * x_2 + 2, s5 >= 0, s5 <= 2 * x_3 + 2, z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s12 + s13 :|: s12 >= 0, s12 <= 2 * z'' + 2, s13 >= 0, s13 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s14 + s15 :|: s14 >= 0, s14 <= 2 * z'' + 2, s15 >= 0, s15 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 4 + 4*s10 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s11 :|: s9 >= 0, s9 <= 2 * z'' + 2, s10 >= 0, s10 <= 2 * z1 + 2, s11 >= 0, s11 <= s10, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s19 + s20 + s21 :|: s19 >= 0, s19 <= 2 * z'' + 2, s20 >= 0, s20 <= 2 * z1 + 2, s21 >= 0, s21 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 1 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s27 :|: s25 >= 0, s25 <= 2 * z'' + 2, s26 >= 0, s26 <= 2 * z1 + 2, s27 >= 0, s27 <= s26, z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s16 + s17 + s18 :|: s16 >= 0, s16 <= 2 * z'' + 2, s17 >= 0, s17 <= 2 * z1 + 2, s18 >= 0, s18 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [4*z'' + 8*z''^2], size: O(n^1) [2 + 2*z'']
  encode_admit: runtime: O(n^2) [12 + 4*z'' + 8*z''^2 + 12*z1 + 8*z1^2], size: O(n^1) [2 + 2*z1]
  encode_=: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2], size: O(n^1) [5 + 2*z'' + 2*z1]

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

(71) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_carry
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 7 + 2*z'' + 2*z1 + 2*z2

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

(72)
Obligation:
Complexity RNTS consisting of the following rules:

   admit(z'', z1) -{ 8 + 4*z' }-> s'' :|: s >= 0, s <= z', s' >= 0, s' <= 1 + u' + (1 + v' + (1 + 2 + s)), s'' >= 0, s'' <= 1 + u + (1 + v + (1 + 2 + s')), v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 1 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s24 :|: s22 >= 0, s22 <= 2 * x_1 + 2, s23 >= 0, s23 <= 2 * x_2 + 2, s24 >= 0, s24 <= s23, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4 + 4*s7 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s8 :|: s6 >= 0, s6 <= 2 * x_1 + 2, s7 >= 0, s7 <= 2 * x_2 + 2, s8 >= 0, s8 <= s7, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= 2 * x_1 + 2, s2 >= 0, s2 <= 2 * x_2 + 2, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 + 4*x_3 + 8*x_3^2 }-> 1 + s3 + s4 + s5 :|: s3 >= 0, s3 <= 2 * x_1 + 2, s4 >= 0, s4 <= 2 * x_2 + 2, s5 >= 0, s5 <= 2 * x_3 + 2, z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s12 + s13 :|: s12 >= 0, s12 <= 2 * z'' + 2, s13 >= 0, s13 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s14 + s15 :|: s14 >= 0, s14 <= 2 * z'' + 2, s15 >= 0, s15 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 4 + 4*s10 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s11 :|: s9 >= 0, s9 <= 2 * z'' + 2, s10 >= 0, s10 <= 2 * z1 + 2, s11 >= 0, s11 <= s10, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s19 + s20 + s21 :|: s19 >= 0, s19 <= 2 * z'' + 2, s20 >= 0, s20 <= 2 * z1 + 2, s21 >= 0, s21 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 1 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s27 :|: s25 >= 0, s25 <= 2 * z'' + 2, s26 >= 0, s26 <= 2 * z1 + 2, s27 >= 0, s27 <= s26, z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s16 + s17 + s18 :|: s16 >= 0, s16 <= 2 * z'' + 2, s17 >= 0, s17 <= 2 * z1 + 2, s18 >= 0, s18 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_carry}, {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [4*z'' + 8*z''^2], size: O(n^1) [2 + 2*z'']
  encode_admit: runtime: O(n^2) [12 + 4*z'' + 8*z''^2 + 12*z1 + 8*z1^2], size: O(n^1) [2 + 2*z1]
  encode_=: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2], size: O(n^1) [5 + 2*z'' + 2*z1]
  encode_carry: runtime: ?, size: O(n^1) [7 + 2*z'' + 2*z1 + 2*z2]

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

(73) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_carry
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2

