WORST_CASE(Omega(n^1),O(n^2))
proof of input_4iHNg7zAIv.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxWeightedTrs
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedTrs
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
    (15) NarrowingProof [BOTH BOUNDS(ID, ID), 26 ms]
    (16) CpxTypedWeightedCompleteTrs
    (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) InliningProof [UPPER BOUND(ID), 1567 ms]
    (20) CpxRNTS
    (21) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxRNTS
    (23) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 3 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 767 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 316 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 167 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 1909 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 226 ms]
    (42) CpxRNTS
    (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 4183 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 296 ms]
    (48) CpxRNTS
    (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 11.7 s]
    (52) CpxRNTS
    (53) IntTrsBoundProof [UPPER BOUND(ID), 5128 ms]
    (54) CpxRNTS
    (55) FinalProof [FINISHED, 0 ms]
    (56) BOUNDS(1, n^2)
(57) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(58) CdtProblem
    (59) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CpxRelTRS
    (61) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CpxRelTRS
    (63) TypeInferenceProof [BOTH BOUNDS(ID, ID), 1 ms]
    (64) typed CpxTrs
    (65) OrderProof [LOWER BOUND(ID), 0 ms]
    (66) typed CpxTrs
    (67) RewriteLemmaProof [LOWER BOUND(ID), 416 ms]
    (68) BEST
        (69) proven lower bound
            (70) LowerBoundPropagationProof [FINISHED, 0 ms]
            (71) BOUNDS(n^1, INF)
        (72) typed CpxTrs
            (73) RewriteLemmaProof [LOWER BOUND(ID), 84 ms]
            (74) typed CpxTrs
            (75) RewriteLemmaProof [LOWER BOUND(ID), 59 ms]
            (76) typed CpxTrs
            (77) RewriteLemmaProof [LOWER BOUND(ID), 556 ms]
            (78) BOUNDS(1, INF)


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS 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)))
   U21(tt, M, N) -> U22(tt, activate(M), activate(N))
   U22(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
   plus(N, 0) -> N
   plus(N, s(M)) -> U11(tt, M, N)
   x(N, 0) -> 0
   x(N, s(M)) -> U21(tt, M, N)
   activate(X) -> X

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

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1))
   U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0)))
   U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1))
   U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1))
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> U11(tt, z1, z0)
   x(z0, 0) -> 0
   x(z0, s(z1)) -> U21(tt, z1, z0)
   activate(z0) -> z0
Tuples:
   U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
   U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
   PLUS(z0, 0) -> c9
   PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
   X(z0, 0) -> c11
   X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
   ACTIVATE(z0) -> c13
S tuples:
   U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
   U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
   PLUS(z0, 0) -> c9
   PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
   X(z0, 0) -> c11
   X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
   ACTIVATE(z0) -> c13
K tuples:none
Defined Rule Symbols:   U11_3, U12_3, U21_3, U22_3, plus_2, x_2, activate_1

Defined Pair Symbols:   U11'_3, U12'_3, U21'_3, U22'_3, PLUS_2, X_2, ACTIVATE_1

Compound Symbols:   c_2, c1_2, c2_2, c3_2, c4_2, c5_2, c6_3, c7_3, c8_2, c9, c10_1, c11, c12_1, c13


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   ACTIVATE(z0) -> c13
   X(z0, 0) -> c11
   PLUS(z0, 0) -> c9

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1))
   U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0)))
   U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1))
   U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1))
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> U11(tt, z1, z0)
   x(z0, 0) -> 0
   x(z0, s(z1)) -> U21(tt, z1, z0)
   activate(z0) -> z0
Tuples:
   U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
   U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
   PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
   X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
S tuples:
   U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
   U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
   PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
   X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
K tuples:none
Defined Rule Symbols:   U11_3, U12_3, U21_3, U22_3, plus_2, x_2, activate_1

Defined Pair Symbols:   U11'_3, U12'_3, U21'_3, U22'_3, PLUS_2, X_2

Compound Symbols:   c_2, c1_2, c2_2, c3_2, c4_2, c5_2, c6_3, c7_3, c8_2, c10_1, c12_1


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

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 9 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1))
   U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0)))
   U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1))
   U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1))
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> U11(tt, z1, z0)
   x(z0, 0) -> 0
   x(z0, s(z1)) -> U21(tt, z1, z0)
   activate(z0) -> z0
Tuples:
   PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
   X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
   U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)))
   U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)))
   U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)))
   U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)))
   U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)))
   U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)))
   U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
   U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
   U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)))
S tuples:
   PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
   X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
   U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)))
   U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)))
   U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)))
   U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)))
   U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)))
   U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)))
   U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
   U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
   U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)))
K tuples:none
Defined Rule Symbols:   U11_3, U12_3, U21_3, U22_3, plus_2, x_2, activate_1

Defined Pair Symbols:   PLUS_2, X_2, U11'_3, U12'_3, U21'_3, U22'_3

Compound Symbols:   c10_1, c12_1, c_1, c1_1, c2_1, c3_1, c4_1, c5_1, c6_2, c7_2, c8_1


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

(7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(8)
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:

   PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
   X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
   U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)))
   U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)))
   U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)))
   U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)))
   U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)))
   U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)))
   U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
   U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
   U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)))

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

   U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1))
   U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0)))
   U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1))
   U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1))
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> U11(tt, z1, z0)
   x(z0, 0) -> 0
   x(z0, s(z1)) -> U21(tt, z1, z0)
   activate(z0) -> z0

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

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

(10)
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:

   PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0)) [1]
   X(z0, s(z1)) -> c12(U21'(tt, z1, z0)) [1]
   U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1))) [1]
   U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1))) [1]
   U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0))) [1]
   U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0))) [1]
   U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1))) [1]
   U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1))) [1]
   U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0))) [1]
   U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0))) [1]
   U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1))) [1]
   U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1)) [0]
   U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0))) [0]
   U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1)) [0]
   U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1)) [0]
   plus(z0, 0) -> z0 [0]
   plus(z0, s(z1)) -> U11(tt, z1, z0) [0]
   x(z0, 0) -> 0 [0]
   x(z0, s(z1)) -> U21(tt, z1, z0) [0]
   activate(z0) -> z0 [0]

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

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

(12)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0)) [1]
   X(z0, s(z1)) -> c12(U21'(tt, z1, z0)) [1]
   U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1))) [1]
   U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1))) [1]
   U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0))) [1]
   U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0))) [1]
   U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1))) [1]
   U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1))) [1]
   U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0))) [1]
   U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0))) [1]
   U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1))) [1]
   U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1)) [0]
   U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0))) [0]
   U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1)) [0]
   U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1)) [0]
   plus(z0, 0) -> z0 [0]
   plus(z0, s(z1)) -> U11(tt, z1, z0) [0]
   x(z0, 0) -> 0 [0]
   x(z0, s(z1)) -> U21(tt, z1, z0) [0]
   activate(z0) -> z0 [0]

The TRS has the following type information:
PLUS :: s:0 -> s:0 -> c10
s :: s:0 -> s:0
c10 :: c:c1 -> c10
U11' :: tt -> s:0 -> s:0 -> c:c1
tt :: tt
X :: s:0 -> s:0 -> c12
c12 :: c4:c5 -> c12
U21' :: tt -> s:0 -> s:0 -> c4:c5
c :: c2:c3 -> c:c1
U12' :: tt -> s:0 -> s:0 -> c2:c3
activate :: s:0 -> s:0
c1 :: c2:c3 -> c:c1
c2 :: c10 -> c2:c3
c3 :: c10 -> c2:c3
c4 :: c6:c7:c8 -> c4:c5
U22' :: tt -> s:0 -> s:0 -> c6:c7:c8
c5 :: c6:c7:c8 -> c4:c5
c6 :: c10 -> c12 -> c6:c7:c8
x :: s:0 -> s:0 -> s:0
c7 :: c10 -> c12 -> c6:c7:c8
c8 :: c10 -> c6:c7:c8
U11 :: tt -> s:0 -> s:0 -> s:0
U12 :: tt -> s:0 -> s:0 -> s:0
plus :: s:0 -> s:0 -> s:0
U21 :: tt -> s:0 -> s:0 -> s:0
U22 :: tt -> s:0 -> s:0 -> s:0
0 :: s:0

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

(13) 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:

   PLUS_2
   X_2
   U11'_3
   U12'_3
   U21'_3
   U22'_3

(c) The following functions are completely defined:

   U11_3
   U12_3
   U21_3
   U22_3
   plus_2
   x_2
   activate_1

Due to the following rules being added:

   U11(v0, v1, v2) -> 0 [0]
   U12(v0, v1, v2) -> 0 [0]
   U21(v0, v1, v2) -> 0 [0]
   U22(v0, v1, v2) -> 0 [0]
   plus(v0, v1) -> 0 [0]
   x(v0, v1) -> 0 [0]
   activate(v0) -> 0 [0]

And the following fresh constants:    const, const1, const2, const3, const4, const5

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

(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:

   PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0)) [1]
   X(z0, s(z1)) -> c12(U21'(tt, z1, z0)) [1]
   U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1))) [1]
   U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1))) [1]
   U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0))) [1]
   U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0))) [1]
   U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1))) [1]
   U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1))) [1]
   U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0))) [1]
   U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0))) [1]
   U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1))) [1]
   U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1)) [0]
   U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0))) [0]
   U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1)) [0]
   U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1)) [0]
   plus(z0, 0) -> z0 [0]
   plus(z0, s(z1)) -> U11(tt, z1, z0) [0]
   x(z0, 0) -> 0 [0]
   x(z0, s(z1)) -> U21(tt, z1, z0) [0]
   activate(z0) -> z0 [0]
   U11(v0, v1, v2) -> 0 [0]
   U12(v0, v1, v2) -> 0 [0]
   U21(v0, v1, v2) -> 0 [0]
   U22(v0, v1, v2) -> 0 [0]
   plus(v0, v1) -> 0 [0]
   x(v0, v1) -> 0 [0]
   activate(v0) -> 0 [0]

The TRS has the following type information:
PLUS :: s:0 -> s:0 -> c10
s :: s:0 -> s:0
c10 :: c:c1 -> c10
U11' :: tt -> s:0 -> s:0 -> c:c1
tt :: tt
X :: s:0 -> s:0 -> c12
c12 :: c4:c5 -> c12
U21' :: tt -> s:0 -> s:0 -> c4:c5
c :: c2:c3 -> c:c1
U12' :: tt -> s:0 -> s:0 -> c2:c3
activate :: s:0 -> s:0
c1 :: c2:c3 -> c:c1
c2 :: c10 -> c2:c3
c3 :: c10 -> c2:c3
c4 :: c6:c7:c8 -> c4:c5
U22' :: tt -> s:0 -> s:0 -> c6:c7:c8
c5 :: c6:c7:c8 -> c4:c5
c6 :: c10 -> c12 -> c6:c7:c8
x :: s:0 -> s:0 -> s:0
c7 :: c10 -> c12 -> c6:c7:c8
c8 :: c10 -> c6:c7:c8
U11 :: tt -> s:0 -> s:0 -> s:0
U12 :: tt -> s:0 -> s:0 -> s:0
plus :: s:0 -> s:0 -> s:0
U21 :: tt -> s:0 -> s:0 -> s:0
U22 :: tt -> s:0 -> s:0 -> s:0
0 :: s:0
const :: c10
const1 :: c:c1
const2 :: c12
const3 :: c4:c5
const4 :: c2:c3
const5 :: c6:c7:c8

