/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (full) 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), 219 ms]
(4) CpxRelTRS
(5) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
    (7) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxWeightedTrs
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedTrs
    (11) CompletionProof [UPPER BOUND(ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) NarrowingProof [BOTH BOUNDS(ID, ID), 2651 ms]
    (14) CpxTypedWeightedCompleteTrs
    (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) InliningProof [UPPER BOUND(ID), 143 ms]
    (18) CpxRNTS
    (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxRNTS
    (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 75 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 117 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 22 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 96 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 11 ms]
    (40) CpxRNTS
    (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 764 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 326 ms]
    (46) CpxRNTS
    (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 1221 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 893 ms]
    (52) CpxRNTS
    (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (54) CpxRNTS
    (55) IntTrsBoundProof [UPPER BOUND(ID), 620 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 68 ms]
    (58) CpxRNTS
    (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (60) CpxRNTS
    (61) IntTrsBoundProof [UPPER BOUND(ID), 159 ms]
    (62) CpxRNTS
    (63) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
    (64) CpxRNTS
    (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (66) CpxRNTS
    (67) IntTrsBoundProof [UPPER BOUND(ID), 248 ms]
    (68) CpxRNTS
    (69) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (70) CpxRNTS
    (71) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (72) CpxRNTS
    (73) IntTrsBoundProof [UPPER BOUND(ID), 189 ms]
    (74) CpxRNTS
    (75) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (76) CpxRNTS
    (77) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (78) CpxRNTS
    (79) IntTrsBoundProof [UPPER BOUND(ID), 168 ms]
    (80) CpxRNTS
    (81) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (82) CpxRNTS
    (83) FinalProof [FINISHED, 0 ms]
    (84) BOUNDS(1, n^2)
(85) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(86) CpxRelTRS
    (87) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) typed CpxTrs
    (89) OrderProof [LOWER BOUND(ID), 0 ms]
    (90) typed CpxTrs
    (91) RewriteLemmaProof [LOWER BOUND(ID), 273 ms]
    (92) BEST
        (93) proven lower bound
            (94) LowerBoundPropagationProof [FINISHED, 0 ms]
            (95) BOUNDS(n^1, INF)
        (96) typed CpxTrs
            (97) RewriteLemmaProof [LOWER BOUND(ID), 58 ms]
            (98) typed CpxTrs
            (99) RewriteLemmaProof [LOWER BOUND(ID), 64 ms]
            (100) typed CpxTrs
            (101) RewriteLemmaProof [LOWER BOUND(ID), 58 ms]
            (102) typed CpxTrs
            (103) RewriteLemmaProof [LOWER BOUND(ID), 58 ms]
            (104) typed CpxTrs
            (105) RewriteLemmaProof [LOWER BOUND(ID), 29 ms]
            (106) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   U11(tt, M, N) -> U12(tt, activate(M), activate(N))
   U12(tt, M, N) -> s(plus(activate(N), activate(M)))
   plus(N, 0) -> N
   plus(N, s(M)) -> U11(tt, M, N)
   activate(X) -> X

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

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

   encArg(tt) -> tt
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_tt -> tt
   encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_0 -> 0


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

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


The TRS R consists of the following rules:

   U11(tt, M, N) -> U12(tt, activate(M), activate(N))
   U12(tt, M, N) -> s(plus(activate(N), activate(M)))
   plus(N, 0) -> N
   plus(N, s(M)) -> U11(tt, M, N)
   activate(X) -> X

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

   encArg(tt) -> tt
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_tt -> tt
   encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_0 -> 0

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

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

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


The TRS R consists of the following rules:

   U11(tt, M, N) -> U12(tt, activate(M), activate(N))
   U12(tt, M, N) -> s(plus(activate(N), activate(M)))
   plus(N, 0) -> N
   plus(N, s(M)) -> U11(tt, M, N)
   activate(X) -> X

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

   encArg(tt) -> tt
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_tt -> tt
   encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_0 -> 0

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

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

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

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


The TRS R consists of the following rules:

   U11(tt, M, N) -> U12(tt, activate(M), activate(N))
   U12(tt, M, N) -> s(plus(activate(N), activate(M)))
   plus(N, 0) -> N
   plus(N, s(M)) -> U11(tt, M, N)
   activate(X) -> X

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

   encArg(tt) -> tt
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_tt -> tt
   encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_0 -> 0

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

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

(8)
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:

   U11(tt, M, N) -> U12(tt, activate(M), activate(N)) [1]
   U12(tt, M, N) -> s(plus(activate(N), activate(M))) [1]
   plus(N, 0) -> N [1]
   plus(N, s(M)) -> U11(tt, M, N) [1]
   activate(X) -> X [1]
   encArg(tt) -> tt [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(0) -> 0 [0]
   encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_activate(x_1)) -> activate(encArg(x_1)) [0]
   encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_tt -> tt [0]
   encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_activate(x_1) -> activate(encArg(x_1)) [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]

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

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

(10)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   U11(tt, M, N) -> U12(tt, activate(M), activate(N)) [1]
   U12(tt, M, N) -> s(plus(activate(N), activate(M))) [1]
   plus(N, 0) -> N [1]
   plus(N, s(M)) -> U11(tt, M, N) [1]
   activate(X) -> X [1]
   encArg(tt) -> tt [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(0) -> 0 [0]
   encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_activate(x_1)) -> activate(encArg(x_1)) [0]
   encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_tt -> tt [0]
   encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_activate(x_1) -> activate(encArg(x_1)) [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]

The TRS has the following type information:
U11 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate
tt :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate
U12 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate
activate :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate
s :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate
plus :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate
0 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate
encArg :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate
cons_U11 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate
cons_U12 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate
cons_plus :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate
cons_activate :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate
encode_U11 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate
encode_tt :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate
encode_U12 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate
encode_activate :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate
encode_s :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate
encode_plus :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate
encode_0 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate

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

(11) 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:

   activate_1
   U12_3
   plus_2
   U11_3
   encArg_1
   encode_U11_3
   encode_tt
   encode_U12_3
   encode_activate_1
   encode_s_1
   encode_plus_2
   encode_0

Due to the following rules being added:

   encArg(v0) -> null_encArg [0]
   encode_U11(v0, v1, v2) -> null_encode_U11 [0]
   encode_tt -> null_encode_tt [0]
   encode_U12(v0, v1, v2) -> null_encode_U12 [0]
   encode_activate(v0) -> null_encode_activate [0]
   encode_s(v0) -> null_encode_s [0]
   encode_plus(v0, v1) -> null_encode_plus [0]
   encode_0 -> null_encode_0 [0]
   U12(v0, v1, v2) -> null_U12 [0]
   plus(v0, v1) -> null_plus [0]
   U11(v0, v1, v2) -> null_U11 [0]

And the following fresh constants:    null_encArg, null_encode_U11, null_encode_tt, null_encode_U12, null_encode_activate, null_encode_s, null_encode_plus, null_encode_0, null_U12, null_plus, null_U11

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

(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:

   U11(tt, M, N) -> U12(tt, activate(M), activate(N)) [1]
   U12(tt, M, N) -> s(plus(activate(N), activate(M))) [1]
   plus(N, 0) -> N [1]
   plus(N, s(M)) -> U11(tt, M, N) [1]
   activate(X) -> X [1]
   encArg(tt) -> tt [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(0) -> 0 [0]
   encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_activate(x_1)) -> activate(encArg(x_1)) [0]
   encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_tt -> tt [0]
   encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_activate(x_1) -> activate(encArg(x_1)) [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encArg(v0) -> null_encArg [0]
   encode_U11(v0, v1, v2) -> null_encode_U11 [0]
   encode_tt -> null_encode_tt [0]
   encode_U12(v0, v1, v2) -> null_encode_U12 [0]
   encode_activate(v0) -> null_encode_activate [0]
   encode_s(v0) -> null_encode_s [0]
   encode_plus(v0, v1) -> null_encode_plus [0]
   encode_0 -> null_encode_0 [0]
   U12(v0, v1, v2) -> null_U12 [0]
   plus(v0, v1) -> null_plus [0]
   U11(v0, v1, v2) -> null_U11 [0]

The TRS has the following type information:
U11 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
tt :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
U12 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
activate :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
s :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
plus :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
0 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
encArg :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
cons_U11 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
cons_U12 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
cons_plus :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
cons_activate :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
encode_U11 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
encode_tt :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
encode_U12 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
encode_activate :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
encode_s :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
encode_plus :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
encode_0 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
null_encArg :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
null_encode_U11 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
null_encode_tt :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
null_encode_U12 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
null_encode_activate :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
null_encode_s :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
null_encode_plus :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
null_encode_0 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
null_U12 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
null_plus :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
null_U11 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11

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

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

(14)
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:

   U11(tt, M, N) -> U12(tt, M, N) [3]
   U12(tt, M, N) -> s(plus(N, M)) [3]
   plus(N, 0) -> N [1]
   plus(N, s(M)) -> U11(tt, M, N) [1]
   activate(X) -> X [1]
   encArg(tt) -> tt [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(0) -> 0 [0]
   encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_activate(tt)) -> activate(tt) [0]
   encArg(cons_activate(s(x_1774))) -> activate(s(encArg(x_1774))) [0]
   encArg(cons_activate(0)) -> activate(0) [0]
   encArg(cons_activate(cons_U11(x_1775, x_2464, x_3309))) -> activate(U11(encArg(x_1775), encArg(x_2464), encArg(x_3309))) [0]
   encArg(cons_activate(cons_U12(x_1776, x_2465, x_3310))) -> activate(U12(encArg(x_1776), encArg(x_2465), encArg(x_3310))) [0]
   encArg(cons_activate(cons_plus(x_1777, x_2466))) -> activate(plus(encArg(x_1777), encArg(x_2466))) [0]
   encArg(cons_activate(cons_activate(x_1778))) -> activate(activate(encArg(x_1778))) [0]
   encArg(cons_activate(x_1)) -> activate(null_encArg) [0]
   encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_tt -> tt [0]
   encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_activate(tt) -> activate(tt) [0]
   encode_activate(s(x_11509)) -> activate(s(encArg(x_11509))) [0]
   encode_activate(0) -> activate(0) [0]
   encode_activate(cons_U11(x_11510, x_2905, x_3603)) -> activate(U11(encArg(x_11510), encArg(x_2905), encArg(x_3603))) [0]
   encode_activate(cons_U12(x_11511, x_2906, x_3604)) -> activate(U12(encArg(x_11511), encArg(x_2906), encArg(x_3604))) [0]
   encode_activate(cons_plus(x_11512, x_2907)) -> activate(plus(encArg(x_11512), encArg(x_2907))) [0]
   encode_activate(cons_activate(x_11513)) -> activate(activate(encArg(x_11513))) [0]
   encode_activate(x_1) -> activate(null_encArg) [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encArg(v0) -> null_encArg [0]
   encode_U11(v0, v1, v2) -> null_encode_U11 [0]
   encode_tt -> null_encode_tt [0]
   encode_U12(v0, v1, v2) -> null_encode_U12 [0]
   encode_activate(v0) -> null_encode_activate [0]
   encode_s(v0) -> null_encode_s [0]
   encode_plus(v0, v1) -> null_encode_plus [0]
   encode_0 -> null_encode_0 [0]
   U12(v0, v1, v2) -> null_U12 [0]
   plus(v0, v1) -> null_plus [0]
   U11(v0, v1, v2) -> null_U11 [0]

The TRS has the following type information:
U11 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
tt :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
U12 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
activate :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
s :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
plus :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
0 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
encArg :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
cons_U11 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
cons_U12 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
cons_plus :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
cons_activate :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
encode_U11 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
encode_tt :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
encode_U12 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
encode_activate :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
encode_s :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
encode_plus :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11 -> tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
encode_0 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
null_encArg :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
null_encode_U11 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
null_encode_tt :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
null_encode_U12 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
null_encode_activate :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
null_encode_s :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
null_encode_plus :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
null_encode_0 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
null_U12 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
null_plus :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11
null_U11 :: tt:s:0:cons_U11:cons_U12:cons_plus:cons_activate:null_encArg:null_encode_U11:null_encode_tt:null_encode_U12:null_encode_activate:null_encode_s:null_encode_plus:null_encode_0:null_U12:null_plus:null_U11

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

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

   tt => 1
   0 => 0
   null_encArg => 0
   null_encode_U11 => 0
   null_encode_tt => 0
   null_encode_U12 => 0
   null_encode_activate => 0
   null_encode_s => 0
   null_encode_plus => 0
   null_encode_0 => 0
   null_U12 => 0
   null_plus => 0
   null_U11 => 0

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

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

   U11(z, z', z'') -{ 3 }-> U12(1, M, N) :|: z' = M, z = 1, z'' = N, M >= 0, N >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(N, M) :|: z' = M, z = 1, z'' = N, M >= 0, N >= 0
   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   encArg(z) -{ 0 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> activate(plus(encArg(x_1777), encArg(x_2466))) :|: x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 0 }-> activate(activate(encArg(x_1778))) :|: z = 1 + (1 + x_1778), x_1778 >= 0
   encArg(z) -{ 0 }-> activate(U12(encArg(x_1776), encArg(x_2465), encArg(x_3310))) :|: x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 0 }-> activate(U11(encArg(x_1775), encArg(x_2464), encArg(x_3309))) :|: z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 0 }-> activate(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> activate(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> activate(0) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> activate(1 + encArg(x_1774)) :|: z = 1 + (1 + x_1774), x_1774 >= 0
   encArg(z) -{ 0 }-> U12(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> U11(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 0 }-> U11(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z = x_1, z' = x_2, z'' = x_3
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   encode_U12(z, z', z'') -{ 0 }-> U12(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z = x_1, z' = x_2, z'' = x_3
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   encode_activate(z) -{ 0 }-> activate(plus(encArg(x_11512), encArg(x_2907))) :|: x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ 0 }-> activate(activate(encArg(x_11513))) :|: z = 1 + x_11513, x_11513 >= 0
   encode_activate(z) -{ 0 }-> activate(U12(encArg(x_11511), encArg(x_2906), encArg(x_3604))) :|: x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 0 }-> activate(U11(encArg(x_11510), encArg(x_2905), encArg(x_3603))) :|: z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 0 }-> activate(1) :|: z = 1
   encode_activate(z) -{ 0 }-> activate(0) :|: z = 0
   encode_activate(z) -{ 0 }-> activate(0) :|: x_1 >= 0, z = x_1
   encode_activate(z) -{ 0 }-> activate(1 + encArg(x_11509)) :|: z = 1 + x_11509, x_11509 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_plus(z, z') -{ 0 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_s(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_s(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 1 }-> N :|: z = N, z' = 0, N >= 0
   plus(z, z') -{ 1 }-> U11(1, M, N) :|: z' = 1 + M, z = N, M >= 0, N >= 0
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X

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

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

   U11(z, z', z'') -{ 3 }-> U12(1, M, N) :|: z' = M, z = 1, z'' = N, M >= 0, N >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(N, M) :|: z' = M, z = 1, z'' = N, M >= 0, N >= 0
   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + x_1, x_1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 0 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> activate(plus(encArg(x_1777), encArg(x_2466))) :|: x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 0 }-> activate(activate(encArg(x_1778))) :|: z = 1 + (1 + x_1778), x_1778 >= 0
   encArg(z) -{ 0 }-> activate(U12(encArg(x_1776), encArg(x_2465), encArg(x_3310))) :|: x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 0 }-> activate(U11(encArg(x_1775), encArg(x_2464), encArg(x_3309))) :|: z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 0 }-> activate(1 + encArg(x_1774)) :|: z = 1 + (1 + x_1774), x_1774 >= 0
   encArg(z) -{ 0 }-> U12(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> U11(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 0 }-> U11(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z = x_1, z' = x_2, z'' = x_3
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   encode_U12(z, z', z'') -{ 0 }-> U12(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z = x_1, z' = x_2, z'' = x_3
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: x_1 >= 0, z = x_1, X >= 0, 0 = X
   encode_activate(z) -{ 0 }-> activate(plus(encArg(x_11512), encArg(x_2907))) :|: x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ 0 }-> activate(activate(encArg(x_11513))) :|: z = 1 + x_11513, x_11513 >= 0
   encode_activate(z) -{ 0 }-> activate(U12(encArg(x_11511), encArg(x_2906), encArg(x_3604))) :|: x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 0 }-> activate(U11(encArg(x_11510), encArg(x_2905), encArg(x_3603))) :|: z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 0 }-> activate(1 + encArg(x_11509)) :|: z = 1 + x_11509, x_11509 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_plus(z, z') -{ 0 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_s(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_s(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 1 }-> N :|: z = N, z' = 0, N >= 0
   plus(z, z') -{ 1 }-> U11(1, M, N) :|: z' = 1 + M, z = N, M >= 0, N >= 0
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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

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

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

   U11(z, z', z'') -{ 3 }-> U12(1, z', z'') :|: z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(z'', z') :|: z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 0 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> activate(plus(encArg(x_1777), encArg(x_2466))) :|: x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 0 }-> activate(activate(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> activate(U12(encArg(x_1776), encArg(x_2465), encArg(x_3310))) :|: x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 0 }-> activate(U11(encArg(x_1775), encArg(x_2464), encArg(x_3309))) :|: z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 0 }-> activate(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> U12(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> U11(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 0 }-> U11(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 0 }-> U12(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ 0 }-> activate(plus(encArg(x_11512), encArg(x_2907))) :|: x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ 0 }-> activate(activate(encArg(z - 1))) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> activate(U12(encArg(x_11511), encArg(x_2906), encArg(x_3604))) :|: x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 0 }-> activate(U11(encArg(x_11510), encArg(x_2905), encArg(x_3603))) :|: z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 0 }-> activate(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 0 }-> plus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> U11(1, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0