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

(74)
Obligation:
Complexity RNTS consisting of the following rules:

   admit(z'', z1) -{ 8 + 4*z' }-> s'' :|: s >= 0, s <= z', s' >= 0, s' <= 1 + u' + (1 + v' + (1 + 2 + s)), s'' >= 0, s'' <= 1 + u + (1 + v + (1 + 2 + s')), v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 1 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s24 :|: s22 >= 0, s22 <= 2 * x_1 + 2, s23 >= 0, s23 <= 2 * x_2 + 2, s24 >= 0, s24 <= s23, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4 + 4*s7 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s8 :|: s6 >= 0, s6 <= 2 * x_1 + 2, s7 >= 0, s7 <= 2 * x_2 + 2, s8 >= 0, s8 <= s7, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= 2 * x_1 + 2, s2 >= 0, s2 <= 2 * x_2 + 2, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 + 4*x_3 + 8*x_3^2 }-> 1 + s3 + s4 + s5 :|: s3 >= 0, s3 <= 2 * x_1 + 2, s4 >= 0, s4 <= 2 * x_2 + 2, s5 >= 0, s5 <= 2 * x_3 + 2, z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s12 + s13 :|: s12 >= 0, s12 <= 2 * z'' + 2, s13 >= 0, s13 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s14 + s15 :|: s14 >= 0, s14 <= 2 * z'' + 2, s15 >= 0, s15 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 4 + 4*s10 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s11 :|: s9 >= 0, s9 <= 2 * z'' + 2, s10 >= 0, s10 <= 2 * z1 + 2, s11 >= 0, s11 <= s10, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s19 + s20 + s21 :|: s19 >= 0, s19 <= 2 * z'' + 2, s20 >= 0, s20 <= 2 * z1 + 2, s21 >= 0, s21 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 1 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s27 :|: s25 >= 0, s25 <= 2 * z'' + 2, s26 >= 0, s26 <= 2 * z1 + 2, s27 >= 0, s27 <= s26, z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s16 + s17 + s18 :|: s16 >= 0, s16 <= 2 * z'' + 2, s17 >= 0, s17 <= 2 * z1 + 2, s18 >= 0, s18 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [4*z'' + 8*z''^2], size: O(n^1) [2 + 2*z'']
  encode_admit: runtime: O(n^2) [12 + 4*z'' + 8*z''^2 + 12*z1 + 8*z1^2], size: O(n^1) [2 + 2*z1]
  encode_=: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2], size: O(n^1) [5 + 2*z'' + 2*z1]
  encode_carry: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2], size: O(n^1) [7 + 2*z'' + 2*z1 + 2*z2]

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

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

(76)
Obligation:
Complexity RNTS consisting of the following rules:

   admit(z'', z1) -{ 8 + 4*z' }-> s'' :|: s >= 0, s <= z', s' >= 0, s' <= 1 + u' + (1 + v' + (1 + 2 + s)), s'' >= 0, s'' <= 1 + u + (1 + v + (1 + 2 + s')), v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 1 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s24 :|: s22 >= 0, s22 <= 2 * x_1 + 2, s23 >= 0, s23 <= 2 * x_2 + 2, s24 >= 0, s24 <= s23, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4 + 4*s7 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s8 :|: s6 >= 0, s6 <= 2 * x_1 + 2, s7 >= 0, s7 <= 2 * x_2 + 2, s8 >= 0, s8 <= s7, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= 2 * x_1 + 2, s2 >= 0, s2 <= 2 * x_2 + 2, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 + 4*x_3 + 8*x_3^2 }-> 1 + s3 + s4 + s5 :|: s3 >= 0, s3 <= 2 * x_1 + 2, s4 >= 0, s4 <= 2 * x_2 + 2, s5 >= 0, s5 <= 2 * x_3 + 2, z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s12 + s13 :|: s12 >= 0, s12 <= 2 * z'' + 2, s13 >= 0, s13 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s14 + s15 :|: s14 >= 0, s14 <= 2 * z'' + 2, s15 >= 0, s15 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 4 + 4*s10 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s11 :|: s9 >= 0, s9 <= 2 * z'' + 2, s10 >= 0, s10 <= 2 * z1 + 2, s11 >= 0, s11 <= s10, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s19 + s20 + s21 :|: s19 >= 0, s19 <= 2 * z'' + 2, s20 >= 0, s20 <= 2 * z1 + 2, s21 >= 0, s21 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 1 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s27 :|: s25 >= 0, s25 <= 2 * z'' + 2, s26 >= 0, s26 <= 2 * z1 + 2, s27 >= 0, s27 <= s26, z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s16 + s17 + s18 :|: s16 >= 0, s16 <= 2 * z'' + 2, s17 >= 0, s17 <= 2 * z1 + 2, s18 >= 0, s18 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [4*z'' + 8*z''^2], size: O(n^1) [2 + 2*z'']
  encode_admit: runtime: O(n^2) [12 + 4*z'' + 8*z''^2 + 12*z1 + 8*z1^2], size: O(n^1) [2 + 2*z1]
  encode_=: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2], size: O(n^1) [5 + 2*z'' + 2*z1]
  encode_carry: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2], size: O(n^1) [7 + 2*z'' + 2*z1 + 2*z2]