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

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

(16)
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:

   PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0)) [1]
   X(z0, s(z1)) -> c12(U21'(tt, z1, z0)) [1]
   U11'(tt, z0, z1) -> c(U12'(tt, z0, z1)) [1]
   U11'(tt, z0, z1) -> c(U12'(tt, z0, 0)) [1]
   U11'(tt, z0, z1) -> c(U12'(tt, 0, z1)) [1]
   U11'(tt, z0, z1) -> c(U12'(tt, 0, 0)) [1]
   U11'(tt, z0, z1) -> c1(U12'(tt, z0, z1)) [1]
   U11'(tt, z0, z1) -> c1(U12'(tt, z0, 0)) [1]
   U11'(tt, z0, z1) -> c1(U12'(tt, 0, z1)) [1]
   U11'(tt, z0, z1) -> c1(U12'(tt, 0, 0)) [1]
   U12'(tt, z0, z1) -> c2(PLUS(z1, z0)) [1]
   U12'(tt, z0, z1) -> c2(PLUS(z1, 0)) [1]
   U12'(tt, z0, z1) -> c2(PLUS(0, z0)) [1]
   U12'(tt, z0, z1) -> c2(PLUS(0, 0)) [1]
   U12'(tt, z0, z1) -> c3(PLUS(z1, z0)) [1]
   U12'(tt, z0, z1) -> c3(PLUS(z1, 0)) [1]
   U12'(tt, z0, z1) -> c3(PLUS(0, z0)) [1]
   U12'(tt, z0, z1) -> c3(PLUS(0, 0)) [1]
   U21'(tt, z0, z1) -> c4(U22'(tt, z0, z1)) [1]
   U21'(tt, z0, z1) -> c4(U22'(tt, z0, 0)) [1]
   U21'(tt, z0, z1) -> c4(U22'(tt, 0, z1)) [1]
   U21'(tt, z0, z1) -> c4(U22'(tt, 0, 0)) [1]
   U21'(tt, z0, z1) -> c5(U22'(tt, z0, z1)) [1]
   U21'(tt, z0, z1) -> c5(U22'(tt, z0, 0)) [1]
   U21'(tt, z0, z1) -> c5(U22'(tt, 0, z1)) [1]
   U21'(tt, z0, z1) -> c5(U22'(tt, 0, 0)) [1]
   U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0))) [1]
   U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0))) [1]
   U22'(tt, z0, z1) -> c8(PLUS(x(z1, z0), z1)) [1]
   U22'(tt, z0, z1) -> c8(PLUS(x(z1, z0), 0)) [1]
   U22'(tt, z0, z1) -> c8(PLUS(x(z1, 0), z1)) [1]
   U22'(tt, z0, z1) -> c8(PLUS(x(z1, 0), 0)) [1]
   U22'(tt, z0, z1) -> c8(PLUS(x(0, z0), z1)) [1]
   U22'(tt, z0, z1) -> c8(PLUS(x(0, z0), 0)) [1]
   U22'(tt, z0, z1) -> c8(PLUS(x(0, 0), z1)) [1]
   U22'(tt, z0, z1) -> c8(PLUS(x(0, 0), 0)) [1]
   U11(tt, z0, z1) -> U12(tt, z0, z1) [0]
   U11(tt, z0, z1) -> U12(tt, z0, 0) [0]
   U11(tt, z0, z1) -> U12(tt, 0, z1) [0]
   U11(tt, z0, z1) -> U12(tt, 0, 0) [0]
   U12(tt, z0, z1) -> s(plus(z1, z0)) [0]
   U12(tt, z0, z1) -> s(plus(z1, 0)) [0]
   U12(tt, z0, z1) -> s(plus(0, z0)) [0]
   U12(tt, z0, z1) -> s(plus(0, 0)) [0]
   U21(tt, z0, z1) -> U22(tt, z0, z1) [0]
   U21(tt, z0, z1) -> U22(tt, z0, 0) [0]
   U21(tt, z0, z1) -> U22(tt, 0, z1) [0]
   U21(tt, z0, z1) -> U22(tt, 0, 0) [0]
   U22(tt, z0, z1) -> plus(x(z1, z0), z1) [0]
   U22(tt, z0, z1) -> plus(x(z1, z0), 0) [0]
   U22(tt, z0, z1) -> plus(x(z1, 0), z1) [0]
   U22(tt, z0, z1) -> plus(x(z1, 0), 0) [0]
   U22(tt, z0, z1) -> plus(x(0, z0), z1) [0]
   U22(tt, z0, z1) -> plus(x(0, z0), 0) [0]
   U22(tt, z0, z1) -> plus(x(0, 0), z1) [0]
   U22(tt, z0, z1) -> plus(x(0, 0), 0) [0]
   plus(z0, 0) -> z0 [0]
   plus(z0, s(z1)) -> U11(tt, z1, z0) [0]
   x(z0, 0) -> 0 [0]
   x(z0, s(z1)) -> U21(tt, z1, z0) [0]
   activate(z0) -> z0 [0]
   U11(v0, v1, v2) -> 0 [0]
   U12(v0, v1, v2) -> 0 [0]
   U21(v0, v1, v2) -> 0 [0]
   U22(v0, v1, v2) -> 0 [0]
   plus(v0, v1) -> 0 [0]
   x(v0, v1) -> 0 [0]
   activate(v0) -> 0 [0]