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

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

   { encode_0 }
   { activate }
   { encode_tt }
   { plus, U12, U11 }
   { encArg }
   { encode_activate }
   { encode_plus }
   { encode_U12 }
   { encode_U11 }
   { encode_s }

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

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

   U11(z, z', z'') -{ 3 }-> U12(1, z', z'') :|: z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(z'', z') :|: z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 0 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> activate(plus(encArg(x_1777), encArg(x_2466))) :|: x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 0 }-> activate(activate(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> activate(U12(encArg(x_1776), encArg(x_2465), encArg(x_3310))) :|: x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 0 }-> activate(U11(encArg(x_1775), encArg(x_2464), encArg(x_3309))) :|: z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 0 }-> activate(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> U12(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> U11(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 0 }-> U11(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 0 }-> U12(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ 0 }-> activate(plus(encArg(x_11512), encArg(x_2907))) :|: x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ 0 }-> activate(activate(encArg(z - 1))) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> activate(U12(encArg(x_11511), encArg(x_2906), encArg(x_3604))) :|: x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 0 }-> activate(U11(encArg(x_11510), encArg(x_2905), encArg(x_3603))) :|: z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 0 }-> activate(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 0 }-> plus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> U11(1, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_0}, {activate}, {encode_tt}, {plus,U12,U11}, {encArg}, {encode_activate}, {encode_plus}, {encode_U12}, {encode_U11}, {encode_s}

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

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

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

   U11(z, z', z'') -{ 3 }-> U12(1, z', z'') :|: z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(z'', z') :|: z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 0 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> activate(plus(encArg(x_1777), encArg(x_2466))) :|: x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 0 }-> activate(activate(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> activate(U12(encArg(x_1776), encArg(x_2465), encArg(x_3310))) :|: x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 0 }-> activate(U11(encArg(x_1775), encArg(x_2464), encArg(x_3309))) :|: z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 0 }-> activate(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> U12(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> U11(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 0 }-> U11(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 0 }-> U12(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ 0 }-> activate(plus(encArg(x_11512), encArg(x_2907))) :|: x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ 0 }-> activate(activate(encArg(z - 1))) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> activate(U12(encArg(x_11511), encArg(x_2906), encArg(x_3604))) :|: x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 0 }-> activate(U11(encArg(x_11510), encArg(x_2905), encArg(x_3603))) :|: z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 0 }-> activate(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 0 }-> plus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> U11(1, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_0}, {activate}, {encode_tt}, {plus,U12,U11}, {encArg}, {encode_activate}, {encode_plus}, {encode_U12}, {encode_U11}, {encode_s}

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_0
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   U11(z, z', z'') -{ 3 }-> U12(1, z', z'') :|: z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(z'', z') :|: z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 0 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> activate(plus(encArg(x_1777), encArg(x_2466))) :|: x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 0 }-> activate(activate(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> activate(U12(encArg(x_1776), encArg(x_2465), encArg(x_3310))) :|: x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 0 }-> activate(U11(encArg(x_1775), encArg(x_2464), encArg(x_3309))) :|: z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 0 }-> activate(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> U12(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> U11(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 0 }-> U11(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 0 }-> U12(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ 0 }-> activate(plus(encArg(x_11512), encArg(x_2907))) :|: x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ 0 }-> activate(activate(encArg(z - 1))) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> activate(U12(encArg(x_11511), encArg(x_2906), encArg(x_3604))) :|: x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 0 }-> activate(U11(encArg(x_11510), encArg(x_2905), encArg(x_3603))) :|: z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 0 }-> activate(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 0 }-> plus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> U11(1, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_0}, {activate}, {encode_tt}, {plus,U12,U11}, {encArg}, {encode_activate}, {encode_plus}, {encode_U12}, {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: ?, size: O(1) [0]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: encode_0
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   U11(z, z', z'') -{ 3 }-> U12(1, z', z'') :|: z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(z'', z') :|: z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 0 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> activate(plus(encArg(x_1777), encArg(x_2466))) :|: x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 0 }-> activate(activate(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> activate(U12(encArg(x_1776), encArg(x_2465), encArg(x_3310))) :|: x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 0 }-> activate(U11(encArg(x_1775), encArg(x_2464), encArg(x_3309))) :|: z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 0 }-> activate(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> U12(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> U11(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 0 }-> U11(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 0 }-> U12(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ 0 }-> activate(plus(encArg(x_11512), encArg(x_2907))) :|: x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ 0 }-> activate(activate(encArg(z - 1))) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> activate(U12(encArg(x_11511), encArg(x_2906), encArg(x_3604))) :|: x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 0 }-> activate(U11(encArg(x_11510), encArg(x_2905), encArg(x_3603))) :|: z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 0 }-> activate(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 0 }-> plus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> U11(1, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {activate}, {encode_tt}, {plus,U12,U11}, {encArg}, {encode_activate}, {encode_plus}, {encode_U12}, {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]

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

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

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

   U11(z, z', z'') -{ 3 }-> U12(1, z', z'') :|: z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(z'', z') :|: z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 0 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> activate(plus(encArg(x_1777), encArg(x_2466))) :|: x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 0 }-> activate(activate(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> activate(U12(encArg(x_1776), encArg(x_2465), encArg(x_3310))) :|: x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 0 }-> activate(U11(encArg(x_1775), encArg(x_2464), encArg(x_3309))) :|: z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 0 }-> activate(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> U12(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> U11(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 0 }-> U11(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 0 }-> U12(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ 0 }-> activate(plus(encArg(x_11512), encArg(x_2907))) :|: x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ 0 }-> activate(activate(encArg(z - 1))) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> activate(U12(encArg(x_11511), encArg(x_2906), encArg(x_3604))) :|: x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 0 }-> activate(U11(encArg(x_11510), encArg(x_2905), encArg(x_3603))) :|: z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 0 }-> activate(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 0 }-> plus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> U11(1, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {activate}, {encode_tt}, {plus,U12,U11}, {encArg}, {encode_activate}, {encode_plus}, {encode_U12}, {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   U11(z, z', z'') -{ 3 }-> U12(1, z', z'') :|: z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(z'', z') :|: z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 0 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> activate(plus(encArg(x_1777), encArg(x_2466))) :|: x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 0 }-> activate(activate(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> activate(U12(encArg(x_1776), encArg(x_2465), encArg(x_3310))) :|: x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 0 }-> activate(U11(encArg(x_1775), encArg(x_2464), encArg(x_3309))) :|: z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 0 }-> activate(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> U12(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> U11(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 0 }-> U11(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 0 }-> U12(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ 0 }-> activate(plus(encArg(x_11512), encArg(x_2907))) :|: x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ 0 }-> activate(activate(encArg(z - 1))) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> activate(U12(encArg(x_11511), encArg(x_2906), encArg(x_3604))) :|: x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 0 }-> activate(U11(encArg(x_11510), encArg(x_2905), encArg(x_3603))) :|: z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 0 }-> activate(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 0 }-> plus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> U11(1, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {activate}, {encode_tt}, {plus,U12,U11}, {encArg}, {encode_activate}, {encode_plus}, {encode_U12}, {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: ?, size: O(n^1) [z]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   U11(z, z', z'') -{ 3 }-> U12(1, z', z'') :|: z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(z'', z') :|: z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 0 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> activate(plus(encArg(x_1777), encArg(x_2466))) :|: x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 0 }-> activate(activate(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> activate(U12(encArg(x_1776), encArg(x_2465), encArg(x_3310))) :|: x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 0 }-> activate(U11(encArg(x_1775), encArg(x_2464), encArg(x_3309))) :|: z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 0 }-> activate(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> U12(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> U11(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 0 }-> U11(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 0 }-> U12(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ 0 }-> activate(plus(encArg(x_11512), encArg(x_2907))) :|: x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ 0 }-> activate(activate(encArg(z - 1))) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> activate(U12(encArg(x_11511), encArg(x_2906), encArg(x_3604))) :|: x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 0 }-> activate(U11(encArg(x_11510), encArg(x_2905), encArg(x_3603))) :|: z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 0 }-> activate(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 0 }-> plus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> U11(1, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_tt}, {plus,U12,U11}, {encArg}, {encode_activate}, {encode_plus}, {encode_U12}, {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]

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

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

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

   U11(z, z', z'') -{ 3 }-> U12(1, z', z'') :|: z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(z'', z') :|: z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 0 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> activate(plus(encArg(x_1777), encArg(x_2466))) :|: x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 0 }-> activate(activate(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> activate(U12(encArg(x_1776), encArg(x_2465), encArg(x_3310))) :|: x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 0 }-> activate(U11(encArg(x_1775), encArg(x_2464), encArg(x_3309))) :|: z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 0 }-> activate(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> U12(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> U11(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 0 }-> U11(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 0 }-> U12(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ 0 }-> activate(plus(encArg(x_11512), encArg(x_2907))) :|: x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ 0 }-> activate(activate(encArg(z - 1))) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> activate(U12(encArg(x_11511), encArg(x_2906), encArg(x_3604))) :|: x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 0 }-> activate(U11(encArg(x_11510), encArg(x_2905), encArg(x_3603))) :|: z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 0 }-> activate(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 0 }-> plus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> U11(1, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_tt}, {plus,U12,U11}, {encArg}, {encode_activate}, {encode_plus}, {encode_U12}, {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   U11(z, z', z'') -{ 3 }-> U12(1, z', z'') :|: z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(z'', z') :|: z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 0 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> activate(plus(encArg(x_1777), encArg(x_2466))) :|: x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 0 }-> activate(activate(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> activate(U12(encArg(x_1776), encArg(x_2465), encArg(x_3310))) :|: x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 0 }-> activate(U11(encArg(x_1775), encArg(x_2464), encArg(x_3309))) :|: z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 0 }-> activate(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> U12(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> U11(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 0 }-> U11(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 0 }-> U12(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ 0 }-> activate(plus(encArg(x_11512), encArg(x_2907))) :|: x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ 0 }-> activate(activate(encArg(z - 1))) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> activate(U12(encArg(x_11511), encArg(x_2906), encArg(x_3604))) :|: x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 0 }-> activate(U11(encArg(x_11510), encArg(x_2905), encArg(x_3603))) :|: z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 0 }-> activate(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 0 }-> plus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> U11(1, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_tt}, {plus,U12,U11}, {encArg}, {encode_activate}, {encode_plus}, {encode_U12}, {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: ?, size: O(1) [1]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   U11(z, z', z'') -{ 3 }-> U12(1, z', z'') :|: z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(z'', z') :|: z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 0 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> activate(plus(encArg(x_1777), encArg(x_2466))) :|: x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 0 }-> activate(activate(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> activate(U12(encArg(x_1776), encArg(x_2465), encArg(x_3310))) :|: x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 0 }-> activate(U11(encArg(x_1775), encArg(x_2464), encArg(x_3309))) :|: z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 0 }-> activate(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> U12(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> U11(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 0 }-> U11(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 0 }-> U12(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ 0 }-> activate(plus(encArg(x_11512), encArg(x_2907))) :|: x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ 0 }-> activate(activate(encArg(z - 1))) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> activate(U12(encArg(x_11511), encArg(x_2906), encArg(x_3604))) :|: x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 0 }-> activate(U11(encArg(x_11510), encArg(x_2905), encArg(x_3603))) :|: z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 0 }-> activate(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 0 }-> plus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> U11(1, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {plus,U12,U11}, {encArg}, {encode_activate}, {encode_plus}, {encode_U12}, {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: O(1) [0], size: O(1) [1]

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

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

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

   U11(z, z', z'') -{ 3 }-> U12(1, z', z'') :|: z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(z'', z') :|: z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 0 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> activate(plus(encArg(x_1777), encArg(x_2466))) :|: x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 0 }-> activate(activate(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> activate(U12(encArg(x_1776), encArg(x_2465), encArg(x_3310))) :|: x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 0 }-> activate(U11(encArg(x_1775), encArg(x_2464), encArg(x_3309))) :|: z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 0 }-> activate(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> U12(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> U11(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 0 }-> U11(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 0 }-> U12(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ 0 }-> activate(plus(encArg(x_11512), encArg(x_2907))) :|: x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ 0 }-> activate(activate(encArg(z - 1))) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> activate(U12(encArg(x_11511), encArg(x_2906), encArg(x_3604))) :|: x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 0 }-> activate(U11(encArg(x_11510), encArg(x_2905), encArg(x_3603))) :|: z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 0 }-> activate(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 0 }-> plus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> U11(1, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {plus,U12,U11}, {encArg}, {encode_activate}, {encode_plus}, {encode_U12}, {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: O(1) [0], size: O(1) [1]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed SIZE bound using CoFloCo for: U12
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z' + z''

Computed SIZE bound using CoFloCo for: U11
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z' + z''

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

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

   U11(z, z', z'') -{ 3 }-> U12(1, z', z'') :|: z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(z'', z') :|: z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 0 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> activate(plus(encArg(x_1777), encArg(x_2466))) :|: x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 0 }-> activate(activate(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> activate(U12(encArg(x_1776), encArg(x_2465), encArg(x_3310))) :|: x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 0 }-> activate(U11(encArg(x_1775), encArg(x_2464), encArg(x_3309))) :|: z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 0 }-> activate(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> U12(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> U11(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 0 }-> U11(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 0 }-> U12(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ 0 }-> activate(plus(encArg(x_11512), encArg(x_2907))) :|: x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ 0 }-> activate(activate(encArg(z - 1))) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> activate(U12(encArg(x_11511), encArg(x_2906), encArg(x_3604))) :|: x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 0 }-> activate(U11(encArg(x_11510), encArg(x_2905), encArg(x_3603))) :|: z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 0 }-> activate(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 0 }-> plus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> U11(1, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {plus,U12,U11}, {encArg}, {encode_activate}, {encode_plus}, {encode_U12}, {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: O(1) [0], size: O(1) [1]
  plus: runtime: ?, size: O(n^1) [z + z']
  U12: runtime: ?, size: O(n^1) [1 + z' + z'']
  U11: runtime: ?, size: O(n^1) [1 + z' + z'']

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: plus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 11 + 7*z'

Computed RUNTIME bound using CoFloCo for: U12
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 14 + 7*z'

Computed RUNTIME bound using CoFloCo for: U11
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 17 + 7*z'

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

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

   U11(z, z', z'') -{ 3 }-> U12(1, z', z'') :|: z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 3 }-> 1 + plus(z'', z') :|: z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 0 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> activate(plus(encArg(x_1777), encArg(x_2466))) :|: x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 0 }-> activate(activate(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> activate(U12(encArg(x_1776), encArg(x_2465), encArg(x_3310))) :|: x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 0 }-> activate(U11(encArg(x_1775), encArg(x_2464), encArg(x_3309))) :|: z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 0 }-> activate(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> U12(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> U11(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 0 }-> U11(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 0 }-> U12(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ 0 }-> activate(plus(encArg(x_11512), encArg(x_2907))) :|: x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ 0 }-> activate(activate(encArg(z - 1))) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> activate(U12(encArg(x_11511), encArg(x_2906), encArg(x_3604))) :|: x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 0 }-> activate(U11(encArg(x_11510), encArg(x_2905), encArg(x_3603))) :|: z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 0 }-> activate(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 0 }-> plus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> U11(1, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encArg}, {encode_activate}, {encode_plus}, {encode_U12}, {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: O(1) [0], size: O(1) [1]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']

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

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

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

   U11(z, z', z'') -{ 17 + 7*z' }-> s :|: s >= 0, s <= z' + z'' + 1, z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 14 + 7*z' }-> 1 + s' :|: s' >= 0, s' <= z'' + z', z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 0 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> activate(plus(encArg(x_1777), encArg(x_2466))) :|: x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 0 }-> activate(activate(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> activate(U12(encArg(x_1776), encArg(x_2465), encArg(x_3310))) :|: x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 0 }-> activate(U11(encArg(x_1775), encArg(x_2464), encArg(x_3309))) :|: z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 0 }-> activate(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> U12(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> U11(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 0 }-> U11(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 0 }-> U12(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ 0 }-> activate(plus(encArg(x_11512), encArg(x_2907))) :|: x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ 0 }-> activate(activate(encArg(z - 1))) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> activate(U12(encArg(x_11511), encArg(x_2906), encArg(x_3604))) :|: x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 0 }-> activate(U11(encArg(x_11510), encArg(x_2905), encArg(x_3603))) :|: z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 0 }-> activate(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 0 }-> plus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 11 + 7*z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encArg}, {encode_activate}, {encode_plus}, {encode_U12}, {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: O(1) [0], size: O(1) [1]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']

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

(49) 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: 1 + 11*z

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

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

   U11(z, z', z'') -{ 17 + 7*z' }-> s :|: s >= 0, s <= z' + z'' + 1, z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 14 + 7*z' }-> 1 + s' :|: s' >= 0, s' <= z'' + z', z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 0 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> activate(plus(encArg(x_1777), encArg(x_2466))) :|: x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 0 }-> activate(activate(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> activate(U12(encArg(x_1776), encArg(x_2465), encArg(x_3310))) :|: x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 0 }-> activate(U11(encArg(x_1775), encArg(x_2464), encArg(x_3309))) :|: z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 0 }-> activate(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> U12(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> U11(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 0 }-> U11(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 0 }-> U12(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ 0 }-> activate(plus(encArg(x_11512), encArg(x_2907))) :|: x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ 0 }-> activate(activate(encArg(z - 1))) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> activate(U12(encArg(x_11511), encArg(x_2906), encArg(x_3604))) :|: x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 0 }-> activate(U11(encArg(x_11510), encArg(x_2905), encArg(x_3603))) :|: z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 0 }-> activate(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 0 }-> plus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 11 + 7*z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encArg}, {encode_activate}, {encode_plus}, {encode_U12}, {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: O(1) [0], size: O(1) [1]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: ?, size: O(n^1) [1 + 11*z]