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

(77) 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*z1

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

(78)
Obligation:
Complexity RNTS consisting of the following rules:

   admit(z'', z1) -{ 8 + 4*z' }-> s'' :|: s >= 0, s <= z', s' >= 0, s' <= 1 + u' + (1 + v' + (1 + 2 + s)), s'' >= 0, s'' <= 1 + u + (1 + v + (1 + 2 + s')), v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 1 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s24 :|: s22 >= 0, s22 <= 2 * x_1 + 2, s23 >= 0, s23 <= 2 * x_2 + 2, s24 >= 0, s24 <= s23, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4 + 4*s7 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s8 :|: s6 >= 0, s6 <= 2 * x_1 + 2, s7 >= 0, s7 <= 2 * x_2 + 2, s8 >= 0, s8 <= s7, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= 2 * x_1 + 2, s2 >= 0, s2 <= 2 * x_2 + 2, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 + 4*x_3 + 8*x_3^2 }-> 1 + s3 + s4 + s5 :|: s3 >= 0, s3 <= 2 * x_1 + 2, s4 >= 0, s4 <= 2 * x_2 + 2, s5 >= 0, s5 <= 2 * x_3 + 2, z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s12 + s13 :|: s12 >= 0, s12 <= 2 * z'' + 2, s13 >= 0, s13 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s14 + s15 :|: s14 >= 0, s14 <= 2 * z'' + 2, s15 >= 0, s15 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 4 + 4*s10 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s11 :|: s9 >= 0, s9 <= 2 * z'' + 2, s10 >= 0, s10 <= 2 * z1 + 2, s11 >= 0, s11 <= s10, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s19 + s20 + s21 :|: s19 >= 0, s19 <= 2 * z'' + 2, s20 >= 0, s20 <= 2 * z1 + 2, s21 >= 0, s21 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 1 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s27 :|: s25 >= 0, s25 <= 2 * z'' + 2, s26 >= 0, s26 <= 2 * z1 + 2, s27 >= 0, s27 <= s26, z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s16 + s17 + s18 :|: s16 >= 0, s16 <= 2 * z'' + 2, s17 >= 0, s17 <= 2 * z1 + 2, s18 >= 0, s18 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_.}, {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [4*z'' + 8*z''^2], size: O(n^1) [2 + 2*z'']
  encode_admit: runtime: O(n^2) [12 + 4*z'' + 8*z''^2 + 12*z1 + 8*z1^2], size: O(n^1) [2 + 2*z1]
  encode_=: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2], size: O(n^1) [5 + 2*z'' + 2*z1]
  encode_carry: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2], size: O(n^1) [7 + 2*z'' + 2*z1 + 2*z2]
  encode_.: runtime: ?, size: O(n^1) [5 + 2*z'' + 2*z1]

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

(79) 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: 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2

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

(80)
Obligation:
Complexity RNTS consisting of the following rules:

   admit(z'', z1) -{ 8 + 4*z' }-> s'' :|: s >= 0, s <= z', s' >= 0, s' <= 1 + u' + (1 + v' + (1 + 2 + s)), s'' >= 0, s'' <= 1 + u + (1 + v + (1 + 2 + s')), v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 1 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s24 :|: s22 >= 0, s22 <= 2 * x_1 + 2, s23 >= 0, s23 <= 2 * x_2 + 2, s24 >= 0, s24 <= s23, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4 + 4*s7 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s8 :|: s6 >= 0, s6 <= 2 * x_1 + 2, s7 >= 0, s7 <= 2 * x_2 + 2, s8 >= 0, s8 <= s7, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= 2 * x_1 + 2, s2 >= 0, s2 <= 2 * x_2 + 2, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 + 4*x_3 + 8*x_3^2 }-> 1 + s3 + s4 + s5 :|: s3 >= 0, s3 <= 2 * x_1 + 2, s4 >= 0, s4 <= 2 * x_2 + 2, s5 >= 0, s5 <= 2 * x_3 + 2, z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s12 + s13 :|: s12 >= 0, s12 <= 2 * z'' + 2, s13 >= 0, s13 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s14 + s15 :|: s14 >= 0, s14 <= 2 * z'' + 2, s15 >= 0, s15 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 4 + 4*s10 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s11 :|: s9 >= 0, s9 <= 2 * z'' + 2, s10 >= 0, s10 <= 2 * z1 + 2, s11 >= 0, s11 <= s10, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s19 + s20 + s21 :|: s19 >= 0, s19 <= 2 * z'' + 2, s20 >= 0, s20 <= 2 * z1 + 2, s21 >= 0, s21 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 1 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s27 :|: s25 >= 0, s25 <= 2 * z'' + 2, s26 >= 0, s26 <= 2 * z1 + 2, s27 >= 0, s27 <= s26, z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s16 + s17 + s18 :|: s16 >= 0, s16 <= 2 * z'' + 2, s17 >= 0, s17 <= 2 * z1 + 2, s18 >= 0, s18 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [4*z'' + 8*z''^2], size: O(n^1) [2 + 2*z'']
  encode_admit: runtime: O(n^2) [12 + 4*z'' + 8*z''^2 + 12*z1 + 8*z1^2], size: O(n^1) [2 + 2*z1]
  encode_=: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2], size: O(n^1) [5 + 2*z'' + 2*z1]
  encode_carry: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2], size: O(n^1) [7 + 2*z'' + 2*z1 + 2*z2]
  encode_.: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2], size: O(n^1) [5 + 2*z'' + 2*z1]