The TRS has the following type information:
PLUS :: s:0 -> s:0 -> c10
s :: s:0 -> s:0
c10 :: c:c1 -> c10
U11' :: tt -> s:0 -> s:0 -> c:c1
tt :: tt
X :: s:0 -> s:0 -> c12
c12 :: c4:c5 -> c12
U21' :: tt -> s:0 -> s:0 -> c4:c5
c :: c2:c3 -> c:c1
U12' :: tt -> s:0 -> s:0 -> c2:c3
activate :: s:0 -> s:0
c1 :: c2:c3 -> c:c1
c2 :: c10 -> c2:c3
c3 :: c10 -> c2:c3
c4 :: c6:c7:c8 -> c4:c5
U22' :: tt -> s:0 -> s:0 -> c6:c7:c8
c5 :: c6:c7:c8 -> c4:c5
c6 :: c10 -> c12 -> c6:c7:c8
x :: s:0 -> s:0 -> s:0
c7 :: c10 -> c12 -> c6:c7:c8
c8 :: c10 -> c6:c7:c8
U11 :: tt -> s:0 -> s:0 -> s:0
U12 :: tt -> s:0 -> s:0 -> s:0
plus :: s:0 -> s:0 -> s:0
U21 :: tt -> s:0 -> s:0 -> s:0
U22 :: tt -> s:0 -> s:0 -> s:0
0 :: s:0
const :: c10
const1 :: c:c1
const2 :: c12
const3 :: c4:c5
const4 :: c2:c3
const5 :: c6:c7:c8

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

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

   tt => 0
   0 => 0
   const => 0
   const1 => 0
   const2 => 0
   const3 => 0
   const4 => 0
   const5 => 0

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

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

   PLUS(z, z') -{ 1 }-> 1 + U11'(0, z1, z0) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   U11(z, z', z'') -{ 0 }-> U12(0, z0, z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U11(z, z', z'') -{ 0 }-> U12(0, z0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U11(z, z', z'') -{ 0 }-> U12(0, 0, z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U11(z, z', z'') -{ 0 }-> U12(0, 0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U11(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, z0, z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, z0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, 0, z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, 0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U12(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z1, z0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U12(z, z', z'') -{ 0 }-> 1 + plus(z1, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, z0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(z1, z0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(z1, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(0, z0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U21(z, z', z'') -{ 0 }-> U22(0, z0, z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U21(z, z', z'') -{ 0 }-> U22(0, z0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U21(z, z', z'') -{ 0 }-> U22(0, 0, z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U21(z, z', z'') -{ 0 }-> U22(0, 0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U21(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z0, z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22(z, z', z'') -{ 0 }-> plus(x(z1, z0), z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22(z, z', z'') -{ 0 }-> plus(x(z1, z0), 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22(z, z', z'') -{ 0 }-> plus(x(z1, 0), z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22(z, z', z'') -{ 0 }-> plus(x(z1, 0), 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22(z, z', z'') -{ 0 }-> plus(x(0, z0), z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22(z, z', z'') -{ 0 }-> plus(x(0, z0), 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z1, z0), z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z1, z0), 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z1, 0), z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z1, 0), 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z0), z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z0), 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(activate(z1), activate(z0)), activate(z1)) + X(activate(z1), activate(z0)) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   X(z, z') -{ 1 }-> 1 + U21'(0, z1, z0) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   activate(z) -{ 0 }-> z0 :|: z = z0, z0 >= 0
   activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   plus(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   plus(z, z') -{ 0 }-> U11(0, z1, z0) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   x(z, z') -{ 0 }-> U21(0, z1, z0) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   x(z, z') -{ 0 }-> 0 :|: z = z0, z0 >= 0, z' = 0
   x(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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

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

   activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   activate(z) -{ 0 }-> z0 :|: z = z0, z0 >= 0