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

(51) 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: 1 + 135*z + 77*z^2

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

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

   U11(z, z', z'') -{ 17 + 7*z' }-> s :|: s >= 0, s <= z' + z'' + 1, z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 14 + 7*z' }-> 1 + s' :|: s' >= 0, s' <= z'' + z', z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 0 }-> plus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> activate(plus(encArg(x_1777), encArg(x_2466))) :|: x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 0 }-> activate(activate(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> activate(U12(encArg(x_1776), encArg(x_2465), encArg(x_3310))) :|: x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 0 }-> activate(U11(encArg(x_1775), encArg(x_2464), encArg(x_3309))) :|: z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 0 }-> activate(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> U12(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> U11(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 0 }-> U11(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 0 }-> U12(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ 0 }-> activate(plus(encArg(x_11512), encArg(x_2907))) :|: x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ 0 }-> activate(activate(encArg(z - 1))) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> activate(U12(encArg(x_11511), encArg(x_2906), encArg(x_3604))) :|: x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 0 }-> activate(U11(encArg(x_11510), encArg(x_2905), encArg(x_3603))) :|: z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 0 }-> activate(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 0 }-> plus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 11 + 7*z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_activate}, {encode_plus}, {encode_U12}, {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: O(1) [0], size: O(1) [1]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [1 + 135*z + 77*z^2], size: O(n^1) [1 + 11*z]

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

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

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

   U11(z, z', z'') -{ 17 + 7*z' }-> s :|: s >= 0, s <= z' + z'' + 1, z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 14 + 7*z' }-> 1 + s' :|: s' >= 0, s' <= z'' + z', z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 13 + 7*s11 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 }-> s12 :|: s10 >= 0, s10 <= 11 * x_1 + 1, s11 >= 0, s11 <= 11 * x_2 + 1, s12 >= 0, s12 <= s10 + s11, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 40 + -173*z + 77*z^2 }-> s26 :|: s25 >= 0, s25 <= 11 * (z - 2) + 1, s26 >= 0, s26 <= 1 + s25, z - 2 >= 0
   encArg(z) -{ 21 + 7*s28 + 135*x_1775 + 77*x_1775^2 + 135*x_2464 + 77*x_2464^2 + 135*x_3309 + 77*x_3309^2 }-> s31 :|: s27 >= 0, s27 <= 11 * x_1775 + 1, s28 >= 0, s28 <= 11 * x_2464 + 1, s29 >= 0, s29 <= 11 * x_3309 + 1, s30 >= 0, s30 <= s28 + s29 + 1, s31 >= 0, s31 <= s30, z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 18 + 7*s33 + 135*x_1776 + 77*x_1776^2 + 135*x_2465 + 77*x_2465^2 + 135*x_3310 + 77*x_3310^2 }-> s36 :|: s32 >= 0, s32 <= 11 * x_1776 + 1, s33 >= 0, s33 <= 11 * x_2465 + 1, s34 >= 0, s34 <= 11 * x_3310 + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= s35, x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 14 + 7*s38 + 135*x_1777 + 77*x_1777^2 + 135*x_2466 + 77*x_2466^2 }-> s40 :|: s37 >= 0, s37 <= 11 * x_1777 + 1, s38 >= 0, s38 <= 11 * x_2466 + 1, s39 >= 0, s39 <= s37 + s38, s40 >= 0, s40 <= s39, x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 41 + -173*z + 77*z^2 }-> s43 :|: s41 >= 0, s41 <= 11 * (z - 2) + 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 2 >= 0
   encArg(z) -{ 20 + 7*s3 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s5 :|: s2 >= 0, s2 <= 11 * x_1 + 1, s3 >= 0, s3 <= 11 * x_2 + 1, s4 >= 0, s4 <= 11 * x_3 + 1, s5 >= 0, s5 <= s3 + s4 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 17 + 7*s7 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s9 :|: s6 >= 0, s6 <= 11 * x_1 + 1, s7 >= 0, s7 <= 11 * x_2 + 1, s8 >= 0, s8 <= 11 * x_3 + 1, s9 >= 0, s9 <= s7 + s8 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -57 + -19*z + 77*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= 11 * (z - 1) + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 20 + 7*s14 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s16 :|: s13 >= 0, s13 <= 11 * z + 1, s14 >= 0, s14 <= 11 * z' + 1, s15 >= 0, s15 <= 11 * z'' + 1, s16 >= 0, s16 <= s14 + s15 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 17 + 7*s18 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s20 :|: s17 >= 0, s17 <= 11 * z + 1, s18 >= 0, s18 <= 11 * z' + 1, s19 >= 0, s19 <= 11 * z'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ -56 + -19*z + 77*z^2 }-> s45 :|: s44 >= 0, s44 <= 11 * (z - 1) + 1, s45 >= 0, s45 <= 1 + s44, z - 1 >= 0
   encode_activate(z) -{ 21 + 7*s47 + 135*x_11510 + 77*x_11510^2 + 135*x_2905 + 77*x_2905^2 + 135*x_3603 + 77*x_3603^2 }-> s50 :|: s46 >= 0, s46 <= 11 * x_11510 + 1, s47 >= 0, s47 <= 11 * x_2905 + 1, s48 >= 0, s48 <= 11 * x_3603 + 1, s49 >= 0, s49 <= s47 + s48 + 1, s50 >= 0, s50 <= s49, z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 18 + 7*s52 + 135*x_11511 + 77*x_11511^2 + 135*x_2906 + 77*x_2906^2 + 135*x_3604 + 77*x_3604^2 }-> s55 :|: s51 >= 0, s51 <= 11 * x_11511 + 1, s52 >= 0, s52 <= 11 * x_2906 + 1, s53 >= 0, s53 <= 11 * x_3604 + 1, s54 >= 0, s54 <= s52 + s53 + 1, s55 >= 0, s55 <= s54, x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 14 + 7*s57 + 135*x_11512 + 77*x_11512^2 + 135*x_2907 + 77*x_2907^2 }-> s59 :|: s56 >= 0, s56 <= 11 * x_11512 + 1, s57 >= 0, s57 <= 11 * x_2907 + 1, s58 >= 0, s58 <= s56 + s57, s59 >= 0, s59 <= s58, x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ -55 + -19*z + 77*z^2 }-> s62 :|: s60 >= 0, s60 <= 11 * (z - 1) + 1, s61 >= 0, s61 <= s60, s62 >= 0, s62 <= s61, z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 13 + 7*s23 + 135*z + 77*z^2 + 135*z' + 77*z'^2 }-> s24 :|: s22 >= 0, s22 <= 11 * z + 1, s23 >= 0, s23 <= 11 * z' + 1, s24 >= 0, s24 <= s22 + s23, z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 135*z + 77*z^2 }-> 1 + s21 :|: s21 >= 0, s21 <= 11 * z + 1, z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 11 + 7*z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_activate}, {encode_plus}, {encode_U12}, {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: O(1) [0], size: O(1) [1]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [1 + 135*z + 77*z^2], size: O(n^1) [1 + 11*z]

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: encode_activate
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 11*z

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

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

   U11(z, z', z'') -{ 17 + 7*z' }-> s :|: s >= 0, s <= z' + z'' + 1, z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 14 + 7*z' }-> 1 + s' :|: s' >= 0, s' <= z'' + z', z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 13 + 7*s11 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 }-> s12 :|: s10 >= 0, s10 <= 11 * x_1 + 1, s11 >= 0, s11 <= 11 * x_2 + 1, s12 >= 0, s12 <= s10 + s11, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 40 + -173*z + 77*z^2 }-> s26 :|: s25 >= 0, s25 <= 11 * (z - 2) + 1, s26 >= 0, s26 <= 1 + s25, z - 2 >= 0
   encArg(z) -{ 21 + 7*s28 + 135*x_1775 + 77*x_1775^2 + 135*x_2464 + 77*x_2464^2 + 135*x_3309 + 77*x_3309^2 }-> s31 :|: s27 >= 0, s27 <= 11 * x_1775 + 1, s28 >= 0, s28 <= 11 * x_2464 + 1, s29 >= 0, s29 <= 11 * x_3309 + 1, s30 >= 0, s30 <= s28 + s29 + 1, s31 >= 0, s31 <= s30, z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 18 + 7*s33 + 135*x_1776 + 77*x_1776^2 + 135*x_2465 + 77*x_2465^2 + 135*x_3310 + 77*x_3310^2 }-> s36 :|: s32 >= 0, s32 <= 11 * x_1776 + 1, s33 >= 0, s33 <= 11 * x_2465 + 1, s34 >= 0, s34 <= 11 * x_3310 + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= s35, x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 14 + 7*s38 + 135*x_1777 + 77*x_1777^2 + 135*x_2466 + 77*x_2466^2 }-> s40 :|: s37 >= 0, s37 <= 11 * x_1777 + 1, s38 >= 0, s38 <= 11 * x_2466 + 1, s39 >= 0, s39 <= s37 + s38, s40 >= 0, s40 <= s39, x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 41 + -173*z + 77*z^2 }-> s43 :|: s41 >= 0, s41 <= 11 * (z - 2) + 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 2 >= 0
   encArg(z) -{ 20 + 7*s3 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s5 :|: s2 >= 0, s2 <= 11 * x_1 + 1, s3 >= 0, s3 <= 11 * x_2 + 1, s4 >= 0, s4 <= 11 * x_3 + 1, s5 >= 0, s5 <= s3 + s4 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 17 + 7*s7 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s9 :|: s6 >= 0, s6 <= 11 * x_1 + 1, s7 >= 0, s7 <= 11 * x_2 + 1, s8 >= 0, s8 <= 11 * x_3 + 1, s9 >= 0, s9 <= s7 + s8 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -57 + -19*z + 77*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= 11 * (z - 1) + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 20 + 7*s14 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s16 :|: s13 >= 0, s13 <= 11 * z + 1, s14 >= 0, s14 <= 11 * z' + 1, s15 >= 0, s15 <= 11 * z'' + 1, s16 >= 0, s16 <= s14 + s15 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 17 + 7*s18 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s20 :|: s17 >= 0, s17 <= 11 * z + 1, s18 >= 0, s18 <= 11 * z' + 1, s19 >= 0, s19 <= 11 * z'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ -56 + -19*z + 77*z^2 }-> s45 :|: s44 >= 0, s44 <= 11 * (z - 1) + 1, s45 >= 0, s45 <= 1 + s44, z - 1 >= 0
   encode_activate(z) -{ 21 + 7*s47 + 135*x_11510 + 77*x_11510^2 + 135*x_2905 + 77*x_2905^2 + 135*x_3603 + 77*x_3603^2 }-> s50 :|: s46 >= 0, s46 <= 11 * x_11510 + 1, s47 >= 0, s47 <= 11 * x_2905 + 1, s48 >= 0, s48 <= 11 * x_3603 + 1, s49 >= 0, s49 <= s47 + s48 + 1, s50 >= 0, s50 <= s49, z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 18 + 7*s52 + 135*x_11511 + 77*x_11511^2 + 135*x_2906 + 77*x_2906^2 + 135*x_3604 + 77*x_3604^2 }-> s55 :|: s51 >= 0, s51 <= 11 * x_11511 + 1, s52 >= 0, s52 <= 11 * x_2906 + 1, s53 >= 0, s53 <= 11 * x_3604 + 1, s54 >= 0, s54 <= s52 + s53 + 1, s55 >= 0, s55 <= s54, x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 14 + 7*s57 + 135*x_11512 + 77*x_11512^2 + 135*x_2907 + 77*x_2907^2 }-> s59 :|: s56 >= 0, s56 <= 11 * x_11512 + 1, s57 >= 0, s57 <= 11 * x_2907 + 1, s58 >= 0, s58 <= s56 + s57, s59 >= 0, s59 <= s58, x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ -55 + -19*z + 77*z^2 }-> s62 :|: s60 >= 0, s60 <= 11 * (z - 1) + 1, s61 >= 0, s61 <= s60, s62 >= 0, s62 <= s61, z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 13 + 7*s23 + 135*z + 77*z^2 + 135*z' + 77*z'^2 }-> s24 :|: s22 >= 0, s22 <= 11 * z + 1, s23 >= 0, s23 <= 11 * z' + 1, s24 >= 0, s24 <= s22 + s23, z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 135*z + 77*z^2 }-> 1 + s21 :|: s21 >= 0, s21 <= 11 * z + 1, z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 11 + 7*z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_activate}, {encode_plus}, {encode_U12}, {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: O(1) [0], size: O(1) [1]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [1 + 135*z + 77*z^2], size: O(n^1) [1 + 11*z]
  encode_activate: runtime: ?, size: O(n^1) [11*z]

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_activate
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 77 + 1311*z + 770*z^2

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

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

   U11(z, z', z'') -{ 17 + 7*z' }-> s :|: s >= 0, s <= z' + z'' + 1, z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 14 + 7*z' }-> 1 + s' :|: s' >= 0, s' <= z'' + z', z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 13 + 7*s11 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 }-> s12 :|: s10 >= 0, s10 <= 11 * x_1 + 1, s11 >= 0, s11 <= 11 * x_2 + 1, s12 >= 0, s12 <= s10 + s11, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 40 + -173*z + 77*z^2 }-> s26 :|: s25 >= 0, s25 <= 11 * (z - 2) + 1, s26 >= 0, s26 <= 1 + s25, z - 2 >= 0
   encArg(z) -{ 21 + 7*s28 + 135*x_1775 + 77*x_1775^2 + 135*x_2464 + 77*x_2464^2 + 135*x_3309 + 77*x_3309^2 }-> s31 :|: s27 >= 0, s27 <= 11 * x_1775 + 1, s28 >= 0, s28 <= 11 * x_2464 + 1, s29 >= 0, s29 <= 11 * x_3309 + 1, s30 >= 0, s30 <= s28 + s29 + 1, s31 >= 0, s31 <= s30, z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 18 + 7*s33 + 135*x_1776 + 77*x_1776^2 + 135*x_2465 + 77*x_2465^2 + 135*x_3310 + 77*x_3310^2 }-> s36 :|: s32 >= 0, s32 <= 11 * x_1776 + 1, s33 >= 0, s33 <= 11 * x_2465 + 1, s34 >= 0, s34 <= 11 * x_3310 + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= s35, x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 14 + 7*s38 + 135*x_1777 + 77*x_1777^2 + 135*x_2466 + 77*x_2466^2 }-> s40 :|: s37 >= 0, s37 <= 11 * x_1777 + 1, s38 >= 0, s38 <= 11 * x_2466 + 1, s39 >= 0, s39 <= s37 + s38, s40 >= 0, s40 <= s39, x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 41 + -173*z + 77*z^2 }-> s43 :|: s41 >= 0, s41 <= 11 * (z - 2) + 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 2 >= 0
   encArg(z) -{ 20 + 7*s3 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s5 :|: s2 >= 0, s2 <= 11 * x_1 + 1, s3 >= 0, s3 <= 11 * x_2 + 1, s4 >= 0, s4 <= 11 * x_3 + 1, s5 >= 0, s5 <= s3 + s4 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 17 + 7*s7 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s9 :|: s6 >= 0, s6 <= 11 * x_1 + 1, s7 >= 0, s7 <= 11 * x_2 + 1, s8 >= 0, s8 <= 11 * x_3 + 1, s9 >= 0, s9 <= s7 + s8 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -57 + -19*z + 77*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= 11 * (z - 1) + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 20 + 7*s14 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s16 :|: s13 >= 0, s13 <= 11 * z + 1, s14 >= 0, s14 <= 11 * z' + 1, s15 >= 0, s15 <= 11 * z'' + 1, s16 >= 0, s16 <= s14 + s15 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 17 + 7*s18 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s20 :|: s17 >= 0, s17 <= 11 * z + 1, s18 >= 0, s18 <= 11 * z' + 1, s19 >= 0, s19 <= 11 * z'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ -56 + -19*z + 77*z^2 }-> s45 :|: s44 >= 0, s44 <= 11 * (z - 1) + 1, s45 >= 0, s45 <= 1 + s44, z - 1 >= 0
   encode_activate(z) -{ 21 + 7*s47 + 135*x_11510 + 77*x_11510^2 + 135*x_2905 + 77*x_2905^2 + 135*x_3603 + 77*x_3603^2 }-> s50 :|: s46 >= 0, s46 <= 11 * x_11510 + 1, s47 >= 0, s47 <= 11 * x_2905 + 1, s48 >= 0, s48 <= 11 * x_3603 + 1, s49 >= 0, s49 <= s47 + s48 + 1, s50 >= 0, s50 <= s49, z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 18 + 7*s52 + 135*x_11511 + 77*x_11511^2 + 135*x_2906 + 77*x_2906^2 + 135*x_3604 + 77*x_3604^2 }-> s55 :|: s51 >= 0, s51 <= 11 * x_11511 + 1, s52 >= 0, s52 <= 11 * x_2906 + 1, s53 >= 0, s53 <= 11 * x_3604 + 1, s54 >= 0, s54 <= s52 + s53 + 1, s55 >= 0, s55 <= s54, x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 14 + 7*s57 + 135*x_11512 + 77*x_11512^2 + 135*x_2907 + 77*x_2907^2 }-> s59 :|: s56 >= 0, s56 <= 11 * x_11512 + 1, s57 >= 0, s57 <= 11 * x_2907 + 1, s58 >= 0, s58 <= s56 + s57, s59 >= 0, s59 <= s58, x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ -55 + -19*z + 77*z^2 }-> s62 :|: s60 >= 0, s60 <= 11 * (z - 1) + 1, s61 >= 0, s61 <= s60, s62 >= 0, s62 <= s61, z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 13 + 7*s23 + 135*z + 77*z^2 + 135*z' + 77*z'^2 }-> s24 :|: s22 >= 0, s22 <= 11 * z + 1, s23 >= 0, s23 <= 11 * z' + 1, s24 >= 0, s24 <= s22 + s23, z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 135*z + 77*z^2 }-> 1 + s21 :|: s21 >= 0, s21 <= 11 * z + 1, z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 11 + 7*z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_plus}, {encode_U12}, {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: O(1) [0], size: O(1) [1]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [1 + 135*z + 77*z^2], size: O(n^1) [1 + 11*z]
  encode_activate: runtime: O(n^2) [77 + 1311*z + 770*z^2], size: O(n^1) [11*z]