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

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

(82)
Obligation:
Complexity RNTS consisting of the following rules:

   admit(z'', z1) -{ 8 + 4*z' }-> s'' :|: s >= 0, s <= z', s' >= 0, s' <= 1 + u' + (1 + v' + (1 + 2 + s)), s'' >= 0, s'' <= 1 + u + (1 + v + (1 + 2 + s')), v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 1 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s24 :|: s22 >= 0, s22 <= 2 * x_1 + 2, s23 >= 0, s23 <= 2 * x_2 + 2, s24 >= 0, s24 <= s23, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4 + 4*s7 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s8 :|: s6 >= 0, s6 <= 2 * x_1 + 2, s7 >= 0, s7 <= 2 * x_2 + 2, s8 >= 0, s8 <= s7, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= 2 * x_1 + 2, s2 >= 0, s2 <= 2 * x_2 + 2, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 + 4*x_3 + 8*x_3^2 }-> 1 + s3 + s4 + s5 :|: s3 >= 0, s3 <= 2 * x_1 + 2, s4 >= 0, s4 <= 2 * x_2 + 2, s5 >= 0, s5 <= 2 * x_3 + 2, z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s12 + s13 :|: s12 >= 0, s12 <= 2 * z'' + 2, s13 >= 0, s13 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s14 + s15 :|: s14 >= 0, s14 <= 2 * z'' + 2, s15 >= 0, s15 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 4 + 4*s10 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s11 :|: s9 >= 0, s9 <= 2 * z'' + 2, s10 >= 0, s10 <= 2 * z1 + 2, s11 >= 0, s11 <= s10, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s19 + s20 + s21 :|: s19 >= 0, s19 <= 2 * z'' + 2, s20 >= 0, s20 <= 2 * z1 + 2, s21 >= 0, s21 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 1 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s27 :|: s25 >= 0, s25 <= 2 * z'' + 2, s26 >= 0, s26 <= 2 * z1 + 2, s27 >= 0, s27 <= s26, z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s16 + s17 + s18 :|: s16 >= 0, s16 <= 2 * z'' + 2, s17 >= 0, s17 <= 2 * z1 + 2, s18 >= 0, s18 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [4*z'' + 8*z''^2], size: O(n^1) [2 + 2*z'']
  encode_admit: runtime: O(n^2) [12 + 4*z'' + 8*z''^2 + 12*z1 + 8*z1^2], size: O(n^1) [2 + 2*z1]
  encode_=: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2], size: O(n^1) [5 + 2*z'' + 2*z1]
  encode_carry: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2], size: O(n^1) [7 + 2*z'' + 2*z1 + 2*z2]
  encode_.: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2], size: O(n^1) [5 + 2*z'' + 2*z1]

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

(83) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_sum
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 7 + 2*z'' + 2*z1 + 2*z2