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

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

   PLUS(z, z') -{ 1 }-> 1 + U11'(0, z1, z0) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   U11(z, z', z'') -{ 0 }-> U12(0, z0, z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U11(z, z', z'') -{ 0 }-> U12(0, z0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U11(z, z', z'') -{ 0 }-> U12(0, 0, z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U11(z, z', z'') -{ 0 }-> U12(0, 0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U11(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, z0, z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, z0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, 0, z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, 0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U12(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z1, z0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U12(z, z', z'') -{ 0 }-> 1 + plus(z1, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, z0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(z1, z0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(z1, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(0, z0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U21(z, z', z'') -{ 0 }-> U22(0, z0, z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U21(z, z', z'') -{ 0 }-> U22(0, z0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U21(z, z', z'') -{ 0 }-> U22(0, 0, z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U21(z, z', z'') -{ 0 }-> U22(0, 0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U21(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z0, z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22(z, z', z'') -{ 0 }-> plus(x(z1, z0), z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22(z, z', z'') -{ 0 }-> plus(x(z1, z0), 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22(z, z', z'') -{ 0 }-> plus(x(z1, 0), z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22(z, z', z'') -{ 0 }-> plus(x(z1, 0), 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22(z, z', z'') -{ 0 }-> plus(x(0, z0), z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22(z, z', z'') -{ 0 }-> plus(x(0, z0), 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z1, z0), z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z1, z0), 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z1, 0), z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z1, 0), 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z0), z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z0), 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z1) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z0', z0''), z01) + X(z02, z03) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, z1 = z0', z0' >= 0, z0 = z0'', z0'' >= 0, z1 = z01, z01 >= 0, z1 = z02, z02 >= 0, z0 = z03, z03 >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z0', z0''), z01) + X(z02, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, z1 = z0', z0' >= 0, z0 = z0'', z0'' >= 0, z1 = z01, z01 >= 0, z1 = z02, z02 >= 0, v0 >= 0, z0 = v0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z0', z0''), z01) + X(0, z02) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, z1 = z0', z0' >= 0, z0 = z0'', z0'' >= 0, z1 = z01, z01 >= 0, v0 >= 0, z1 = v0, z0 = z02, z02 >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z0', z0''), z01) + X(0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, z1 = z0', z0' >= 0, z0 = z0'', z0'' >= 0, z1 = z01, z01 >= 0, v0 >= 0, z1 = v0, v0' >= 0, z0 = v0'
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z0', z0''), 0) + X(z01, z02) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, z1 = z0', z0' >= 0, z0 = z0'', z0'' >= 0, v0 >= 0, z1 = v0, z1 = z01, z01 >= 0, z0 = z02, z02 >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z0', z0''), 0) + X(z01, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, z1 = z0', z0' >= 0, z0 = z0'', z0'' >= 0, v0 >= 0, z1 = v0, z1 = z01, z01 >= 0, v0' >= 0, z0 = v0'
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z0', z0''), 0) + X(0, z01) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, z1 = z0', z0' >= 0, z0 = z0'', z0'' >= 0, v0 >= 0, z1 = v0, v0' >= 0, z1 = v0', z0 = z01, z01 >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z0', z0''), 0) + X(0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, z1 = z0', z0' >= 0, z0 = z0'', z0'' >= 0, v0 >= 0, z1 = v0, v0' >= 0, z1 = v0', v0'' >= 0, z0 = v0''
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z0', 0), z0'') + X(z01, z02) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, z1 = z0', z0' >= 0, v0 >= 0, z0 = v0, z1 = z0'', z0'' >= 0, z1 = z01, z01 >= 0, z0 = z02, z02 >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z0', 0), z0'') + X(z01, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, z1 = z0', z0' >= 0, v0 >= 0, z0 = v0, z1 = z0'', z0'' >= 0, z1 = z01, z01 >= 0, v0' >= 0, z0 = v0'
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z0', 0), z0'') + X(0, z01) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, z1 = z0', z0' >= 0, v0 >= 0, z0 = v0, z1 = z0'', z0'' >= 0, v0' >= 0, z1 = v0', z0 = z01, z01 >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z0', 0), z0'') + X(0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, z1 = z0', z0' >= 0, v0 >= 0, z0 = v0, z1 = z0'', z0'' >= 0, v0' >= 0, z1 = v0', v0'' >= 0, z0 = v0''
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z0', 0), 0) + X(z0'', z01) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, z1 = z0', z0' >= 0, v0 >= 0, z0 = v0, v0' >= 0, z1 = v0', z1 = z0'', z0'' >= 0, z0 = z01, z01 >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z0', 0), 0) + X(z0'', 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, z1 = z0', z0' >= 0, v0 >= 0, z0 = v0, v0' >= 0, z1 = v0', z1 = z0'', z0'' >= 0, v0'' >= 0, z0 = v0''
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z0', 0), 0) + X(0, z0'') :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, z1 = z0', z0' >= 0, v0 >= 0, z0 = v0, v0' >= 0, z1 = v0', v0'' >= 0, z1 = v0'', z0 = z0'', z0'' >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z0', 0), 0) + X(0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, z1 = z0', z0' >= 0, v0 >= 0, z0 = v0, v0' >= 0, z1 = v0', v0'' >= 0, z1 = v0'', v01 >= 0, z0 = v01
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z0'), z0'') + X(z01, z02) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, v0 >= 0, z1 = v0, z0 = z0', z0' >= 0, z1 = z0'', z0'' >= 0, z1 = z01, z01 >= 0, z0 = z02, z02 >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z0'), z0'') + X(z01, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, v0 >= 0, z1 = v0, z0 = z0', z0' >= 0, z1 = z0'', z0'' >= 0, z1 = z01, z01 >= 0, v0' >= 0, z0 = v0'
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z0'), z0'') + X(0, z01) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, v0 >= 0, z1 = v0, z0 = z0', z0' >= 0, z1 = z0'', z0'' >= 0, v0' >= 0, z1 = v0', z0 = z01, z01 >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z0'), z0'') + X(0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, v0 >= 0, z1 = v0, z0 = z0', z0' >= 0, z1 = z0'', z0'' >= 0, v0' >= 0, z1 = v0', v0'' >= 0, z0 = v0''
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z0'), 0) + X(z0'', z01) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, v0 >= 0, z1 = v0, z0 = z0', z0' >= 0, v0' >= 0, z1 = v0', z1 = z0'', z0'' >= 0, z0 = z01, z01 >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z0'), 0) + X(z0'', 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, v0 >= 0, z1 = v0, z0 = z0', z0' >= 0, v0' >= 0, z1 = v0', z1 = z0'', z0'' >= 0, v0'' >= 0, z0 = v0''
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z0'), 0) + X(0, z0'') :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, v0 >= 0, z1 = v0, z0 = z0', z0' >= 0, v0' >= 0, z1 = v0', v0'' >= 0, z1 = v0'', z0 = z0'', z0'' >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z0'), 0) + X(0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, v0 >= 0, z1 = v0, z0 = z0', z0' >= 0, v0' >= 0, z1 = v0', v0'' >= 0, z1 = v0'', v01 >= 0, z0 = v01
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z0') + X(z0'', z01) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, v0 >= 0, z1 = v0, v0' >= 0, z0 = v0', z1 = z0', z0' >= 0, z1 = z0'', z0'' >= 0, z0 = z01, z01 >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z0') + X(z0'', 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, v0 >= 0, z1 = v0, v0' >= 0, z0 = v0', z1 = z0', z0' >= 0, z1 = z0'', z0'' >= 0, v0'' >= 0, z0 = v0''
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z0') + X(0, z0'') :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, v0 >= 0, z1 = v0, v0' >= 0, z0 = v0', z1 = z0', z0' >= 0, v0'' >= 0, z1 = v0'', z0 = z0'', z0'' >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z0') + X(0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, v0 >= 0, z1 = v0, v0' >= 0, z0 = v0', z1 = z0', z0' >= 0, v0'' >= 0, z1 = v0'', v01 >= 0, z0 = v01
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z0', z0'') :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, v0 >= 0, z1 = v0, v0' >= 0, z0 = v0', v0'' >= 0, z1 = v0'', z1 = z0', z0' >= 0, z0 = z0'', z0'' >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z0', 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, v0 >= 0, z1 = v0, v0' >= 0, z0 = v0', v0'' >= 0, z1 = v0'', z1 = z0', z0' >= 0, v01 >= 0, z0 = v01
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, z0') :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, v0 >= 0, z1 = v0, v0' >= 0, z0 = v0', v0'' >= 0, z1 = v0'', v01 >= 0, z1 = v01, z0 = z0', z0' >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, 0) :|: z1 >= 0, z0 >= 0, z = 0, z' = z0, z'' = z1, v0 >= 0, z1 = v0, v0' >= 0, z0 = v0', v0'' >= 0, z1 = v0'', v01 >= 0, z1 = v01, v02 >= 0, z0 = v02
   X(z, z') -{ 1 }-> 1 + U21'(0, z1, z0) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   activate(z) -{ 0 }-> z0 :|: z = z0, z0 >= 0
   activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   plus(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   plus(z, z') -{ 0 }-> U11(0, z1, z0) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   x(z, z') -{ 0 }-> U21(0, z1, z0) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   x(z, z') -{ 0 }-> 0 :|: z = z0, z0 >= 0, z' = 0
   x(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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