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

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

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

   U11(z, z', z'') -{ 17 + 7*z' }-> s :|: s >= 0, s <= z' + z'' + 1, z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 14 + 7*z' }-> 1 + s' :|: s' >= 0, s' <= z'' + z', z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 13 + 7*s11 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 }-> s12 :|: s10 >= 0, s10 <= 11 * x_1 + 1, s11 >= 0, s11 <= 11 * x_2 + 1, s12 >= 0, s12 <= s10 + s11, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 40 + -173*z + 77*z^2 }-> s26 :|: s25 >= 0, s25 <= 11 * (z - 2) + 1, s26 >= 0, s26 <= 1 + s25, z - 2 >= 0
   encArg(z) -{ 21 + 7*s28 + 135*x_1775 + 77*x_1775^2 + 135*x_2464 + 77*x_2464^2 + 135*x_3309 + 77*x_3309^2 }-> s31 :|: s27 >= 0, s27 <= 11 * x_1775 + 1, s28 >= 0, s28 <= 11 * x_2464 + 1, s29 >= 0, s29 <= 11 * x_3309 + 1, s30 >= 0, s30 <= s28 + s29 + 1, s31 >= 0, s31 <= s30, z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 18 + 7*s33 + 135*x_1776 + 77*x_1776^2 + 135*x_2465 + 77*x_2465^2 + 135*x_3310 + 77*x_3310^2 }-> s36 :|: s32 >= 0, s32 <= 11 * x_1776 + 1, s33 >= 0, s33 <= 11 * x_2465 + 1, s34 >= 0, s34 <= 11 * x_3310 + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= s35, x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 14 + 7*s38 + 135*x_1777 + 77*x_1777^2 + 135*x_2466 + 77*x_2466^2 }-> s40 :|: s37 >= 0, s37 <= 11 * x_1777 + 1, s38 >= 0, s38 <= 11 * x_2466 + 1, s39 >= 0, s39 <= s37 + s38, s40 >= 0, s40 <= s39, x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 41 + -173*z + 77*z^2 }-> s43 :|: s41 >= 0, s41 <= 11 * (z - 2) + 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 2 >= 0
   encArg(z) -{ 20 + 7*s3 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s5 :|: s2 >= 0, s2 <= 11 * x_1 + 1, s3 >= 0, s3 <= 11 * x_2 + 1, s4 >= 0, s4 <= 11 * x_3 + 1, s5 >= 0, s5 <= s3 + s4 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 17 + 7*s7 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s9 :|: s6 >= 0, s6 <= 11 * x_1 + 1, s7 >= 0, s7 <= 11 * x_2 + 1, s8 >= 0, s8 <= 11 * x_3 + 1, s9 >= 0, s9 <= s7 + s8 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -57 + -19*z + 77*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= 11 * (z - 1) + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 20 + 7*s14 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s16 :|: s13 >= 0, s13 <= 11 * z + 1, s14 >= 0, s14 <= 11 * z' + 1, s15 >= 0, s15 <= 11 * z'' + 1, s16 >= 0, s16 <= s14 + s15 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 17 + 7*s18 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s20 :|: s17 >= 0, s17 <= 11 * z + 1, s18 >= 0, s18 <= 11 * z' + 1, s19 >= 0, s19 <= 11 * z'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ -56 + -19*z + 77*z^2 }-> s45 :|: s44 >= 0, s44 <= 11 * (z - 1) + 1, s45 >= 0, s45 <= 1 + s44, z - 1 >= 0
   encode_activate(z) -{ 21 + 7*s47 + 135*x_11510 + 77*x_11510^2 + 135*x_2905 + 77*x_2905^2 + 135*x_3603 + 77*x_3603^2 }-> s50 :|: s46 >= 0, s46 <= 11 * x_11510 + 1, s47 >= 0, s47 <= 11 * x_2905 + 1, s48 >= 0, s48 <= 11 * x_3603 + 1, s49 >= 0, s49 <= s47 + s48 + 1, s50 >= 0, s50 <= s49, z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 18 + 7*s52 + 135*x_11511 + 77*x_11511^2 + 135*x_2906 + 77*x_2906^2 + 135*x_3604 + 77*x_3604^2 }-> s55 :|: s51 >= 0, s51 <= 11 * x_11511 + 1, s52 >= 0, s52 <= 11 * x_2906 + 1, s53 >= 0, s53 <= 11 * x_3604 + 1, s54 >= 0, s54 <= s52 + s53 + 1, s55 >= 0, s55 <= s54, x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 14 + 7*s57 + 135*x_11512 + 77*x_11512^2 + 135*x_2907 + 77*x_2907^2 }-> s59 :|: s56 >= 0, s56 <= 11 * x_11512 + 1, s57 >= 0, s57 <= 11 * x_2907 + 1, s58 >= 0, s58 <= s56 + s57, s59 >= 0, s59 <= s58, x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ -55 + -19*z + 77*z^2 }-> s62 :|: s60 >= 0, s60 <= 11 * (z - 1) + 1, s61 >= 0, s61 <= s60, s62 >= 0, s62 <= s61, z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 13 + 7*s23 + 135*z + 77*z^2 + 135*z' + 77*z'^2 }-> s24 :|: s22 >= 0, s22 <= 11 * z + 1, s23 >= 0, s23 <= 11 * z' + 1, s24 >= 0, s24 <= s22 + s23, z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 135*z + 77*z^2 }-> 1 + s21 :|: s21 >= 0, s21 <= 11 * z + 1, z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 11 + 7*z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_plus}, {encode_U12}, {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: O(1) [0], size: O(1) [1]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [1 + 135*z + 77*z^2], size: O(n^1) [1 + 11*z]
  encode_activate: runtime: O(n^2) [77 + 1311*z + 770*z^2], size: O(n^1) [11*z]

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

(61) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_plus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + 11*z + 11*z'

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

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

   U11(z, z', z'') -{ 17 + 7*z' }-> s :|: s >= 0, s <= z' + z'' + 1, z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 14 + 7*z' }-> 1 + s' :|: s' >= 0, s' <= z'' + z', z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 13 + 7*s11 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 }-> s12 :|: s10 >= 0, s10 <= 11 * x_1 + 1, s11 >= 0, s11 <= 11 * x_2 + 1, s12 >= 0, s12 <= s10 + s11, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 40 + -173*z + 77*z^2 }-> s26 :|: s25 >= 0, s25 <= 11 * (z - 2) + 1, s26 >= 0, s26 <= 1 + s25, z - 2 >= 0
   encArg(z) -{ 21 + 7*s28 + 135*x_1775 + 77*x_1775^2 + 135*x_2464 + 77*x_2464^2 + 135*x_3309 + 77*x_3309^2 }-> s31 :|: s27 >= 0, s27 <= 11 * x_1775 + 1, s28 >= 0, s28 <= 11 * x_2464 + 1, s29 >= 0, s29 <= 11 * x_3309 + 1, s30 >= 0, s30 <= s28 + s29 + 1, s31 >= 0, s31 <= s30, z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 18 + 7*s33 + 135*x_1776 + 77*x_1776^2 + 135*x_2465 + 77*x_2465^2 + 135*x_3310 + 77*x_3310^2 }-> s36 :|: s32 >= 0, s32 <= 11 * x_1776 + 1, s33 >= 0, s33 <= 11 * x_2465 + 1, s34 >= 0, s34 <= 11 * x_3310 + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= s35, x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 14 + 7*s38 + 135*x_1777 + 77*x_1777^2 + 135*x_2466 + 77*x_2466^2 }-> s40 :|: s37 >= 0, s37 <= 11 * x_1777 + 1, s38 >= 0, s38 <= 11 * x_2466 + 1, s39 >= 0, s39 <= s37 + s38, s40 >= 0, s40 <= s39, x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 41 + -173*z + 77*z^2 }-> s43 :|: s41 >= 0, s41 <= 11 * (z - 2) + 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 2 >= 0
   encArg(z) -{ 20 + 7*s3 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s5 :|: s2 >= 0, s2 <= 11 * x_1 + 1, s3 >= 0, s3 <= 11 * x_2 + 1, s4 >= 0, s4 <= 11 * x_3 + 1, s5 >= 0, s5 <= s3 + s4 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 17 + 7*s7 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s9 :|: s6 >= 0, s6 <= 11 * x_1 + 1, s7 >= 0, s7 <= 11 * x_2 + 1, s8 >= 0, s8 <= 11 * x_3 + 1, s9 >= 0, s9 <= s7 + s8 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -57 + -19*z + 77*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= 11 * (z - 1) + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 20 + 7*s14 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s16 :|: s13 >= 0, s13 <= 11 * z + 1, s14 >= 0, s14 <= 11 * z' + 1, s15 >= 0, s15 <= 11 * z'' + 1, s16 >= 0, s16 <= s14 + s15 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 17 + 7*s18 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s20 :|: s17 >= 0, s17 <= 11 * z + 1, s18 >= 0, s18 <= 11 * z' + 1, s19 >= 0, s19 <= 11 * z'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ -56 + -19*z + 77*z^2 }-> s45 :|: s44 >= 0, s44 <= 11 * (z - 1) + 1, s45 >= 0, s45 <= 1 + s44, z - 1 >= 0
   encode_activate(z) -{ 21 + 7*s47 + 135*x_11510 + 77*x_11510^2 + 135*x_2905 + 77*x_2905^2 + 135*x_3603 + 77*x_3603^2 }-> s50 :|: s46 >= 0, s46 <= 11 * x_11510 + 1, s47 >= 0, s47 <= 11 * x_2905 + 1, s48 >= 0, s48 <= 11 * x_3603 + 1, s49 >= 0, s49 <= s47 + s48 + 1, s50 >= 0, s50 <= s49, z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 18 + 7*s52 + 135*x_11511 + 77*x_11511^2 + 135*x_2906 + 77*x_2906^2 + 135*x_3604 + 77*x_3604^2 }-> s55 :|: s51 >= 0, s51 <= 11 * x_11511 + 1, s52 >= 0, s52 <= 11 * x_2906 + 1, s53 >= 0, s53 <= 11 * x_3604 + 1, s54 >= 0, s54 <= s52 + s53 + 1, s55 >= 0, s55 <= s54, x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 14 + 7*s57 + 135*x_11512 + 77*x_11512^2 + 135*x_2907 + 77*x_2907^2 }-> s59 :|: s56 >= 0, s56 <= 11 * x_11512 + 1, s57 >= 0, s57 <= 11 * x_2907 + 1, s58 >= 0, s58 <= s56 + s57, s59 >= 0, s59 <= s58, x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ -55 + -19*z + 77*z^2 }-> s62 :|: s60 >= 0, s60 <= 11 * (z - 1) + 1, s61 >= 0, s61 <= s60, s62 >= 0, s62 <= s61, z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 13 + 7*s23 + 135*z + 77*z^2 + 135*z' + 77*z'^2 }-> s24 :|: s22 >= 0, s22 <= 11 * z + 1, s23 >= 0, s23 <= 11 * z' + 1, s24 >= 0, s24 <= s22 + s23, z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 135*z + 77*z^2 }-> 1 + s21 :|: s21 >= 0, s21 <= 11 * z + 1, z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 11 + 7*z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_plus}, {encode_U12}, {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: O(1) [0], size: O(1) [1]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [1 + 135*z + 77*z^2], size: O(n^1) [1 + 11*z]
  encode_activate: runtime: O(n^2) [77 + 1311*z + 770*z^2], size: O(n^1) [11*z]
  encode_plus: runtime: ?, size: O(n^1) [2 + 11*z + 11*z']

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_plus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 20 + 135*z + 77*z^2 + 212*z' + 77*z'^2

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

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

   U11(z, z', z'') -{ 17 + 7*z' }-> s :|: s >= 0, s <= z' + z'' + 1, z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 14 + 7*z' }-> 1 + s' :|: s' >= 0, s' <= z'' + z', z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 13 + 7*s11 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 }-> s12 :|: s10 >= 0, s10 <= 11 * x_1 + 1, s11 >= 0, s11 <= 11 * x_2 + 1, s12 >= 0, s12 <= s10 + s11, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 40 + -173*z + 77*z^2 }-> s26 :|: s25 >= 0, s25 <= 11 * (z - 2) + 1, s26 >= 0, s26 <= 1 + s25, z - 2 >= 0
   encArg(z) -{ 21 + 7*s28 + 135*x_1775 + 77*x_1775^2 + 135*x_2464 + 77*x_2464^2 + 135*x_3309 + 77*x_3309^2 }-> s31 :|: s27 >= 0, s27 <= 11 * x_1775 + 1, s28 >= 0, s28 <= 11 * x_2464 + 1, s29 >= 0, s29 <= 11 * x_3309 + 1, s30 >= 0, s30 <= s28 + s29 + 1, s31 >= 0, s31 <= s30, z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 18 + 7*s33 + 135*x_1776 + 77*x_1776^2 + 135*x_2465 + 77*x_2465^2 + 135*x_3310 + 77*x_3310^2 }-> s36 :|: s32 >= 0, s32 <= 11 * x_1776 + 1, s33 >= 0, s33 <= 11 * x_2465 + 1, s34 >= 0, s34 <= 11 * x_3310 + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= s35, x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 14 + 7*s38 + 135*x_1777 + 77*x_1777^2 + 135*x_2466 + 77*x_2466^2 }-> s40 :|: s37 >= 0, s37 <= 11 * x_1777 + 1, s38 >= 0, s38 <= 11 * x_2466 + 1, s39 >= 0, s39 <= s37 + s38, s40 >= 0, s40 <= s39, x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 41 + -173*z + 77*z^2 }-> s43 :|: s41 >= 0, s41 <= 11 * (z - 2) + 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 2 >= 0
   encArg(z) -{ 20 + 7*s3 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s5 :|: s2 >= 0, s2 <= 11 * x_1 + 1, s3 >= 0, s3 <= 11 * x_2 + 1, s4 >= 0, s4 <= 11 * x_3 + 1, s5 >= 0, s5 <= s3 + s4 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 17 + 7*s7 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s9 :|: s6 >= 0, s6 <= 11 * x_1 + 1, s7 >= 0, s7 <= 11 * x_2 + 1, s8 >= 0, s8 <= 11 * x_3 + 1, s9 >= 0, s9 <= s7 + s8 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -57 + -19*z + 77*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= 11 * (z - 1) + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 20 + 7*s14 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s16 :|: s13 >= 0, s13 <= 11 * z + 1, s14 >= 0, s14 <= 11 * z' + 1, s15 >= 0, s15 <= 11 * z'' + 1, s16 >= 0, s16 <= s14 + s15 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 17 + 7*s18 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s20 :|: s17 >= 0, s17 <= 11 * z + 1, s18 >= 0, s18 <= 11 * z' + 1, s19 >= 0, s19 <= 11 * z'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ -56 + -19*z + 77*z^2 }-> s45 :|: s44 >= 0, s44 <= 11 * (z - 1) + 1, s45 >= 0, s45 <= 1 + s44, z - 1 >= 0
   encode_activate(z) -{ 21 + 7*s47 + 135*x_11510 + 77*x_11510^2 + 135*x_2905 + 77*x_2905^2 + 135*x_3603 + 77*x_3603^2 }-> s50 :|: s46 >= 0, s46 <= 11 * x_11510 + 1, s47 >= 0, s47 <= 11 * x_2905 + 1, s48 >= 0, s48 <= 11 * x_3603 + 1, s49 >= 0, s49 <= s47 + s48 + 1, s50 >= 0, s50 <= s49, z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 18 + 7*s52 + 135*x_11511 + 77*x_11511^2 + 135*x_2906 + 77*x_2906^2 + 135*x_3604 + 77*x_3604^2 }-> s55 :|: s51 >= 0, s51 <= 11 * x_11511 + 1, s52 >= 0, s52 <= 11 * x_2906 + 1, s53 >= 0, s53 <= 11 * x_3604 + 1, s54 >= 0, s54 <= s52 + s53 + 1, s55 >= 0, s55 <= s54, x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 14 + 7*s57 + 135*x_11512 + 77*x_11512^2 + 135*x_2907 + 77*x_2907^2 }-> s59 :|: s56 >= 0, s56 <= 11 * x_11512 + 1, s57 >= 0, s57 <= 11 * x_2907 + 1, s58 >= 0, s58 <= s56 + s57, s59 >= 0, s59 <= s58, x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ -55 + -19*z + 77*z^2 }-> s62 :|: s60 >= 0, s60 <= 11 * (z - 1) + 1, s61 >= 0, s61 <= s60, s62 >= 0, s62 <= s61, z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 13 + 7*s23 + 135*z + 77*z^2 + 135*z' + 77*z'^2 }-> s24 :|: s22 >= 0, s22 <= 11 * z + 1, s23 >= 0, s23 <= 11 * z' + 1, s24 >= 0, s24 <= s22 + s23, z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 135*z + 77*z^2 }-> 1 + s21 :|: s21 >= 0, s21 <= 11 * z + 1, z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 11 + 7*z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_U12}, {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: O(1) [0], size: O(1) [1]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [1 + 135*z + 77*z^2], size: O(n^1) [1 + 11*z]
  encode_activate: runtime: O(n^2) [77 + 1311*z + 770*z^2], size: O(n^1) [11*z]
  encode_plus: runtime: O(n^2) [20 + 135*z + 77*z^2 + 212*z' + 77*z'^2], size: O(n^1) [2 + 11*z + 11*z']