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

(84)
Obligation:
Complexity RNTS consisting of the following rules:

   admit(z'', z1) -{ 8 + 4*z' }-> s'' :|: s >= 0, s <= z', s' >= 0, s' <= 1 + u' + (1 + v' + (1 + 2 + s)), s'' >= 0, s'' <= 1 + u + (1 + v + (1 + 2 + s')), v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 1 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s24 :|: s22 >= 0, s22 <= 2 * x_1 + 2, s23 >= 0, s23 <= 2 * x_2 + 2, s24 >= 0, s24 <= s23, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4 + 4*s7 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s8 :|: s6 >= 0, s6 <= 2 * x_1 + 2, s7 >= 0, s7 <= 2 * x_2 + 2, s8 >= 0, s8 <= s7, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= 2 * x_1 + 2, s2 >= 0, s2 <= 2 * x_2 + 2, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 + 4*x_3 + 8*x_3^2 }-> 1 + s3 + s4 + s5 :|: s3 >= 0, s3 <= 2 * x_1 + 2, s4 >= 0, s4 <= 2 * x_2 + 2, s5 >= 0, s5 <= 2 * x_3 + 2, z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s12 + s13 :|: s12 >= 0, s12 <= 2 * z'' + 2, s13 >= 0, s13 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s14 + s15 :|: s14 >= 0, s14 <= 2 * z'' + 2, s15 >= 0, s15 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 4 + 4*s10 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s11 :|: s9 >= 0, s9 <= 2 * z'' + 2, s10 >= 0, s10 <= 2 * z1 + 2, s11 >= 0, s11 <= s10, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s19 + s20 + s21 :|: s19 >= 0, s19 <= 2 * z'' + 2, s20 >= 0, s20 <= 2 * z1 + 2, s21 >= 0, s21 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 1 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s27 :|: s25 >= 0, s25 <= 2 * z'' + 2, s26 >= 0, s26 <= 2 * z1 + 2, s27 >= 0, s27 <= s26, z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s16 + s17 + s18 :|: s16 >= 0, s16 <= 2 * z'' + 2, s17 >= 0, s17 <= 2 * z1 + 2, s18 >= 0, s18 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_sum}, {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [4*z'' + 8*z''^2], size: O(n^1) [2 + 2*z'']
  encode_admit: runtime: O(n^2) [12 + 4*z'' + 8*z''^2 + 12*z1 + 8*z1^2], size: O(n^1) [2 + 2*z1]
  encode_=: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2], size: O(n^1) [5 + 2*z'' + 2*z1]
  encode_carry: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2], size: O(n^1) [7 + 2*z'' + 2*z1 + 2*z2]
  encode_.: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2], size: O(n^1) [5 + 2*z'' + 2*z1]
  encode_sum: runtime: ?, size: O(n^1) [7 + 2*z'' + 2*z1 + 2*z2]

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

(85) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_sum
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2

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

(86)
Obligation:
Complexity RNTS consisting of the following rules:

   admit(z'', z1) -{ 8 + 4*z' }-> s'' :|: s >= 0, s <= z', s' >= 0, s' <= 1 + u' + (1 + v' + (1 + 2 + s)), s'' >= 0, s'' <= 1 + u + (1 + v + (1 + 2 + s')), v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 1 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s24 :|: s22 >= 0, s22 <= 2 * x_1 + 2, s23 >= 0, s23 <= 2 * x_2 + 2, s24 >= 0, s24 <= s23, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4 + 4*s7 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s8 :|: s6 >= 0, s6 <= 2 * x_1 + 2, s7 >= 0, s7 <= 2 * x_2 + 2, s8 >= 0, s8 <= s7, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= 2 * x_1 + 2, s2 >= 0, s2 <= 2 * x_2 + 2, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 + 4*x_3 + 8*x_3^2 }-> 1 + s3 + s4 + s5 :|: s3 >= 0, s3 <= 2 * x_1 + 2, s4 >= 0, s4 <= 2 * x_2 + 2, s5 >= 0, s5 <= 2 * x_3 + 2, z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s12 + s13 :|: s12 >= 0, s12 <= 2 * z'' + 2, s13 >= 0, s13 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s14 + s15 :|: s14 >= 0, s14 <= 2 * z'' + 2, s15 >= 0, s15 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 4 + 4*s10 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s11 :|: s9 >= 0, s9 <= 2 * z'' + 2, s10 >= 0, s10 <= 2 * z1 + 2, s11 >= 0, s11 <= s10, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s19 + s20 + s21 :|: s19 >= 0, s19 <= 2 * z'' + 2, s20 >= 0, s20 <= 2 * z1 + 2, s21 >= 0, s21 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 1 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s27 :|: s25 >= 0, s25 <= 2 * z'' + 2, s26 >= 0, s26 <= 2 * z1 + 2, s27 >= 0, s27 <= s26, z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s16 + s17 + s18 :|: s16 >= 0, s16 <= 2 * z'' + 2, s17 >= 0, s17 <= 2 * z1 + 2, s18 >= 0, s18 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [4*z'' + 8*z''^2], size: O(n^1) [2 + 2*z'']
  encode_admit: runtime: O(n^2) [12 + 4*z'' + 8*z''^2 + 12*z1 + 8*z1^2], size: O(n^1) [2 + 2*z1]
  encode_=: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2], size: O(n^1) [5 + 2*z'' + 2*z1]
  encode_carry: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2], size: O(n^1) [7 + 2*z'' + 2*z1 + 2*z2]
  encode_.: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2], size: O(n^1) [5 + 2*z'' + 2*z1]
  encode_sum: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2], size: O(n^1) [7 + 2*z'' + 2*z1 + 2*z2]