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

   PLUS(z, z') -{ 1 }-> 1 + U11'(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   U11(z, z', z'') -{ 0 }-> U12(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   X(z, z') -{ 1 }-> 1 + U21'(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> U11(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   x(z, z') -{ 0 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0


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

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

   { U11', PLUS, U12' }
   { activate }
   { plus, U12, U11 }
   { U22, x, U21 }
   { U21', U22', X }

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

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

   PLUS(z, z') -{ 1 }-> 1 + U11'(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   U11(z, z', z'') -{ 0 }-> U12(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   X(z, z') -{ 1 }-> 1 + U21'(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> U11(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   x(z, z') -{ 0 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {U11',PLUS,U12'}, {activate}, {plus,U12,U11}, {U22,x,U21}, {U21',U22',X}

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

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

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

   PLUS(z, z') -{ 1 }-> 1 + U11'(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   U11(z, z', z'') -{ 0 }-> U12(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   X(z, z') -{ 1 }-> 1 + U21'(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> U11(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   x(z, z') -{ 0 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {U11',PLUS,U12'}, {activate}, {plus,U12,U11}, {U22,x,U21}, {U21',U22',X}

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: U11'
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

Computed SIZE bound using CoFloCo for: U12'
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

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

   PLUS(z, z') -{ 1 }-> 1 + U11'(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   U11(z, z', z'') -{ 0 }-> U12(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   X(z, z') -{ 1 }-> 1 + U21'(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> U11(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   x(z, z') -{ 0 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {U11',PLUS,U12'}, {activate}, {plus,U12,U11}, {U22,x,U21}, {U21',U22',X}
Previous analysis results are:
  U11': runtime: ?, size: O(1) [0]
  PLUS: runtime: ?, size: O(1) [1]
  U12': runtime: ?, size: O(1) [2]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: U11'
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + 3*z'

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

Computed RUNTIME bound using CoFloCo for: U12'
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + 3*z'

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

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

   PLUS(z, z') -{ 1 }-> 1 + U11'(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   U11(z, z', z'') -{ 0 }-> U12(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 1 }-> 1 + U12'(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + PLUS(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   X(z, z') -{ 1 }-> 1 + U21'(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> U11(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   x(z, z') -{ 0 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {activate}, {plus,U12,U11}, {U22,x,U21}, {U21',U22',X}
Previous analysis results are:
  U11': runtime: O(n^1) [2 + 3*z'], size: O(1) [0]
  PLUS: runtime: O(n^1) [3*z'], size: O(1) [1]
  U12': runtime: O(n^1) [1 + 3*z'], size: O(1) [2]

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

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

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

   PLUS(z, z') -{ 3*z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z >= 0
   U11(z, z', z'') -{ 0 }-> U12(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s' :|: s' >= 0, s' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s'' :|: s'' >= 0, s'' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s1 :|: s1 >= 0, s1 <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s2 :|: s2 >= 0, s2 <= 2, z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s6 :|: s6 >= 0, s6 <= 1, z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   X(z, z') -{ 1 }-> 1 + U21'(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> U11(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   x(z, z') -{ 0 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {activate}, {plus,U12,U11}, {U22,x,U21}, {U21',U22',X}
Previous analysis results are:
  U11': runtime: O(n^1) [2 + 3*z'], size: O(1) [0]
  PLUS: runtime: O(n^1) [3*z'], size: O(1) [1]
  U12': runtime: O(n^1) [1 + 3*z'], size: O(1) [2]