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

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

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

   U11(z, z', z'') -{ 17 + 7*z' }-> s :|: s >= 0, s <= z' + z'' + 1, z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 14 + 7*z' }-> 1 + s' :|: s' >= 0, s' <= z'' + z', z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 13 + 7*s11 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 }-> s12 :|: s10 >= 0, s10 <= 11 * x_1 + 1, s11 >= 0, s11 <= 11 * x_2 + 1, s12 >= 0, s12 <= s10 + s11, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 40 + -173*z + 77*z^2 }-> s26 :|: s25 >= 0, s25 <= 11 * (z - 2) + 1, s26 >= 0, s26 <= 1 + s25, z - 2 >= 0
   encArg(z) -{ 21 + 7*s28 + 135*x_1775 + 77*x_1775^2 + 135*x_2464 + 77*x_2464^2 + 135*x_3309 + 77*x_3309^2 }-> s31 :|: s27 >= 0, s27 <= 11 * x_1775 + 1, s28 >= 0, s28 <= 11 * x_2464 + 1, s29 >= 0, s29 <= 11 * x_3309 + 1, s30 >= 0, s30 <= s28 + s29 + 1, s31 >= 0, s31 <= s30, z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 18 + 7*s33 + 135*x_1776 + 77*x_1776^2 + 135*x_2465 + 77*x_2465^2 + 135*x_3310 + 77*x_3310^2 }-> s36 :|: s32 >= 0, s32 <= 11 * x_1776 + 1, s33 >= 0, s33 <= 11 * x_2465 + 1, s34 >= 0, s34 <= 11 * x_3310 + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= s35, x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 14 + 7*s38 + 135*x_1777 + 77*x_1777^2 + 135*x_2466 + 77*x_2466^2 }-> s40 :|: s37 >= 0, s37 <= 11 * x_1777 + 1, s38 >= 0, s38 <= 11 * x_2466 + 1, s39 >= 0, s39 <= s37 + s38, s40 >= 0, s40 <= s39, x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 41 + -173*z + 77*z^2 }-> s43 :|: s41 >= 0, s41 <= 11 * (z - 2) + 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 2 >= 0
   encArg(z) -{ 20 + 7*s3 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s5 :|: s2 >= 0, s2 <= 11 * x_1 + 1, s3 >= 0, s3 <= 11 * x_2 + 1, s4 >= 0, s4 <= 11 * x_3 + 1, s5 >= 0, s5 <= s3 + s4 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 17 + 7*s7 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s9 :|: s6 >= 0, s6 <= 11 * x_1 + 1, s7 >= 0, s7 <= 11 * x_2 + 1, s8 >= 0, s8 <= 11 * x_3 + 1, s9 >= 0, s9 <= s7 + s8 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -57 + -19*z + 77*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= 11 * (z - 1) + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 20 + 7*s14 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s16 :|: s13 >= 0, s13 <= 11 * z + 1, s14 >= 0, s14 <= 11 * z' + 1, s15 >= 0, s15 <= 11 * z'' + 1, s16 >= 0, s16 <= s14 + s15 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 17 + 7*s18 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s20 :|: s17 >= 0, s17 <= 11 * z + 1, s18 >= 0, s18 <= 11 * z' + 1, s19 >= 0, s19 <= 11 * z'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ -56 + -19*z + 77*z^2 }-> s45 :|: s44 >= 0, s44 <= 11 * (z - 1) + 1, s45 >= 0, s45 <= 1 + s44, z - 1 >= 0
   encode_activate(z) -{ 21 + 7*s47 + 135*x_11510 + 77*x_11510^2 + 135*x_2905 + 77*x_2905^2 + 135*x_3603 + 77*x_3603^2 }-> s50 :|: s46 >= 0, s46 <= 11 * x_11510 + 1, s47 >= 0, s47 <= 11 * x_2905 + 1, s48 >= 0, s48 <= 11 * x_3603 + 1, s49 >= 0, s49 <= s47 + s48 + 1, s50 >= 0, s50 <= s49, z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 18 + 7*s52 + 135*x_11511 + 77*x_11511^2 + 135*x_2906 + 77*x_2906^2 + 135*x_3604 + 77*x_3604^2 }-> s55 :|: s51 >= 0, s51 <= 11 * x_11511 + 1, s52 >= 0, s52 <= 11 * x_2906 + 1, s53 >= 0, s53 <= 11 * x_3604 + 1, s54 >= 0, s54 <= s52 + s53 + 1, s55 >= 0, s55 <= s54, x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 14 + 7*s57 + 135*x_11512 + 77*x_11512^2 + 135*x_2907 + 77*x_2907^2 }-> s59 :|: s56 >= 0, s56 <= 11 * x_11512 + 1, s57 >= 0, s57 <= 11 * x_2907 + 1, s58 >= 0, s58 <= s56 + s57, s59 >= 0, s59 <= s58, x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ -55 + -19*z + 77*z^2 }-> s62 :|: s60 >= 0, s60 <= 11 * (z - 1) + 1, s61 >= 0, s61 <= s60, s62 >= 0, s62 <= s61, z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 13 + 7*s23 + 135*z + 77*z^2 + 135*z' + 77*z'^2 }-> s24 :|: s22 >= 0, s22 <= 11 * z + 1, s23 >= 0, s23 <= 11 * z' + 1, s24 >= 0, s24 <= s22 + s23, z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 135*z + 77*z^2 }-> 1 + s21 :|: s21 >= 0, s21 <= 11 * z + 1, z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 11 + 7*z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_U12}, {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: O(1) [0], size: O(1) [1]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [1 + 135*z + 77*z^2], size: O(n^1) [1 + 11*z]
  encode_activate: runtime: O(n^2) [77 + 1311*z + 770*z^2], size: O(n^1) [11*z]
  encode_plus: runtime: O(n^2) [20 + 135*z + 77*z^2 + 212*z' + 77*z'^2], size: O(n^1) [2 + 11*z + 11*z']

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

(67) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_U12
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + 11*z' + 11*z''

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

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

   U11(z, z', z'') -{ 17 + 7*z' }-> s :|: s >= 0, s <= z' + z'' + 1, z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 14 + 7*z' }-> 1 + s' :|: s' >= 0, s' <= z'' + z', z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 13 + 7*s11 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 }-> s12 :|: s10 >= 0, s10 <= 11 * x_1 + 1, s11 >= 0, s11 <= 11 * x_2 + 1, s12 >= 0, s12 <= s10 + s11, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 40 + -173*z + 77*z^2 }-> s26 :|: s25 >= 0, s25 <= 11 * (z - 2) + 1, s26 >= 0, s26 <= 1 + s25, z - 2 >= 0
   encArg(z) -{ 21 + 7*s28 + 135*x_1775 + 77*x_1775^2 + 135*x_2464 + 77*x_2464^2 + 135*x_3309 + 77*x_3309^2 }-> s31 :|: s27 >= 0, s27 <= 11 * x_1775 + 1, s28 >= 0, s28 <= 11 * x_2464 + 1, s29 >= 0, s29 <= 11 * x_3309 + 1, s30 >= 0, s30 <= s28 + s29 + 1, s31 >= 0, s31 <= s30, z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 18 + 7*s33 + 135*x_1776 + 77*x_1776^2 + 135*x_2465 + 77*x_2465^2 + 135*x_3310 + 77*x_3310^2 }-> s36 :|: s32 >= 0, s32 <= 11 * x_1776 + 1, s33 >= 0, s33 <= 11 * x_2465 + 1, s34 >= 0, s34 <= 11 * x_3310 + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= s35, x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 14 + 7*s38 + 135*x_1777 + 77*x_1777^2 + 135*x_2466 + 77*x_2466^2 }-> s40 :|: s37 >= 0, s37 <= 11 * x_1777 + 1, s38 >= 0, s38 <= 11 * x_2466 + 1, s39 >= 0, s39 <= s37 + s38, s40 >= 0, s40 <= s39, x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 41 + -173*z + 77*z^2 }-> s43 :|: s41 >= 0, s41 <= 11 * (z - 2) + 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 2 >= 0
   encArg(z) -{ 20 + 7*s3 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s5 :|: s2 >= 0, s2 <= 11 * x_1 + 1, s3 >= 0, s3 <= 11 * x_2 + 1, s4 >= 0, s4 <= 11 * x_3 + 1, s5 >= 0, s5 <= s3 + s4 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 17 + 7*s7 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s9 :|: s6 >= 0, s6 <= 11 * x_1 + 1, s7 >= 0, s7 <= 11 * x_2 + 1, s8 >= 0, s8 <= 11 * x_3 + 1, s9 >= 0, s9 <= s7 + s8 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -57 + -19*z + 77*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= 11 * (z - 1) + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 20 + 7*s14 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s16 :|: s13 >= 0, s13 <= 11 * z + 1, s14 >= 0, s14 <= 11 * z' + 1, s15 >= 0, s15 <= 11 * z'' + 1, s16 >= 0, s16 <= s14 + s15 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 17 + 7*s18 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s20 :|: s17 >= 0, s17 <= 11 * z + 1, s18 >= 0, s18 <= 11 * z' + 1, s19 >= 0, s19 <= 11 * z'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ -56 + -19*z + 77*z^2 }-> s45 :|: s44 >= 0, s44 <= 11 * (z - 1) + 1, s45 >= 0, s45 <= 1 + s44, z - 1 >= 0
   encode_activate(z) -{ 21 + 7*s47 + 135*x_11510 + 77*x_11510^2 + 135*x_2905 + 77*x_2905^2 + 135*x_3603 + 77*x_3603^2 }-> s50 :|: s46 >= 0, s46 <= 11 * x_11510 + 1, s47 >= 0, s47 <= 11 * x_2905 + 1, s48 >= 0, s48 <= 11 * x_3603 + 1, s49 >= 0, s49 <= s47 + s48 + 1, s50 >= 0, s50 <= s49, z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 18 + 7*s52 + 135*x_11511 + 77*x_11511^2 + 135*x_2906 + 77*x_2906^2 + 135*x_3604 + 77*x_3604^2 }-> s55 :|: s51 >= 0, s51 <= 11 * x_11511 + 1, s52 >= 0, s52 <= 11 * x_2906 + 1, s53 >= 0, s53 <= 11 * x_3604 + 1, s54 >= 0, s54 <= s52 + s53 + 1, s55 >= 0, s55 <= s54, x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 14 + 7*s57 + 135*x_11512 + 77*x_11512^2 + 135*x_2907 + 77*x_2907^2 }-> s59 :|: s56 >= 0, s56 <= 11 * x_11512 + 1, s57 >= 0, s57 <= 11 * x_2907 + 1, s58 >= 0, s58 <= s56 + s57, s59 >= 0, s59 <= s58, x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ -55 + -19*z + 77*z^2 }-> s62 :|: s60 >= 0, s60 <= 11 * (z - 1) + 1, s61 >= 0, s61 <= s60, s62 >= 0, s62 <= s61, z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 13 + 7*s23 + 135*z + 77*z^2 + 135*z' + 77*z'^2 }-> s24 :|: s22 >= 0, s22 <= 11 * z + 1, s23 >= 0, s23 <= 11 * z' + 1, s24 >= 0, s24 <= s22 + s23, z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 135*z + 77*z^2 }-> 1 + s21 :|: s21 >= 0, s21 <= 11 * z + 1, z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 11 + 7*z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_U12}, {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: O(1) [0], size: O(1) [1]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [1 + 135*z + 77*z^2], size: O(n^1) [1 + 11*z]
  encode_activate: runtime: O(n^2) [77 + 1311*z + 770*z^2], size: O(n^1) [11*z]
  encode_plus: runtime: O(n^2) [20 + 135*z + 77*z^2 + 212*z' + 77*z'^2], size: O(n^1) [2 + 11*z + 11*z']
  encode_U12: runtime: ?, size: O(n^1) [3 + 11*z' + 11*z'']

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

(69) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_U12
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 24 + 135*z + 77*z^2 + 212*z' + 77*z'^2 + 135*z'' + 77*z''^2

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

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

   U11(z, z', z'') -{ 17 + 7*z' }-> s :|: s >= 0, s <= z' + z'' + 1, z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 14 + 7*z' }-> 1 + s' :|: s' >= 0, s' <= z'' + z', z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 13 + 7*s11 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 }-> s12 :|: s10 >= 0, s10 <= 11 * x_1 + 1, s11 >= 0, s11 <= 11 * x_2 + 1, s12 >= 0, s12 <= s10 + s11, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 40 + -173*z + 77*z^2 }-> s26 :|: s25 >= 0, s25 <= 11 * (z - 2) + 1, s26 >= 0, s26 <= 1 + s25, z - 2 >= 0
   encArg(z) -{ 21 + 7*s28 + 135*x_1775 + 77*x_1775^2 + 135*x_2464 + 77*x_2464^2 + 135*x_3309 + 77*x_3309^2 }-> s31 :|: s27 >= 0, s27 <= 11 * x_1775 + 1, s28 >= 0, s28 <= 11 * x_2464 + 1, s29 >= 0, s29 <= 11 * x_3309 + 1, s30 >= 0, s30 <= s28 + s29 + 1, s31 >= 0, s31 <= s30, z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 18 + 7*s33 + 135*x_1776 + 77*x_1776^2 + 135*x_2465 + 77*x_2465^2 + 135*x_3310 + 77*x_3310^2 }-> s36 :|: s32 >= 0, s32 <= 11 * x_1776 + 1, s33 >= 0, s33 <= 11 * x_2465 + 1, s34 >= 0, s34 <= 11 * x_3310 + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= s35, x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 14 + 7*s38 + 135*x_1777 + 77*x_1777^2 + 135*x_2466 + 77*x_2466^2 }-> s40 :|: s37 >= 0, s37 <= 11 * x_1777 + 1, s38 >= 0, s38 <= 11 * x_2466 + 1, s39 >= 0, s39 <= s37 + s38, s40 >= 0, s40 <= s39, x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 41 + -173*z + 77*z^2 }-> s43 :|: s41 >= 0, s41 <= 11 * (z - 2) + 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 2 >= 0
   encArg(z) -{ 20 + 7*s3 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s5 :|: s2 >= 0, s2 <= 11 * x_1 + 1, s3 >= 0, s3 <= 11 * x_2 + 1, s4 >= 0, s4 <= 11 * x_3 + 1, s5 >= 0, s5 <= s3 + s4 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 17 + 7*s7 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s9 :|: s6 >= 0, s6 <= 11 * x_1 + 1, s7 >= 0, s7 <= 11 * x_2 + 1, s8 >= 0, s8 <= 11 * x_3 + 1, s9 >= 0, s9 <= s7 + s8 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -57 + -19*z + 77*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= 11 * (z - 1) + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 20 + 7*s14 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s16 :|: s13 >= 0, s13 <= 11 * z + 1, s14 >= 0, s14 <= 11 * z' + 1, s15 >= 0, s15 <= 11 * z'' + 1, s16 >= 0, s16 <= s14 + s15 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 17 + 7*s18 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s20 :|: s17 >= 0, s17 <= 11 * z + 1, s18 >= 0, s18 <= 11 * z' + 1, s19 >= 0, s19 <= 11 * z'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ -56 + -19*z + 77*z^2 }-> s45 :|: s44 >= 0, s44 <= 11 * (z - 1) + 1, s45 >= 0, s45 <= 1 + s44, z - 1 >= 0
   encode_activate(z) -{ 21 + 7*s47 + 135*x_11510 + 77*x_11510^2 + 135*x_2905 + 77*x_2905^2 + 135*x_3603 + 77*x_3603^2 }-> s50 :|: s46 >= 0, s46 <= 11 * x_11510 + 1, s47 >= 0, s47 <= 11 * x_2905 + 1, s48 >= 0, s48 <= 11 * x_3603 + 1, s49 >= 0, s49 <= s47 + s48 + 1, s50 >= 0, s50 <= s49, z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 18 + 7*s52 + 135*x_11511 + 77*x_11511^2 + 135*x_2906 + 77*x_2906^2 + 135*x_3604 + 77*x_3604^2 }-> s55 :|: s51 >= 0, s51 <= 11 * x_11511 + 1, s52 >= 0, s52 <= 11 * x_2906 + 1, s53 >= 0, s53 <= 11 * x_3604 + 1, s54 >= 0, s54 <= s52 + s53 + 1, s55 >= 0, s55 <= s54, x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 14 + 7*s57 + 135*x_11512 + 77*x_11512^2 + 135*x_2907 + 77*x_2907^2 }-> s59 :|: s56 >= 0, s56 <= 11 * x_11512 + 1, s57 >= 0, s57 <= 11 * x_2907 + 1, s58 >= 0, s58 <= s56 + s57, s59 >= 0, s59 <= s58, x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ -55 + -19*z + 77*z^2 }-> s62 :|: s60 >= 0, s60 <= 11 * (z - 1) + 1, s61 >= 0, s61 <= s60, s62 >= 0, s62 <= s61, z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 13 + 7*s23 + 135*z + 77*z^2 + 135*z' + 77*z'^2 }-> s24 :|: s22 >= 0, s22 <= 11 * z + 1, s23 >= 0, s23 <= 11 * z' + 1, s24 >= 0, s24 <= s22 + s23, z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 135*z + 77*z^2 }-> 1 + s21 :|: s21 >= 0, s21 <= 11 * z + 1, z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 11 + 7*z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: O(1) [0], size: O(1) [1]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [1 + 135*z + 77*z^2], size: O(n^1) [1 + 11*z]
  encode_activate: runtime: O(n^2) [77 + 1311*z + 770*z^2], size: O(n^1) [11*z]
  encode_plus: runtime: O(n^2) [20 + 135*z + 77*z^2 + 212*z' + 77*z'^2], size: O(n^1) [2 + 11*z + 11*z']
  encode_U12: runtime: O(n^2) [24 + 135*z + 77*z^2 + 212*z' + 77*z'^2 + 135*z'' + 77*z''^2], size: O(n^1) [3 + 11*z' + 11*z'']