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

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

(88)
Obligation:
Complexity RNTS consisting of the following rules:

   admit(z'', z1) -{ 8 + 4*z' }-> s'' :|: s >= 0, s <= z', s' >= 0, s' <= 1 + u' + (1 + v' + (1 + 2 + s)), s'' >= 0, s'' <= 1 + u + (1 + v + (1 + 2 + s')), v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 1 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s24 :|: s22 >= 0, s22 <= 2 * x_1 + 2, s23 >= 0, s23 <= 2 * x_2 + 2, s24 >= 0, s24 <= s23, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4 + 4*s7 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s8 :|: s6 >= 0, s6 <= 2 * x_1 + 2, s7 >= 0, s7 <= 2 * x_2 + 2, s8 >= 0, s8 <= s7, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= 2 * x_1 + 2, s2 >= 0, s2 <= 2 * x_2 + 2, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 + 4*x_3 + 8*x_3^2 }-> 1 + s3 + s4 + s5 :|: s3 >= 0, s3 <= 2 * x_1 + 2, s4 >= 0, s4 <= 2 * x_2 + 2, s5 >= 0, s5 <= 2 * x_3 + 2, z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s12 + s13 :|: s12 >= 0, s12 <= 2 * z'' + 2, s13 >= 0, s13 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s14 + s15 :|: s14 >= 0, s14 <= 2 * z'' + 2, s15 >= 0, s15 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 4 + 4*s10 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s11 :|: s9 >= 0, s9 <= 2 * z'' + 2, s10 >= 0, s10 <= 2 * z1 + 2, s11 >= 0, s11 <= s10, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s19 + s20 + s21 :|: s19 >= 0, s19 <= 2 * z'' + 2, s20 >= 0, s20 <= 2 * z1 + 2, s21 >= 0, s21 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 1 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s27 :|: s25 >= 0, s25 <= 2 * z'' + 2, s26 >= 0, s26 <= 2 * z1 + 2, s27 >= 0, s27 <= s26, z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s16 + s17 + s18 :|: s16 >= 0, s16 <= 2 * z'' + 2, s17 >= 0, s17 <= 2 * z1 + 2, s18 >= 0, s18 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [4*z'' + 8*z''^2], size: O(n^1) [2 + 2*z'']
  encode_admit: runtime: O(n^2) [12 + 4*z'' + 8*z''^2 + 12*z1 + 8*z1^2], size: O(n^1) [2 + 2*z1]
  encode_=: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2], size: O(n^1) [5 + 2*z'' + 2*z1]
  encode_carry: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2], size: O(n^1) [7 + 2*z'' + 2*z1 + 2*z2]
  encode_.: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2], size: O(n^1) [5 + 2*z'' + 2*z1]
  encode_sum: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2], size: O(n^1) [7 + 2*z'' + 2*z1 + 2*z2]

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

(89) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_cond
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + 2*z1

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

(90)
Obligation:
Complexity RNTS consisting of the following rules:

   admit(z'', z1) -{ 8 + 4*z' }-> s'' :|: s >= 0, s <= z', s' >= 0, s' <= 1 + u' + (1 + v' + (1 + 2 + s)), s'' >= 0, s'' <= 1 + u + (1 + v + (1 + 2 + s')), v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 1 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s24 :|: s22 >= 0, s22 <= 2 * x_1 + 2, s23 >= 0, s23 <= 2 * x_2 + 2, s24 >= 0, s24 <= s23, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4 + 4*s7 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s8 :|: s6 >= 0, s6 <= 2 * x_1 + 2, s7 >= 0, s7 <= 2 * x_2 + 2, s8 >= 0, s8 <= s7, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= 2 * x_1 + 2, s2 >= 0, s2 <= 2 * x_2 + 2, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 + 4*x_3 + 8*x_3^2 }-> 1 + s3 + s4 + s5 :|: s3 >= 0, s3 <= 2 * x_1 + 2, s4 >= 0, s4 <= 2 * x_2 + 2, s5 >= 0, s5 <= 2 * x_3 + 2, z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s12 + s13 :|: s12 >= 0, s12 <= 2 * z'' + 2, s13 >= 0, s13 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s14 + s15 :|: s14 >= 0, s14 <= 2 * z'' + 2, s15 >= 0, s15 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 4 + 4*s10 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s11 :|: s9 >= 0, s9 <= 2 * z'' + 2, s10 >= 0, s10 <= 2 * z1 + 2, s11 >= 0, s11 <= s10, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s19 + s20 + s21 :|: s19 >= 0, s19 <= 2 * z'' + 2, s20 >= 0, s20 <= 2 * z1 + 2, s21 >= 0, s21 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 1 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s27 :|: s25 >= 0, s25 <= 2 * z'' + 2, s26 >= 0, s26 <= 2 * z1 + 2, s27 >= 0, s27 <= s26, z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s16 + s17 + s18 :|: s16 >= 0, s16 <= 2 * z'' + 2, s17 >= 0, s17 <= 2 * z1 + 2, s18 >= 0, s18 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: {encode_cond}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [4*z'' + 8*z''^2], size: O(n^1) [2 + 2*z'']
  encode_admit: runtime: O(n^2) [12 + 4*z'' + 8*z''^2 + 12*z1 + 8*z1^2], size: O(n^1) [2 + 2*z1]
  encode_=: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2], size: O(n^1) [5 + 2*z'' + 2*z1]
  encode_carry: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2], size: O(n^1) [7 + 2*z'' + 2*z1 + 2*z2]
  encode_.: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2], size: O(n^1) [5 + 2*z'' + 2*z1]
  encode_sum: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2], size: O(n^1) [7 + 2*z'' + 2*z1 + 2*z2]
  encode_cond: runtime: ?, size: O(n^1) [2 + 2*z1]

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

(91) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_cond
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 1 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2

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

(92)
Obligation:
Complexity RNTS consisting of the following rules:

   admit(z'', z1) -{ 8 + 4*z' }-> s'' :|: s >= 0, s <= z', s' >= 0, s' <= 1 + u' + (1 + v' + (1 + 2 + s)), s'' >= 0, s'' <= 1 + u + (1 + v + (1 + 2 + s')), v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + (1 + u' + (1 + v' + (1 + 2 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 2 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z1 = 1 + u + (1 + v + (1 + 2 + z)), z'' >= 0, u >= 0, 1 + (1 + z'' + u + v) + 2 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 2 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' = 1
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encArg(z'') -{ 1 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s24 :|: s22 >= 0, s22 <= 2 * x_1 + 2, s23 >= 0, s23 <= 2 * x_2 + 2, s24 >= 0, s24 <= s23, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4 + 4*s7 + 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> s8 :|: s6 >= 0, s6 <= 2 * x_1 + 2, s7 >= 0, s7 <= 2 * x_2 + 2, s8 >= 0, s8 <= s7, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   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'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= 2 * x_1 + 2, s2 >= 0, s2 <= 2 * x_2 + 2, x_1 >= 0, x_2 >= 0, z'' = 1 + x_1 + x_2
   encArg(z'') -{ 4*x_1 + 8*x_1^2 + 4*x_2 + 8*x_2^2 + 4*x_3 + 8*x_3^2 }-> 1 + s3 + s4 + s5 :|: s3 >= 0, s3 <= 2 * x_1 + 2, s4 >= 0, s4 <= 2 * x_2 + 2, s5 >= 0, s5 <= 2 * x_3 + 2, z'' = 1 + x_1 + x_2 + x_3, x_1 >= 0, x_3 >= 0, x_2 >= 0
   encode_.(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_.(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s12 + s13 :|: s12 >= 0, s12 <= 2 * z'' + 2, s13 >= 0, s13 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_=(z'', z1) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> 1 + s14 + s15 :|: s14 >= 0, s14 <= 2 * z'' + 2, s15 >= 0, s15 <= 2 * z1 + 2, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 4 + 4*s10 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s11 :|: s9 >= 0, s9 <= 2 * z'' + 2, s10 >= 0, s10 <= 2 * z1 + 2, s11 >= 0, s11 <= s10, z'' >= 0, z1 >= 0
   encode_admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_carry(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_carry(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s19 + s20 + s21 :|: s19 >= 0, s19 <= 2 * z'' + 2, s20 >= 0, s20 <= 2 * z1 + 2, s21 >= 0, s21 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 1 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 }-> s27 :|: s25 >= 0, s25 <= 2 * z'' + 2, s26 >= 0, s26 <= 2 * z1 + 2, s27 >= 0, s27 <= s26, z'' >= 0, z1 >= 0
   encode_cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   encode_nil -{ 0 }-> 0 :|: 
   encode_sum(z'', z1, z2) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0, z2 >= 0
   encode_sum(z'', z1, z2) -{ 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2 }-> 1 + s16 + s17 + s18 :|: s16 >= 0, s16 <= 2 * z'' + 2, s17 >= 0, s17 <= 2 * z1 + 2, s18 >= 0, s18 <= 2 * z2 + 2, z'' >= 0, z2 >= 0, z1 >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_w -{ 0 }-> 2 :|: 
   encode_w -{ 0 }-> 0 :|: 