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

(33) 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

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

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

   PLUS(z, z') -{ 3*z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z >= 0
   U11(z, z', z'') -{ 0 }-> U12(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s' :|: s' >= 0, s' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s'' :|: s'' >= 0, s'' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s1 :|: s1 >= 0, s1 <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s2 :|: s2 >= 0, s2 <= 2, z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s6 :|: s6 >= 0, s6 <= 1, z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   X(z, z') -{ 1 }-> 1 + U21'(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> U11(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   x(z, z') -{ 0 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {activate}, {plus,U12,U11}, {U22,x,U21}, {U21',U22',X}
Previous analysis results are:
  U11': runtime: O(n^1) [2 + 3*z'], size: O(1) [0]
  PLUS: runtime: O(n^1) [3*z'], size: O(1) [1]
  U12': runtime: O(n^1) [1 + 3*z'], size: O(1) [2]
  activate: runtime: ?, size: O(n^1) [z]

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

(35) 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: 0

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

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

   PLUS(z, z') -{ 3*z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z >= 0
   U11(z, z', z'') -{ 0 }-> U12(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s' :|: s' >= 0, s' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s'' :|: s'' >= 0, s'' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s1 :|: s1 >= 0, s1 <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s2 :|: s2 >= 0, s2 <= 2, z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s6 :|: s6 >= 0, s6 <= 1, z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   X(z, z') -{ 1 }-> 1 + U21'(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> U11(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   x(z, z') -{ 0 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {plus,U12,U11}, {U22,x,U21}, {U21',U22',X}
Previous analysis results are:
  U11': runtime: O(n^1) [2 + 3*z'], size: O(1) [0]
  PLUS: runtime: O(n^1) [3*z'], size: O(1) [1]
  U12': runtime: O(n^1) [1 + 3*z'], size: O(1) [2]
  activate: runtime: O(1) [0], size: O(n^1) [z]

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

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

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

   PLUS(z, z') -{ 3*z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z >= 0
   U11(z, z', z'') -{ 0 }-> U12(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s' :|: s' >= 0, s' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s'' :|: s'' >= 0, s'' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s1 :|: s1 >= 0, s1 <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s2 :|: s2 >= 0, s2 <= 2, z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s6 :|: s6 >= 0, s6 <= 1, z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   X(z, z') -{ 1 }-> 1 + U21'(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> U11(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   x(z, z') -{ 0 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {plus,U12,U11}, {U22,x,U21}, {U21',U22',X}
Previous analysis results are:
  U11': runtime: O(n^1) [2 + 3*z'], size: O(1) [0]
  PLUS: runtime: O(n^1) [3*z'], size: O(1) [1]
  U12': runtime: O(n^1) [1 + 3*z'], size: O(1) [2]
  activate: runtime: O(1) [0], size: O(n^1) [z]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT 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''

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

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

   PLUS(z, z') -{ 3*z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z >= 0
   U11(z, z', z'') -{ 0 }-> U12(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s' :|: s' >= 0, s' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s'' :|: s'' >= 0, s'' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s1 :|: s1 >= 0, s1 <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s2 :|: s2 >= 0, s2 <= 2, z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s6 :|: s6 >= 0, s6 <= 1, z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   X(z, z') -{ 1 }-> 1 + U21'(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> U11(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   x(z, z') -{ 0 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {plus,U12,U11}, {U22,x,U21}, {U21',U22',X}
Previous analysis results are:
  U11': runtime: O(n^1) [2 + 3*z'], size: O(1) [0]
  PLUS: runtime: O(n^1) [3*z'], size: O(1) [1]
  U12': runtime: O(n^1) [1 + 3*z'], size: O(1) [2]
  activate: runtime: O(1) [0], size: O(n^1) [z]
  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'']

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed RUNTIME bound using CoFloCo for: U12
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed RUNTIME bound using CoFloCo for: U11
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   PLUS(z, z') -{ 3*z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z >= 0
   U11(z, z', z'') -{ 0 }-> U12(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> U12(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s' :|: s' >= 0, s' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s'' :|: s'' >= 0, s'' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s1 :|: s1 >= 0, s1 <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s2 :|: s2 >= 0, s2 <= 2, z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + plus(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s6 :|: s6 >= 0, s6 <= 1, z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   X(z, z') -{ 1 }-> 1 + U21'(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> U11(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   x(z, z') -{ 0 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {U22,x,U21}, {U21',U22',X}
Previous analysis results are:
  U11': runtime: O(n^1) [2 + 3*z'], size: O(1) [0]
  PLUS: runtime: O(n^1) [3*z'], size: O(1) [1]
  U12': runtime: O(n^1) [1 + 3*z'], size: O(1) [2]
  activate: runtime: O(1) [0], size: O(n^1) [z]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  U12: runtime: O(1) [0], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(1) [0], size: O(n^1) [1 + z' + z'']