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

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

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

   U11(z, z', z'') -{ 17 + 7*z' }-> s :|: s >= 0, s <= z' + z'' + 1, z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 14 + 7*z' }-> 1 + s' :|: s' >= 0, s' <= z'' + z', z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 13 + 7*s11 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 }-> s12 :|: s10 >= 0, s10 <= 11 * x_1 + 1, s11 >= 0, s11 <= 11 * x_2 + 1, s12 >= 0, s12 <= s10 + s11, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 40 + -173*z + 77*z^2 }-> s26 :|: s25 >= 0, s25 <= 11 * (z - 2) + 1, s26 >= 0, s26 <= 1 + s25, z - 2 >= 0
   encArg(z) -{ 21 + 7*s28 + 135*x_1775 + 77*x_1775^2 + 135*x_2464 + 77*x_2464^2 + 135*x_3309 + 77*x_3309^2 }-> s31 :|: s27 >= 0, s27 <= 11 * x_1775 + 1, s28 >= 0, s28 <= 11 * x_2464 + 1, s29 >= 0, s29 <= 11 * x_3309 + 1, s30 >= 0, s30 <= s28 + s29 + 1, s31 >= 0, s31 <= s30, z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 18 + 7*s33 + 135*x_1776 + 77*x_1776^2 + 135*x_2465 + 77*x_2465^2 + 135*x_3310 + 77*x_3310^2 }-> s36 :|: s32 >= 0, s32 <= 11 * x_1776 + 1, s33 >= 0, s33 <= 11 * x_2465 + 1, s34 >= 0, s34 <= 11 * x_3310 + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= s35, x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 14 + 7*s38 + 135*x_1777 + 77*x_1777^2 + 135*x_2466 + 77*x_2466^2 }-> s40 :|: s37 >= 0, s37 <= 11 * x_1777 + 1, s38 >= 0, s38 <= 11 * x_2466 + 1, s39 >= 0, s39 <= s37 + s38, s40 >= 0, s40 <= s39, x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 41 + -173*z + 77*z^2 }-> s43 :|: s41 >= 0, s41 <= 11 * (z - 2) + 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 2 >= 0
   encArg(z) -{ 20 + 7*s3 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s5 :|: s2 >= 0, s2 <= 11 * x_1 + 1, s3 >= 0, s3 <= 11 * x_2 + 1, s4 >= 0, s4 <= 11 * x_3 + 1, s5 >= 0, s5 <= s3 + s4 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 17 + 7*s7 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s9 :|: s6 >= 0, s6 <= 11 * x_1 + 1, s7 >= 0, s7 <= 11 * x_2 + 1, s8 >= 0, s8 <= 11 * x_3 + 1, s9 >= 0, s9 <= s7 + s8 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -57 + -19*z + 77*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= 11 * (z - 1) + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 20 + 7*s14 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s16 :|: s13 >= 0, s13 <= 11 * z + 1, s14 >= 0, s14 <= 11 * z' + 1, s15 >= 0, s15 <= 11 * z'' + 1, s16 >= 0, s16 <= s14 + s15 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 17 + 7*s18 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s20 :|: s17 >= 0, s17 <= 11 * z + 1, s18 >= 0, s18 <= 11 * z' + 1, s19 >= 0, s19 <= 11 * z'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ -56 + -19*z + 77*z^2 }-> s45 :|: s44 >= 0, s44 <= 11 * (z - 1) + 1, s45 >= 0, s45 <= 1 + s44, z - 1 >= 0
   encode_activate(z) -{ 21 + 7*s47 + 135*x_11510 + 77*x_11510^2 + 135*x_2905 + 77*x_2905^2 + 135*x_3603 + 77*x_3603^2 }-> s50 :|: s46 >= 0, s46 <= 11 * x_11510 + 1, s47 >= 0, s47 <= 11 * x_2905 + 1, s48 >= 0, s48 <= 11 * x_3603 + 1, s49 >= 0, s49 <= s47 + s48 + 1, s50 >= 0, s50 <= s49, z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 18 + 7*s52 + 135*x_11511 + 77*x_11511^2 + 135*x_2906 + 77*x_2906^2 + 135*x_3604 + 77*x_3604^2 }-> s55 :|: s51 >= 0, s51 <= 11 * x_11511 + 1, s52 >= 0, s52 <= 11 * x_2906 + 1, s53 >= 0, s53 <= 11 * x_3604 + 1, s54 >= 0, s54 <= s52 + s53 + 1, s55 >= 0, s55 <= s54, x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 14 + 7*s57 + 135*x_11512 + 77*x_11512^2 + 135*x_2907 + 77*x_2907^2 }-> s59 :|: s56 >= 0, s56 <= 11 * x_11512 + 1, s57 >= 0, s57 <= 11 * x_2907 + 1, s58 >= 0, s58 <= s56 + s57, s59 >= 0, s59 <= s58, x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ -55 + -19*z + 77*z^2 }-> s62 :|: s60 >= 0, s60 <= 11 * (z - 1) + 1, s61 >= 0, s61 <= s60, s62 >= 0, s62 <= s61, z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 13 + 7*s23 + 135*z + 77*z^2 + 135*z' + 77*z'^2 }-> s24 :|: s22 >= 0, s22 <= 11 * z + 1, s23 >= 0, s23 <= 11 * z' + 1, s24 >= 0, s24 <= s22 + s23, z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 135*z + 77*z^2 }-> 1 + s21 :|: s21 >= 0, s21 <= 11 * z + 1, z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 11 + 7*z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: O(1) [0], size: O(1) [1]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [1 + 135*z + 77*z^2], size: O(n^1) [1 + 11*z]
  encode_activate: runtime: O(n^2) [77 + 1311*z + 770*z^2], size: O(n^1) [11*z]
  encode_plus: runtime: O(n^2) [20 + 135*z + 77*z^2 + 212*z' + 77*z'^2], size: O(n^1) [2 + 11*z + 11*z']
  encode_U12: runtime: O(n^2) [24 + 135*z + 77*z^2 + 212*z' + 77*z'^2 + 135*z'' + 77*z''^2], size: O(n^1) [3 + 11*z' + 11*z'']

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

(73) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_U11
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + 11*z' + 11*z''

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

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

   U11(z, z', z'') -{ 17 + 7*z' }-> s :|: s >= 0, s <= z' + z'' + 1, z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 14 + 7*z' }-> 1 + s' :|: s' >= 0, s' <= z'' + z', z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 13 + 7*s11 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 }-> s12 :|: s10 >= 0, s10 <= 11 * x_1 + 1, s11 >= 0, s11 <= 11 * x_2 + 1, s12 >= 0, s12 <= s10 + s11, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 40 + -173*z + 77*z^2 }-> s26 :|: s25 >= 0, s25 <= 11 * (z - 2) + 1, s26 >= 0, s26 <= 1 + s25, z - 2 >= 0
   encArg(z) -{ 21 + 7*s28 + 135*x_1775 + 77*x_1775^2 + 135*x_2464 + 77*x_2464^2 + 135*x_3309 + 77*x_3309^2 }-> s31 :|: s27 >= 0, s27 <= 11 * x_1775 + 1, s28 >= 0, s28 <= 11 * x_2464 + 1, s29 >= 0, s29 <= 11 * x_3309 + 1, s30 >= 0, s30 <= s28 + s29 + 1, s31 >= 0, s31 <= s30, z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 18 + 7*s33 + 135*x_1776 + 77*x_1776^2 + 135*x_2465 + 77*x_2465^2 + 135*x_3310 + 77*x_3310^2 }-> s36 :|: s32 >= 0, s32 <= 11 * x_1776 + 1, s33 >= 0, s33 <= 11 * x_2465 + 1, s34 >= 0, s34 <= 11 * x_3310 + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= s35, x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 14 + 7*s38 + 135*x_1777 + 77*x_1777^2 + 135*x_2466 + 77*x_2466^2 }-> s40 :|: s37 >= 0, s37 <= 11 * x_1777 + 1, s38 >= 0, s38 <= 11 * x_2466 + 1, s39 >= 0, s39 <= s37 + s38, s40 >= 0, s40 <= s39, x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 41 + -173*z + 77*z^2 }-> s43 :|: s41 >= 0, s41 <= 11 * (z - 2) + 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 2 >= 0
   encArg(z) -{ 20 + 7*s3 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s5 :|: s2 >= 0, s2 <= 11 * x_1 + 1, s3 >= 0, s3 <= 11 * x_2 + 1, s4 >= 0, s4 <= 11 * x_3 + 1, s5 >= 0, s5 <= s3 + s4 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 17 + 7*s7 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s9 :|: s6 >= 0, s6 <= 11 * x_1 + 1, s7 >= 0, s7 <= 11 * x_2 + 1, s8 >= 0, s8 <= 11 * x_3 + 1, s9 >= 0, s9 <= s7 + s8 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -57 + -19*z + 77*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= 11 * (z - 1) + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 20 + 7*s14 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s16 :|: s13 >= 0, s13 <= 11 * z + 1, s14 >= 0, s14 <= 11 * z' + 1, s15 >= 0, s15 <= 11 * z'' + 1, s16 >= 0, s16 <= s14 + s15 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 17 + 7*s18 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s20 :|: s17 >= 0, s17 <= 11 * z + 1, s18 >= 0, s18 <= 11 * z' + 1, s19 >= 0, s19 <= 11 * z'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ -56 + -19*z + 77*z^2 }-> s45 :|: s44 >= 0, s44 <= 11 * (z - 1) + 1, s45 >= 0, s45 <= 1 + s44, z - 1 >= 0
   encode_activate(z) -{ 21 + 7*s47 + 135*x_11510 + 77*x_11510^2 + 135*x_2905 + 77*x_2905^2 + 135*x_3603 + 77*x_3603^2 }-> s50 :|: s46 >= 0, s46 <= 11 * x_11510 + 1, s47 >= 0, s47 <= 11 * x_2905 + 1, s48 >= 0, s48 <= 11 * x_3603 + 1, s49 >= 0, s49 <= s47 + s48 + 1, s50 >= 0, s50 <= s49, z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 18 + 7*s52 + 135*x_11511 + 77*x_11511^2 + 135*x_2906 + 77*x_2906^2 + 135*x_3604 + 77*x_3604^2 }-> s55 :|: s51 >= 0, s51 <= 11 * x_11511 + 1, s52 >= 0, s52 <= 11 * x_2906 + 1, s53 >= 0, s53 <= 11 * x_3604 + 1, s54 >= 0, s54 <= s52 + s53 + 1, s55 >= 0, s55 <= s54, x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 14 + 7*s57 + 135*x_11512 + 77*x_11512^2 + 135*x_2907 + 77*x_2907^2 }-> s59 :|: s56 >= 0, s56 <= 11 * x_11512 + 1, s57 >= 0, s57 <= 11 * x_2907 + 1, s58 >= 0, s58 <= s56 + s57, s59 >= 0, s59 <= s58, x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ -55 + -19*z + 77*z^2 }-> s62 :|: s60 >= 0, s60 <= 11 * (z - 1) + 1, s61 >= 0, s61 <= s60, s62 >= 0, s62 <= s61, z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 13 + 7*s23 + 135*z + 77*z^2 + 135*z' + 77*z'^2 }-> s24 :|: s22 >= 0, s22 <= 11 * z + 1, s23 >= 0, s23 <= 11 * z' + 1, s24 >= 0, s24 <= s22 + s23, z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 135*z + 77*z^2 }-> 1 + s21 :|: s21 >= 0, s21 <= 11 * z + 1, z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 11 + 7*z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_U11}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: O(1) [0], size: O(1) [1]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [1 + 135*z + 77*z^2], size: O(n^1) [1 + 11*z]
  encode_activate: runtime: O(n^2) [77 + 1311*z + 770*z^2], size: O(n^1) [11*z]
  encode_plus: runtime: O(n^2) [20 + 135*z + 77*z^2 + 212*z' + 77*z'^2], size: O(n^1) [2 + 11*z + 11*z']
  encode_U12: runtime: O(n^2) [24 + 135*z + 77*z^2 + 212*z' + 77*z'^2 + 135*z'' + 77*z''^2], size: O(n^1) [3 + 11*z' + 11*z'']
  encode_U11: runtime: ?, size: O(n^1) [3 + 11*z' + 11*z'']

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

(75) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_U11
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 27 + 135*z + 77*z^2 + 212*z' + 77*z'^2 + 135*z'' + 77*z''^2

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

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

   U11(z, z', z'') -{ 17 + 7*z' }-> s :|: s >= 0, s <= z' + z'' + 1, z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 14 + 7*z' }-> 1 + s' :|: s' >= 0, s' <= z'' + z', z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 13 + 7*s11 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 }-> s12 :|: s10 >= 0, s10 <= 11 * x_1 + 1, s11 >= 0, s11 <= 11 * x_2 + 1, s12 >= 0, s12 <= s10 + s11, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 40 + -173*z + 77*z^2 }-> s26 :|: s25 >= 0, s25 <= 11 * (z - 2) + 1, s26 >= 0, s26 <= 1 + s25, z - 2 >= 0
   encArg(z) -{ 21 + 7*s28 + 135*x_1775 + 77*x_1775^2 + 135*x_2464 + 77*x_2464^2 + 135*x_3309 + 77*x_3309^2 }-> s31 :|: s27 >= 0, s27 <= 11 * x_1775 + 1, s28 >= 0, s28 <= 11 * x_2464 + 1, s29 >= 0, s29 <= 11 * x_3309 + 1, s30 >= 0, s30 <= s28 + s29 + 1, s31 >= 0, s31 <= s30, z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 18 + 7*s33 + 135*x_1776 + 77*x_1776^2 + 135*x_2465 + 77*x_2465^2 + 135*x_3310 + 77*x_3310^2 }-> s36 :|: s32 >= 0, s32 <= 11 * x_1776 + 1, s33 >= 0, s33 <= 11 * x_2465 + 1, s34 >= 0, s34 <= 11 * x_3310 + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= s35, x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 14 + 7*s38 + 135*x_1777 + 77*x_1777^2 + 135*x_2466 + 77*x_2466^2 }-> s40 :|: s37 >= 0, s37 <= 11 * x_1777 + 1, s38 >= 0, s38 <= 11 * x_2466 + 1, s39 >= 0, s39 <= s37 + s38, s40 >= 0, s40 <= s39, x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 41 + -173*z + 77*z^2 }-> s43 :|: s41 >= 0, s41 <= 11 * (z - 2) + 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 2 >= 0
   encArg(z) -{ 20 + 7*s3 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s5 :|: s2 >= 0, s2 <= 11 * x_1 + 1, s3 >= 0, s3 <= 11 * x_2 + 1, s4 >= 0, s4 <= 11 * x_3 + 1, s5 >= 0, s5 <= s3 + s4 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 17 + 7*s7 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s9 :|: s6 >= 0, s6 <= 11 * x_1 + 1, s7 >= 0, s7 <= 11 * x_2 + 1, s8 >= 0, s8 <= 11 * x_3 + 1, s9 >= 0, s9 <= s7 + s8 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -57 + -19*z + 77*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= 11 * (z - 1) + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 20 + 7*s14 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s16 :|: s13 >= 0, s13 <= 11 * z + 1, s14 >= 0, s14 <= 11 * z' + 1, s15 >= 0, s15 <= 11 * z'' + 1, s16 >= 0, s16 <= s14 + s15 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 17 + 7*s18 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s20 :|: s17 >= 0, s17 <= 11 * z + 1, s18 >= 0, s18 <= 11 * z' + 1, s19 >= 0, s19 <= 11 * z'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ -56 + -19*z + 77*z^2 }-> s45 :|: s44 >= 0, s44 <= 11 * (z - 1) + 1, s45 >= 0, s45 <= 1 + s44, z - 1 >= 0
   encode_activate(z) -{ 21 + 7*s47 + 135*x_11510 + 77*x_11510^2 + 135*x_2905 + 77*x_2905^2 + 135*x_3603 + 77*x_3603^2 }-> s50 :|: s46 >= 0, s46 <= 11 * x_11510 + 1, s47 >= 0, s47 <= 11 * x_2905 + 1, s48 >= 0, s48 <= 11 * x_3603 + 1, s49 >= 0, s49 <= s47 + s48 + 1, s50 >= 0, s50 <= s49, z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 18 + 7*s52 + 135*x_11511 + 77*x_11511^2 + 135*x_2906 + 77*x_2906^2 + 135*x_3604 + 77*x_3604^2 }-> s55 :|: s51 >= 0, s51 <= 11 * x_11511 + 1, s52 >= 0, s52 <= 11 * x_2906 + 1, s53 >= 0, s53 <= 11 * x_3604 + 1, s54 >= 0, s54 <= s52 + s53 + 1, s55 >= 0, s55 <= s54, x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 14 + 7*s57 + 135*x_11512 + 77*x_11512^2 + 135*x_2907 + 77*x_2907^2 }-> s59 :|: s56 >= 0, s56 <= 11 * x_11512 + 1, s57 >= 0, s57 <= 11 * x_2907 + 1, s58 >= 0, s58 <= s56 + s57, s59 >= 0, s59 <= s58, x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ -55 + -19*z + 77*z^2 }-> s62 :|: s60 >= 0, s60 <= 11 * (z - 1) + 1, s61 >= 0, s61 <= s60, s62 >= 0, s62 <= s61, z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 13 + 7*s23 + 135*z + 77*z^2 + 135*z' + 77*z'^2 }-> s24 :|: s22 >= 0, s22 <= 11 * z + 1, s23 >= 0, s23 <= 11 * z' + 1, s24 >= 0, s24 <= s22 + s23, z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 135*z + 77*z^2 }-> 1 + s21 :|: s21 >= 0, s21 <= 11 * z + 1, z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 11 + 7*z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: O(1) [0], size: O(1) [1]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [1 + 135*z + 77*z^2], size: O(n^1) [1 + 11*z]
  encode_activate: runtime: O(n^2) [77 + 1311*z + 770*z^2], size: O(n^1) [11*z]
  encode_plus: runtime: O(n^2) [20 + 135*z + 77*z^2 + 212*z' + 77*z'^2], size: O(n^1) [2 + 11*z + 11*z']
  encode_U12: runtime: O(n^2) [24 + 135*z + 77*z^2 + 212*z' + 77*z'^2 + 135*z'' + 77*z''^2], size: O(n^1) [3 + 11*z' + 11*z'']
  encode_U11: runtime: O(n^2) [27 + 135*z + 77*z^2 + 212*z' + 77*z'^2 + 135*z'' + 77*z''^2], size: O(n^1) [3 + 11*z' + 11*z'']