Function symbols to be analyzed: 
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  encode_nil: runtime: O(1) [0], size: O(1) [0]
  encode_true: runtime: O(1) [0], size: O(1) [1]
  encode_w: runtime: O(1) [0], size: O(1) [2]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]
  encArg: runtime: O(n^2) [4*z'' + 8*z''^2], size: O(n^1) [2 + 2*z'']
  encode_admit: runtime: O(n^2) [12 + 4*z'' + 8*z''^2 + 12*z1 + 8*z1^2], size: O(n^1) [2 + 2*z1]
  encode_=: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2], size: O(n^1) [5 + 2*z'' + 2*z1]
  encode_carry: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2], size: O(n^1) [7 + 2*z'' + 2*z1 + 2*z2]
  encode_.: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2], size: O(n^1) [5 + 2*z'' + 2*z1]
  encode_sum: runtime: O(n^2) [4*z'' + 8*z''^2 + 4*z1 + 8*z1^2 + 4*z2 + 8*z2^2], size: O(n^1) [7 + 2*z'' + 2*z1 + 2*z2]
  encode_cond: runtime: O(n^2) [1 + 4*z'' + 8*z''^2 + 4*z1 + 8*z1^2], size: O(n^1) [2 + 2*z1]

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

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

(94)
BOUNDS(1, n^2)

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

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

(96)
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:

   admit(x, nil) -> nil
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
   cond(true, y) -> y

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

   encArg(nil) -> nil
   encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(w) -> w
   encArg(=(x_1, x_2)) -> =(encArg(x_1), encArg(x_2))
   encArg(sum(x_1, x_2, x_3)) -> sum(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(carry(x_1, x_2, x_3)) -> carry(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(true) -> true
   encArg(cons_admit(x_1, x_2)) -> admit(encArg(x_1), encArg(x_2))
   encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2))
   encode_admit(x_1, x_2) -> admit(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_w -> w
   encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2))
   encode_=(x_1, x_2) -> =(encArg(x_1), encArg(x_2))
   encode_sum(x_1, x_2, x_3) -> sum(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_carry(x_1, x_2, x_3) -> carry(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_true -> true

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

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

The rewrite sequence

admit(x, .(u, .(v, .(w, z)))) ->^+ cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))

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

The pumping substitution is [z / .(u, .(v, .(w, z)))].

The result substitution is [x / carry(x, u, v)].




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

(98)
Complex Obligation (BEST)

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

(99)
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:

   admit(x, nil) -> nil
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
   cond(true, y) -> y

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

   encArg(nil) -> nil
   encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(w) -> w
   encArg(=(x_1, x_2)) -> =(encArg(x_1), encArg(x_2))
   encArg(sum(x_1, x_2, x_3)) -> sum(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(carry(x_1, x_2, x_3)) -> carry(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(true) -> true
   encArg(cons_admit(x_1, x_2)) -> admit(encArg(x_1), encArg(x_2))
   encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2))
   encode_admit(x_1, x_2) -> admit(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_w -> w
   encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2))
   encode_=(x_1, x_2) -> =(encArg(x_1), encArg(x_2))
   encode_sum(x_1, x_2, x_3) -> sum(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_carry(x_1, x_2, x_3) -> carry(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_true -> true

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

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

(101)
BOUNDS(n^1, INF)

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

(102)
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:

   admit(x, nil) -> nil
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
   cond(true, y) -> y

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

   encArg(nil) -> nil
   encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(w) -> w
   encArg(=(x_1, x_2)) -> =(encArg(x_1), encArg(x_2))
   encArg(sum(x_1, x_2, x_3)) -> sum(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(carry(x_1, x_2, x_3)) -> carry(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(true) -> true
   encArg(cons_admit(x_1, x_2)) -> admit(encArg(x_1), encArg(x_2))
   encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2))
   encode_admit(x_1, x_2) -> admit(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_w -> w
   encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2))
   encode_=(x_1, x_2) -> =(encArg(x_1), encArg(x_2))
   encode_sum(x_1, x_2, x_3) -> sum(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_carry(x_1, x_2, x_3) -> carry(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_true -> true

Rewrite Strategy: INNERMOST