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

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

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

   U11(z, z', z'') -{ 17 + 7*z' }-> s :|: s >= 0, s <= z' + z'' + 1, z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 14 + 7*z' }-> 1 + s' :|: s' >= 0, s' <= z'' + z', z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 13 + 7*s11 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 }-> s12 :|: s10 >= 0, s10 <= 11 * x_1 + 1, s11 >= 0, s11 <= 11 * x_2 + 1, s12 >= 0, s12 <= s10 + s11, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 40 + -173*z + 77*z^2 }-> s26 :|: s25 >= 0, s25 <= 11 * (z - 2) + 1, s26 >= 0, s26 <= 1 + s25, z - 2 >= 0
   encArg(z) -{ 21 + 7*s28 + 135*x_1775 + 77*x_1775^2 + 135*x_2464 + 77*x_2464^2 + 135*x_3309 + 77*x_3309^2 }-> s31 :|: s27 >= 0, s27 <= 11 * x_1775 + 1, s28 >= 0, s28 <= 11 * x_2464 + 1, s29 >= 0, s29 <= 11 * x_3309 + 1, s30 >= 0, s30 <= s28 + s29 + 1, s31 >= 0, s31 <= s30, z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 18 + 7*s33 + 135*x_1776 + 77*x_1776^2 + 135*x_2465 + 77*x_2465^2 + 135*x_3310 + 77*x_3310^2 }-> s36 :|: s32 >= 0, s32 <= 11 * x_1776 + 1, s33 >= 0, s33 <= 11 * x_2465 + 1, s34 >= 0, s34 <= 11 * x_3310 + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= s35, x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 14 + 7*s38 + 135*x_1777 + 77*x_1777^2 + 135*x_2466 + 77*x_2466^2 }-> s40 :|: s37 >= 0, s37 <= 11 * x_1777 + 1, s38 >= 0, s38 <= 11 * x_2466 + 1, s39 >= 0, s39 <= s37 + s38, s40 >= 0, s40 <= s39, x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 41 + -173*z + 77*z^2 }-> s43 :|: s41 >= 0, s41 <= 11 * (z - 2) + 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 2 >= 0
   encArg(z) -{ 20 + 7*s3 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s5 :|: s2 >= 0, s2 <= 11 * x_1 + 1, s3 >= 0, s3 <= 11 * x_2 + 1, s4 >= 0, s4 <= 11 * x_3 + 1, s5 >= 0, s5 <= s3 + s4 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 17 + 7*s7 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s9 :|: s6 >= 0, s6 <= 11 * x_1 + 1, s7 >= 0, s7 <= 11 * x_2 + 1, s8 >= 0, s8 <= 11 * x_3 + 1, s9 >= 0, s9 <= s7 + s8 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -57 + -19*z + 77*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= 11 * (z - 1) + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 20 + 7*s14 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s16 :|: s13 >= 0, s13 <= 11 * z + 1, s14 >= 0, s14 <= 11 * z' + 1, s15 >= 0, s15 <= 11 * z'' + 1, s16 >= 0, s16 <= s14 + s15 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 17 + 7*s18 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s20 :|: s17 >= 0, s17 <= 11 * z + 1, s18 >= 0, s18 <= 11 * z' + 1, s19 >= 0, s19 <= 11 * z'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ -56 + -19*z + 77*z^2 }-> s45 :|: s44 >= 0, s44 <= 11 * (z - 1) + 1, s45 >= 0, s45 <= 1 + s44, z - 1 >= 0
   encode_activate(z) -{ 21 + 7*s47 + 135*x_11510 + 77*x_11510^2 + 135*x_2905 + 77*x_2905^2 + 135*x_3603 + 77*x_3603^2 }-> s50 :|: s46 >= 0, s46 <= 11 * x_11510 + 1, s47 >= 0, s47 <= 11 * x_2905 + 1, s48 >= 0, s48 <= 11 * x_3603 + 1, s49 >= 0, s49 <= s47 + s48 + 1, s50 >= 0, s50 <= s49, z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 18 + 7*s52 + 135*x_11511 + 77*x_11511^2 + 135*x_2906 + 77*x_2906^2 + 135*x_3604 + 77*x_3604^2 }-> s55 :|: s51 >= 0, s51 <= 11 * x_11511 + 1, s52 >= 0, s52 <= 11 * x_2906 + 1, s53 >= 0, s53 <= 11 * x_3604 + 1, s54 >= 0, s54 <= s52 + s53 + 1, s55 >= 0, s55 <= s54, x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 14 + 7*s57 + 135*x_11512 + 77*x_11512^2 + 135*x_2907 + 77*x_2907^2 }-> s59 :|: s56 >= 0, s56 <= 11 * x_11512 + 1, s57 >= 0, s57 <= 11 * x_2907 + 1, s58 >= 0, s58 <= s56 + s57, s59 >= 0, s59 <= s58, x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ -55 + -19*z + 77*z^2 }-> s62 :|: s60 >= 0, s60 <= 11 * (z - 1) + 1, s61 >= 0, s61 <= s60, s62 >= 0, s62 <= s61, z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 13 + 7*s23 + 135*z + 77*z^2 + 135*z' + 77*z'^2 }-> s24 :|: s22 >= 0, s22 <= 11 * z + 1, s23 >= 0, s23 <= 11 * z' + 1, s24 >= 0, s24 <= s22 + s23, z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 135*z + 77*z^2 }-> 1 + s21 :|: s21 >= 0, s21 <= 11 * z + 1, z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 11 + 7*z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: O(1) [0], size: O(1) [1]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [1 + 135*z + 77*z^2], size: O(n^1) [1 + 11*z]
  encode_activate: runtime: O(n^2) [77 + 1311*z + 770*z^2], size: O(n^1) [11*z]
  encode_plus: runtime: O(n^2) [20 + 135*z + 77*z^2 + 212*z' + 77*z'^2], size: O(n^1) [2 + 11*z + 11*z']
  encode_U12: runtime: O(n^2) [24 + 135*z + 77*z^2 + 212*z' + 77*z'^2 + 135*z'' + 77*z''^2], size: O(n^1) [3 + 11*z' + 11*z'']
  encode_U11: runtime: O(n^2) [27 + 135*z + 77*z^2 + 212*z' + 77*z'^2 + 135*z'' + 77*z''^2], size: O(n^1) [3 + 11*z' + 11*z'']

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

(79) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_s
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + 11*z

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

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

   U11(z, z', z'') -{ 17 + 7*z' }-> s :|: s >= 0, s <= z' + z'' + 1, z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 14 + 7*z' }-> 1 + s' :|: s' >= 0, s' <= z'' + z', z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 13 + 7*s11 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 }-> s12 :|: s10 >= 0, s10 <= 11 * x_1 + 1, s11 >= 0, s11 <= 11 * x_2 + 1, s12 >= 0, s12 <= s10 + s11, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 40 + -173*z + 77*z^2 }-> s26 :|: s25 >= 0, s25 <= 11 * (z - 2) + 1, s26 >= 0, s26 <= 1 + s25, z - 2 >= 0
   encArg(z) -{ 21 + 7*s28 + 135*x_1775 + 77*x_1775^2 + 135*x_2464 + 77*x_2464^2 + 135*x_3309 + 77*x_3309^2 }-> s31 :|: s27 >= 0, s27 <= 11 * x_1775 + 1, s28 >= 0, s28 <= 11 * x_2464 + 1, s29 >= 0, s29 <= 11 * x_3309 + 1, s30 >= 0, s30 <= s28 + s29 + 1, s31 >= 0, s31 <= s30, z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 18 + 7*s33 + 135*x_1776 + 77*x_1776^2 + 135*x_2465 + 77*x_2465^2 + 135*x_3310 + 77*x_3310^2 }-> s36 :|: s32 >= 0, s32 <= 11 * x_1776 + 1, s33 >= 0, s33 <= 11 * x_2465 + 1, s34 >= 0, s34 <= 11 * x_3310 + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= s35, x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 14 + 7*s38 + 135*x_1777 + 77*x_1777^2 + 135*x_2466 + 77*x_2466^2 }-> s40 :|: s37 >= 0, s37 <= 11 * x_1777 + 1, s38 >= 0, s38 <= 11 * x_2466 + 1, s39 >= 0, s39 <= s37 + s38, s40 >= 0, s40 <= s39, x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 41 + -173*z + 77*z^2 }-> s43 :|: s41 >= 0, s41 <= 11 * (z - 2) + 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 2 >= 0
   encArg(z) -{ 20 + 7*s3 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s5 :|: s2 >= 0, s2 <= 11 * x_1 + 1, s3 >= 0, s3 <= 11 * x_2 + 1, s4 >= 0, s4 <= 11 * x_3 + 1, s5 >= 0, s5 <= s3 + s4 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 17 + 7*s7 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s9 :|: s6 >= 0, s6 <= 11 * x_1 + 1, s7 >= 0, s7 <= 11 * x_2 + 1, s8 >= 0, s8 <= 11 * x_3 + 1, s9 >= 0, s9 <= s7 + s8 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -57 + -19*z + 77*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= 11 * (z - 1) + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 20 + 7*s14 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s16 :|: s13 >= 0, s13 <= 11 * z + 1, s14 >= 0, s14 <= 11 * z' + 1, s15 >= 0, s15 <= 11 * z'' + 1, s16 >= 0, s16 <= s14 + s15 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 17 + 7*s18 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s20 :|: s17 >= 0, s17 <= 11 * z + 1, s18 >= 0, s18 <= 11 * z' + 1, s19 >= 0, s19 <= 11 * z'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ -56 + -19*z + 77*z^2 }-> s45 :|: s44 >= 0, s44 <= 11 * (z - 1) + 1, s45 >= 0, s45 <= 1 + s44, z - 1 >= 0
   encode_activate(z) -{ 21 + 7*s47 + 135*x_11510 + 77*x_11510^2 + 135*x_2905 + 77*x_2905^2 + 135*x_3603 + 77*x_3603^2 }-> s50 :|: s46 >= 0, s46 <= 11 * x_11510 + 1, s47 >= 0, s47 <= 11 * x_2905 + 1, s48 >= 0, s48 <= 11 * x_3603 + 1, s49 >= 0, s49 <= s47 + s48 + 1, s50 >= 0, s50 <= s49, z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 18 + 7*s52 + 135*x_11511 + 77*x_11511^2 + 135*x_2906 + 77*x_2906^2 + 135*x_3604 + 77*x_3604^2 }-> s55 :|: s51 >= 0, s51 <= 11 * x_11511 + 1, s52 >= 0, s52 <= 11 * x_2906 + 1, s53 >= 0, s53 <= 11 * x_3604 + 1, s54 >= 0, s54 <= s52 + s53 + 1, s55 >= 0, s55 <= s54, x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 14 + 7*s57 + 135*x_11512 + 77*x_11512^2 + 135*x_2907 + 77*x_2907^2 }-> s59 :|: s56 >= 0, s56 <= 11 * x_11512 + 1, s57 >= 0, s57 <= 11 * x_2907 + 1, s58 >= 0, s58 <= s56 + s57, s59 >= 0, s59 <= s58, x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ -55 + -19*z + 77*z^2 }-> s62 :|: s60 >= 0, s60 <= 11 * (z - 1) + 1, s61 >= 0, s61 <= s60, s62 >= 0, s62 <= s61, z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 13 + 7*s23 + 135*z + 77*z^2 + 135*z' + 77*z'^2 }-> s24 :|: s22 >= 0, s22 <= 11 * z + 1, s23 >= 0, s23 <= 11 * z' + 1, s24 >= 0, s24 <= s22 + s23, z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 135*z + 77*z^2 }-> 1 + s21 :|: s21 >= 0, s21 <= 11 * z + 1, z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 11 + 7*z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: O(1) [0], size: O(1) [1]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [1 + 135*z + 77*z^2], size: O(n^1) [1 + 11*z]
  encode_activate: runtime: O(n^2) [77 + 1311*z + 770*z^2], size: O(n^1) [11*z]
  encode_plus: runtime: O(n^2) [20 + 135*z + 77*z^2 + 212*z' + 77*z'^2], size: O(n^1) [2 + 11*z + 11*z']
  encode_U12: runtime: O(n^2) [24 + 135*z + 77*z^2 + 212*z' + 77*z'^2 + 135*z'' + 77*z''^2], size: O(n^1) [3 + 11*z' + 11*z'']
  encode_U11: runtime: O(n^2) [27 + 135*z + 77*z^2 + 212*z' + 77*z'^2 + 135*z'' + 77*z''^2], size: O(n^1) [3 + 11*z' + 11*z'']
  encode_s: runtime: ?, size: O(n^1) [2 + 11*z]

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

(81) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_s
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 1 + 135*z + 77*z^2

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

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

   U11(z, z', z'') -{ 17 + 7*z' }-> s :|: s >= 0, s <= z' + z'' + 1, z = 1, z' >= 0, z'' >= 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 14 + 7*z' }-> 1 + s' :|: s' >= 0, s' <= z'' + z', z = 1, z' >= 0, z'' >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   encArg(z) -{ 1 }-> X :|: z = 1 + 1, X >= 0, 1 = X
   encArg(z) -{ 1 }-> X :|: z = 1 + 0, X >= 0, 0 = X
   encArg(z) -{ 1 }-> X :|: z - 1 >= 0, X >= 0, 0 = X
   encArg(z) -{ 13 + 7*s11 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 }-> s12 :|: s10 >= 0, s10 <= 11 * x_1 + 1, s11 >= 0, s11 <= 11 * x_2 + 1, s12 >= 0, s12 <= s10 + s11, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 40 + -173*z + 77*z^2 }-> s26 :|: s25 >= 0, s25 <= 11 * (z - 2) + 1, s26 >= 0, s26 <= 1 + s25, z - 2 >= 0
   encArg(z) -{ 21 + 7*s28 + 135*x_1775 + 77*x_1775^2 + 135*x_2464 + 77*x_2464^2 + 135*x_3309 + 77*x_3309^2 }-> s31 :|: s27 >= 0, s27 <= 11 * x_1775 + 1, s28 >= 0, s28 <= 11 * x_2464 + 1, s29 >= 0, s29 <= 11 * x_3309 + 1, s30 >= 0, s30 <= s28 + s29 + 1, s31 >= 0, s31 <= s30, z = 1 + (1 + x_1775 + x_2464 + x_3309), x_3309 >= 0, x_1775 >= 0, x_2464 >= 0
   encArg(z) -{ 18 + 7*s33 + 135*x_1776 + 77*x_1776^2 + 135*x_2465 + 77*x_2465^2 + 135*x_3310 + 77*x_3310^2 }-> s36 :|: s32 >= 0, s32 <= 11 * x_1776 + 1, s33 >= 0, s33 <= 11 * x_2465 + 1, s34 >= 0, s34 <= 11 * x_3310 + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= s35, x_3310 >= 0, x_1776 >= 0, z = 1 + (1 + x_1776 + x_2465 + x_3310), x_2465 >= 0
   encArg(z) -{ 14 + 7*s38 + 135*x_1777 + 77*x_1777^2 + 135*x_2466 + 77*x_2466^2 }-> s40 :|: s37 >= 0, s37 <= 11 * x_1777 + 1, s38 >= 0, s38 <= 11 * x_2466 + 1, s39 >= 0, s39 <= s37 + s38, s40 >= 0, s40 <= s39, x_2466 >= 0, x_1777 >= 0, z = 1 + (1 + x_1777 + x_2466)
   encArg(z) -{ 41 + -173*z + 77*z^2 }-> s43 :|: s41 >= 0, s41 <= 11 * (z - 2) + 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 2 >= 0
   encArg(z) -{ 20 + 7*s3 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s5 :|: s2 >= 0, s2 <= 11 * x_1 + 1, s3 >= 0, s3 <= 11 * x_2 + 1, s4 >= 0, s4 <= 11 * x_3 + 1, s5 >= 0, s5 <= s3 + s4 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 17 + 7*s7 + 135*x_1 + 77*x_1^2 + 135*x_2 + 77*x_2^2 + 135*x_3 + 77*x_3^2 }-> s9 :|: s6 >= 0, s6 <= 11 * x_1 + 1, s7 >= 0, s7 <= 11 * x_2 + 1, s8 >= 0, s8 <= 11 * x_3 + 1, s9 >= 0, s9 <= s7 + s8 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -57 + -19*z + 77*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= 11 * (z - 1) + 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_U11(z, z', z'') -{ 20 + 7*s14 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s16 :|: s13 >= 0, s13 <= 11 * z + 1, s14 >= 0, s14 <= 11 * z' + 1, s15 >= 0, s15 <= 11 * z'' + 1, s16 >= 0, s16 <= s14 + s15 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_U12(z, z', z'') -{ 17 + 7*s18 + 135*z + 77*z^2 + 135*z' + 77*z'^2 + 135*z'' + 77*z''^2 }-> s20 :|: s17 >= 0, s17 <= 11 * z + 1, s18 >= 0, s18 <= 11 * z' + 1, s19 >= 0, s19 <= 11 * z'' + 1, s20 >= 0, s20 <= s18 + s19 + 1, z >= 0, z'' >= 0, z' >= 0
   encode_U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_activate(z) -{ 1 }-> X :|: z = 1, X >= 0, 1 = X
   encode_activate(z) -{ 1 }-> X :|: z = 0, X >= 0, 0 = X
   encode_activate(z) -{ 1 }-> X :|: z >= 0, X >= 0, 0 = X
   encode_activate(z) -{ -56 + -19*z + 77*z^2 }-> s45 :|: s44 >= 0, s44 <= 11 * (z - 1) + 1, s45 >= 0, s45 <= 1 + s44, z - 1 >= 0
   encode_activate(z) -{ 21 + 7*s47 + 135*x_11510 + 77*x_11510^2 + 135*x_2905 + 77*x_2905^2 + 135*x_3603 + 77*x_3603^2 }-> s50 :|: s46 >= 0, s46 <= 11 * x_11510 + 1, s47 >= 0, s47 <= 11 * x_2905 + 1, s48 >= 0, s48 <= 11 * x_3603 + 1, s49 >= 0, s49 <= s47 + s48 + 1, s50 >= 0, s50 <= s49, z = 1 + x_11510 + x_2905 + x_3603, x_3603 >= 0, x_11510 >= 0, x_2905 >= 0
   encode_activate(z) -{ 18 + 7*s52 + 135*x_11511 + 77*x_11511^2 + 135*x_2906 + 77*x_2906^2 + 135*x_3604 + 77*x_3604^2 }-> s55 :|: s51 >= 0, s51 <= 11 * x_11511 + 1, s52 >= 0, s52 <= 11 * x_2906 + 1, s53 >= 0, s53 <= 11 * x_3604 + 1, s54 >= 0, s54 <= s52 + s53 + 1, s55 >= 0, s55 <= s54, x_3604 >= 0, x_11511 >= 0, z = 1 + x_11511 + x_2906 + x_3604, x_2906 >= 0
   encode_activate(z) -{ 14 + 7*s57 + 135*x_11512 + 77*x_11512^2 + 135*x_2907 + 77*x_2907^2 }-> s59 :|: s56 >= 0, s56 <= 11 * x_11512 + 1, s57 >= 0, s57 <= 11 * x_2907 + 1, s58 >= 0, s58 <= s56 + s57, s59 >= 0, s59 <= s58, x_2907 >= 0, z = 1 + x_11512 + x_2907, x_11512 >= 0
   encode_activate(z) -{ -55 + -19*z + 77*z^2 }-> s62 :|: s60 >= 0, s60 <= 11 * (z - 1) + 1, s61 >= 0, s61 <= s60, s62 >= 0, s62 <= s61, z - 1 >= 0
   encode_activate(z) -{ 0 }-> 0 :|: z >= 0
   encode_plus(z, z') -{ 13 + 7*s23 + 135*z + 77*z^2 + 135*z' + 77*z'^2 }-> s24 :|: s22 >= 0, s22 <= 11 * z + 1, s23 >= 0, s23 <= 11 * z' + 1, s24 >= 0, s24 <= s22 + s23, z >= 0, z' >= 0
   encode_plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 135*z + 77*z^2 }-> 1 + s21 :|: s21 >= 0, s21 <= 11 * z + 1, z >= 0
   encode_tt -{ 0 }-> 1 :|: 
   encode_tt -{ 0 }-> 0 :|: 
   plus(z, z') -{ 11 + 7*z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  activate: runtime: O(1) [1], size: O(n^1) [z]
  encode_tt: runtime: O(1) [0], size: O(1) [1]
  plus: runtime: O(n^1) [11 + 7*z'], size: O(n^1) [z + z']
  U12: runtime: O(n^1) [14 + 7*z'], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(n^1) [17 + 7*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [1 + 135*z + 77*z^2], size: O(n^1) [1 + 11*z]
  encode_activate: runtime: O(n^2) [77 + 1311*z + 770*z^2], size: O(n^1) [11*z]
  encode_plus: runtime: O(n^2) [20 + 135*z + 77*z^2 + 212*z' + 77*z'^2], size: O(n^1) [2 + 11*z + 11*z']
  encode_U12: runtime: O(n^2) [24 + 135*z + 77*z^2 + 212*z' + 77*z'^2 + 135*z'' + 77*z''^2], size: O(n^1) [3 + 11*z' + 11*z'']
  encode_U11: runtime: O(n^2) [27 + 135*z + 77*z^2 + 212*z' + 77*z'^2 + 135*z'' + 77*z''^2], size: O(n^1) [3 + 11*z' + 11*z'']
  encode_s: runtime: O(n^2) [1 + 135*z + 77*z^2], size: O(n^1) [2 + 11*z]

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

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

(84)
BOUNDS(1, n^2)

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

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

(86)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   U11(tt, M, N) -> U12(tt, activate(M), activate(N))
   U12(tt, M, N) -> s(plus(activate(N), activate(M)))
   plus(N, 0') -> N
   plus(N, s(M)) -> U11(tt, M, N)
   activate(X) -> X

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

   encArg(tt) -> tt
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0') -> 0'
   encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_tt -> tt
   encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_0 -> 0'

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

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

(88)
Obligation:
TRS:
Rules:
U11(tt, M, N) -> U12(tt, activate(M), activate(N))
U12(tt, M, N) -> s(plus(activate(N), activate(M)))
plus(N, 0') -> N
plus(N, s(M)) -> U11(tt, M, N)
activate(X) -> X
encArg(tt) -> tt
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
encArg(cons_activate(x_1)) -> activate(encArg(x_1))
encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
encode_tt -> tt
encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
encode_activate(x_1) -> activate(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
encode_0 -> 0'

Types:
U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
tt :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
s :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
0' :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encArg :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_tt :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_s :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_0 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
hole_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate1_4 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4 :: Nat -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate

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

(89) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
U11, U12, plus, encArg

They will be analysed ascendingly in the following order:
U11 = U12
U11 = plus
U11 < encArg
U12 = plus
U12 < encArg
plus < encArg

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

(90)
Obligation:
TRS:
Rules:
U11(tt, M, N) -> U12(tt, activate(M), activate(N))
U12(tt, M, N) -> s(plus(activate(N), activate(M)))
plus(N, 0') -> N
plus(N, s(M)) -> U11(tt, M, N)
activate(X) -> X
encArg(tt) -> tt
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
encArg(cons_activate(x_1)) -> activate(encArg(x_1))
encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
encode_tt -> tt
encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
encode_activate(x_1) -> activate(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
encode_0 -> 0'

Types:
U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
tt :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
s :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
0' :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encArg :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_tt :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_s :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_0 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
hole_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate1_4 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4 :: Nat -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate


Generator Equations:
gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0) <=> tt
gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(x, 1)) <=> s(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(x))


The following defined symbols remain to be analysed:
U12, U11, plus, encArg

They will be analysed ascendingly in the following order:
U11 = U12
U11 = plus
U11 < encArg
U12 = plus
U12 < encArg
plus < encArg

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

(91) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U12(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n4_4), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c)) -> *3_4, rt in Omega(n4_4)

Induction Base:
U12(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c))

Induction Step:
U12(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(n4_4, 1)), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c)) ->_R^Omega(1)
s(plus(activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c)), activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(n4_4, 1))))) ->_R^Omega(1)
s(plus(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c), activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n4_4))))) ->_R^Omega(1)
s(plus(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n4_4)))) ->_R^Omega(1)
s(U11(tt, gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n4_4), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c))) ->_R^Omega(1)
s(U12(tt, activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n4_4)), activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c)))) ->_R^Omega(1)
s(U12(tt, gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n4_4), activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c)))) ->_R^Omega(1)
s(U12(tt, gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n4_4), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c))) ->_IH
s(*3_4)

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

(92)
Complex Obligation (BEST)

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

(93)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
U11(tt, M, N) -> U12(tt, activate(M), activate(N))
U12(tt, M, N) -> s(plus(activate(N), activate(M)))
plus(N, 0') -> N
plus(N, s(M)) -> U11(tt, M, N)
activate(X) -> X
encArg(tt) -> tt
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
encArg(cons_activate(x_1)) -> activate(encArg(x_1))
encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
encode_tt -> tt
encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
encode_activate(x_1) -> activate(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
encode_0 -> 0'

Types:
U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
tt :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
s :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
0' :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encArg :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_tt :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_s :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_0 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
hole_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate1_4 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4 :: Nat -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate


Generator Equations:
gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0) <=> tt
gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(x, 1)) <=> s(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(x))


The following defined symbols remain to be analysed:
U12, U11, plus, encArg

They will be analysed ascendingly in the following order:
U11 = U12
U11 = plus
U11 < encArg
U12 = plus
U12 < encArg
plus < encArg

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

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

(95)
BOUNDS(n^1, INF)

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

(96)
Obligation:
TRS:
Rules:
U11(tt, M, N) -> U12(tt, activate(M), activate(N))
U12(tt, M, N) -> s(plus(activate(N), activate(M)))
plus(N, 0') -> N
plus(N, s(M)) -> U11(tt, M, N)
activate(X) -> X
encArg(tt) -> tt
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
encArg(cons_activate(x_1)) -> activate(encArg(x_1))
encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
encode_tt -> tt
encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
encode_activate(x_1) -> activate(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
encode_0 -> 0'

Types:
U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
tt :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
s :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
0' :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encArg :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_tt :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_s :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_0 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
hole_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate1_4 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4 :: Nat -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate


Lemmas:
U12(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n4_4), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c)) -> *3_4, rt in Omega(n4_4)


Generator Equations:
gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0) <=> tt
gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(x, 1)) <=> s(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(x))


The following defined symbols remain to be analysed:
plus, U11, encArg

They will be analysed ascendingly in the following order:
U11 = U12
U11 = plus
U11 < encArg
U12 = plus
U12 < encArg
plus < encArg

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

(97) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n974_4))) -> *3_4, rt in Omega(n974_4)

Induction Base:
plus(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, 0)))

Induction Step:
plus(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, +(n974_4, 1)))) ->_R^Omega(1)
U11(tt, gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n974_4)), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a)) ->_R^Omega(1)
U12(tt, activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n974_4))), activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a))) ->_R^Omega(1)
U12(tt, gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n974_4)), activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a))) ->_R^Omega(1)
U12(tt, gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n974_4)), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a)) ->_R^Omega(1)
s(plus(activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a)), activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n974_4))))) ->_R^Omega(1)
s(plus(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a), activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n974_4))))) ->_R^Omega(1)
s(plus(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n974_4)))) ->_IH
s(*3_4)

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

(98)
Obligation:
TRS:
Rules:
U11(tt, M, N) -> U12(tt, activate(M), activate(N))
U12(tt, M, N) -> s(plus(activate(N), activate(M)))
plus(N, 0') -> N
plus(N, s(M)) -> U11(tt, M, N)
activate(X) -> X
encArg(tt) -> tt
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
encArg(cons_activate(x_1)) -> activate(encArg(x_1))
encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
encode_tt -> tt
encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
encode_activate(x_1) -> activate(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
encode_0 -> 0'

Types:
U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
tt :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
s :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
0' :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encArg :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_tt :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_s :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_0 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
hole_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate1_4 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4 :: Nat -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate


Lemmas:
U12(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n4_4), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c)) -> *3_4, rt in Omega(n4_4)
plus(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n974_4))) -> *3_4, rt in Omega(n974_4)


Generator Equations:
gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0) <=> tt
gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(x, 1)) <=> s(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(x))


The following defined symbols remain to be analysed:
U11, U12, encArg

They will be analysed ascendingly in the following order:
U11 = U12
U11 = plus
U11 < encArg
U12 = plus
U12 < encArg
plus < encArg

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

(99) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U11(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n2555_4), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c)) -> *3_4, rt in Omega(n2555_4)

Induction Base:
U11(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c))

Induction Step:
U11(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(n2555_4, 1)), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c)) ->_R^Omega(1)
U12(tt, activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(n2555_4, 1))), activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c))) ->_R^Omega(1)
U12(tt, gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n2555_4)), activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c))) ->_R^Omega(1)
U12(tt, gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n2555_4)), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c)) ->_R^Omega(1)
s(plus(activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c)), activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n2555_4))))) ->_R^Omega(1)
s(plus(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c), activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n2555_4))))) ->_R^Omega(1)
s(plus(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n2555_4)))) ->_R^Omega(1)
s(U11(tt, gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n2555_4), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c))) ->_IH
s(*3_4)

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

(100)
Obligation:
TRS:
Rules:
U11(tt, M, N) -> U12(tt, activate(M), activate(N))
U12(tt, M, N) -> s(plus(activate(N), activate(M)))
plus(N, 0') -> N
plus(N, s(M)) -> U11(tt, M, N)
activate(X) -> X
encArg(tt) -> tt
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
encArg(cons_activate(x_1)) -> activate(encArg(x_1))
encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
encode_tt -> tt
encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
encode_activate(x_1) -> activate(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
encode_0 -> 0'

Types:
U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
tt :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
s :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
0' :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encArg :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_tt :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_s :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_0 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
hole_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate1_4 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4 :: Nat -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate


Lemmas:
U12(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n4_4), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c)) -> *3_4, rt in Omega(n4_4)
plus(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n974_4))) -> *3_4, rt in Omega(n974_4)
U11(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n2555_4), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c)) -> *3_4, rt in Omega(n2555_4)


Generator Equations:
gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0) <=> tt
gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(x, 1)) <=> s(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(x))


The following defined symbols remain to be analysed:
U12, plus, encArg

They will be analysed ascendingly in the following order:
U11 = U12
U11 = plus
U11 < encArg
U12 = plus
U12 < encArg
plus < encArg

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

(101) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U12(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n4441_4), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c)) -> *3_4, rt in Omega(n4441_4)

Induction Base:
U12(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c))

Induction Step:
U12(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(n4441_4, 1)), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c)) ->_R^Omega(1)
s(plus(activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c)), activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(n4441_4, 1))))) ->_R^Omega(1)
s(plus(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c), activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n4441_4))))) ->_R^Omega(1)
s(plus(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n4441_4)))) ->_R^Omega(1)
s(U11(tt, gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n4441_4), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c))) ->_R^Omega(1)
s(U12(tt, activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n4441_4)), activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c)))) ->_R^Omega(1)
s(U12(tt, gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n4441_4), activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c)))) ->_R^Omega(1)
s(U12(tt, gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n4441_4), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c))) ->_IH
s(*3_4)

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

(102)
Obligation:
TRS:
Rules:
U11(tt, M, N) -> U12(tt, activate(M), activate(N))
U12(tt, M, N) -> s(plus(activate(N), activate(M)))
plus(N, 0') -> N
plus(N, s(M)) -> U11(tt, M, N)
activate(X) -> X
encArg(tt) -> tt
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
encArg(cons_activate(x_1)) -> activate(encArg(x_1))
encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
encode_tt -> tt
encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
encode_activate(x_1) -> activate(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
encode_0 -> 0'

Types:
U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
tt :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
s :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
0' :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encArg :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_tt :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_s :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_0 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
hole_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate1_4 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4 :: Nat -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate


Lemmas:
U12(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n4441_4), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c)) -> *3_4, rt in Omega(n4441_4)
plus(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n974_4))) -> *3_4, rt in Omega(n974_4)
U11(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n2555_4), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c)) -> *3_4, rt in Omega(n2555_4)


Generator Equations:
gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0) <=> tt
gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(x, 1)) <=> s(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(x))


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

They will be analysed ascendingly in the following order:
U11 = U12
U11 = plus
U11 < encArg
U12 = plus
U12 < encArg
plus < encArg

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

(103) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n6437_4))) -> *3_4, rt in Omega(n6437_4)

Induction Base:
plus(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, 0)))

Induction Step:
plus(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, +(n6437_4, 1)))) ->_R^Omega(1)
U11(tt, gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n6437_4)), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a)) ->_R^Omega(1)
U12(tt, activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n6437_4))), activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a))) ->_R^Omega(1)
U12(tt, gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n6437_4)), activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a))) ->_R^Omega(1)
U12(tt, gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n6437_4)), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a)) ->_R^Omega(1)
s(plus(activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a)), activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n6437_4))))) ->_R^Omega(1)
s(plus(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a), activate(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n6437_4))))) ->_R^Omega(1)
s(plus(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n6437_4)))) ->_IH
s(*3_4)

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

(104)
Obligation:
TRS:
Rules:
U11(tt, M, N) -> U12(tt, activate(M), activate(N))
U12(tt, M, N) -> s(plus(activate(N), activate(M)))
plus(N, 0') -> N
plus(N, s(M)) -> U11(tt, M, N)
activate(X) -> X
encArg(tt) -> tt
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_U11(x_1, x_2, x_3)) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_U12(x_1, x_2, x_3)) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
encArg(cons_activate(x_1)) -> activate(encArg(x_1))
encode_U11(x_1, x_2, x_3) -> U11(encArg(x_1), encArg(x_2), encArg(x_3))
encode_tt -> tt
encode_U12(x_1, x_2, x_3) -> U12(encArg(x_1), encArg(x_2), encArg(x_3))
encode_activate(x_1) -> activate(encArg(x_1))
encode_s(x_1) -> s(encArg(x_1))
encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
encode_0 -> 0'

Types:
U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
tt :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
s :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
0' :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encArg :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
cons_activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_U11 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_tt :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_U12 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_activate :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_s :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_plus :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
encode_0 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
hole_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate1_4 :: tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate
gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4 :: Nat -> tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate


Lemmas:
U12(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n4441_4), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c)) -> *3_4, rt in Omega(n4441_4)
plus(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(a), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(1, n6437_4))) -> *3_4, rt in Omega(n6437_4)
U11(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n2555_4), gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c)) -> *3_4, rt in Omega(n2555_4)


Generator Equations:
gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0) <=> tt
gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(x, 1)) <=> s(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(x))


The following defined symbols remain to be analysed:
encArg
----------------------------------------

(105) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n8718_4)) -> gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n8718_4), rt in Omega(0)

Induction Base:
encArg(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(0)) ->_R^Omega(0)
tt

Induction Step:
encArg(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(+(n8718_4, 1))) ->_R^Omega(0)
s(encArg(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(n8718_4))) ->_IH
s(gen_tt:s:0':cons_U11:cons_U12:cons_plus:cons_activate2_4(c8719_4))

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

(106)
BOUNDS(1, INF)