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

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

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

   PLUS(z, z') -{ 3*z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z >= 0
   U11(z, z', z'') -{ 0 }-> s10 :|: s10 >= 0, s10 <= 0 + 0 + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> s7 :|: s7 >= 0, s7 <= z' + z'' + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> s8 :|: s8 >= 0, s8 <= z' + 0 + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 0 + z'' + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s' :|: s' >= 0, s' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s'' :|: s'' >= 0, s'' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s1 :|: s1 >= 0, s1 <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s2 :|: s2 >= 0, s2 <= 2, z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 1 + s11 :|: s11 >= 0, s11 <= z'' + z', z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= z'' + 0, z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= 0 + z', z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + s14 :|: s14 >= 0, s14 <= 0 + 0, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s6 :|: s6 >= 0, s6 <= 1, z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   X(z, z') -{ 1 }-> 1 + U21'(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   x(z, z') -{ 0 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {U22,x,U21}, {U21',U22',X}
Previous analysis results are:
  U11': runtime: O(n^1) [2 + 3*z'], size: O(1) [0]
  PLUS: runtime: O(n^1) [3*z'], size: O(1) [1]
  U12': runtime: O(n^1) [1 + 3*z'], size: O(1) [2]
  activate: runtime: O(1) [0], size: O(n^1) [z]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  U12: runtime: O(1) [0], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(1) [0], size: O(n^1) [1 + z' + z'']

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: U22
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: z'*z'' + 2*z''

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

Computed SIZE bound using KoAT for: U21
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: z'*z'' + 4*z''

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

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

   PLUS(z, z') -{ 3*z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z >= 0
   U11(z, z', z'') -{ 0 }-> s10 :|: s10 >= 0, s10 <= 0 + 0 + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> s7 :|: s7 >= 0, s7 <= z' + z'' + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> s8 :|: s8 >= 0, s8 <= z' + 0 + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 0 + z'' + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s' :|: s' >= 0, s' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s'' :|: s'' >= 0, s'' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s1 :|: s1 >= 0, s1 <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s2 :|: s2 >= 0, s2 <= 2, z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 1 + s11 :|: s11 >= 0, s11 <= z'' + z', z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= z'' + 0, z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= 0 + z', z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + s14 :|: s14 >= 0, s14 <= 0 + 0, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s6 :|: s6 >= 0, s6 <= 1, z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   X(z, z') -{ 1 }-> 1 + U21'(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   x(z, z') -{ 0 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {U22,x,U21}, {U21',U22',X}
Previous analysis results are:
  U11': runtime: O(n^1) [2 + 3*z'], size: O(1) [0]
  PLUS: runtime: O(n^1) [3*z'], size: O(1) [1]
  U12': runtime: O(n^1) [1 + 3*z'], size: O(1) [2]
  activate: runtime: O(1) [0], size: O(n^1) [z]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  U12: runtime: O(1) [0], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(1) [0], size: O(n^1) [1 + z' + z'']
  U22: runtime: ?, size: O(n^2) [z'*z'' + 2*z'']
  x: runtime: ?, size: O(n^2) [4*z + z*z']
  U21: runtime: ?, size: O(n^2) [z'*z'' + 4*z'']

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: U22
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

Computed RUNTIME bound using CoFloCo for: U21
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   PLUS(z, z') -{ 3*z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z >= 0
   U11(z, z', z'') -{ 0 }-> s10 :|: s10 >= 0, s10 <= 0 + 0 + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> s7 :|: s7 >= 0, s7 <= z' + z'' + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> s8 :|: s8 >= 0, s8 <= z' + 0 + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 0 + z'' + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s' :|: s' >= 0, s' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s'' :|: s'' >= 0, s'' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s1 :|: s1 >= 0, s1 <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s2 :|: s2 >= 0, s2 <= 2, z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 1 + s11 :|: s11 >= 0, s11 <= z'' + z', z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= z'' + 0, z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= 0 + z', z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + s14 :|: s14 >= 0, s14 <= 0 + 0, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s6 :|: s6 >= 0, s6 <= 1, z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> U22(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> plus(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(z'', 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, z'), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), z'') + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(z'', 0) :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, z') :|: z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + PLUS(x(0, 0), 0) + X(0, 0) :|: z'' >= 0, z' >= 0, z = 0
   X(z, z') -{ 1 }-> 1 + U21'(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   x(z, z') -{ 0 }-> U21(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {U21',U22',X}
Previous analysis results are:
  U11': runtime: O(n^1) [2 + 3*z'], size: O(1) [0]
  PLUS: runtime: O(n^1) [3*z'], size: O(1) [1]
  U12': runtime: O(n^1) [1 + 3*z'], size: O(1) [2]
  activate: runtime: O(1) [0], size: O(n^1) [z]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  U12: runtime: O(1) [0], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(1) [0], size: O(n^1) [1 + z' + z'']
  U22: runtime: O(1) [0], size: O(n^2) [z'*z'' + 2*z'']
  x: runtime: O(1) [0], size: O(n^2) [4*z + z*z']
  U21: runtime: O(1) [0], size: O(n^2) [z'*z'' + 4*z'']

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

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

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

   PLUS(z, z') -{ 3*z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z >= 0
   U11(z, z', z'') -{ 0 }-> s10 :|: s10 >= 0, s10 <= 0 + 0 + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> s7 :|: s7 >= 0, s7 <= z' + z'' + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> s8 :|: s8 >= 0, s8 <= z' + 0 + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 0 + z'' + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s' :|: s' >= 0, s' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s'' :|: s'' >= 0, s'' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s1 :|: s1 >= 0, s1 <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s2 :|: s2 >= 0, s2 <= 2, z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 1 + s11 :|: s11 >= 0, s11 <= z'' + z', z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= z'' + 0, z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= 0 + z', z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + s14 :|: s14 >= 0, s14 <= 0 + 0, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s6 :|: s6 >= 0, s6 <= 1, z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> s32 :|: s32 >= 0, s32 <= 2 * z'' + z' * z'', z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 2 * 0 + z' * 0, z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> s34 :|: s34 >= 0, s34 <= 2 * z'' + 0 * z'', z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 2 * 0 + 0 * 0, z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s37 :|: s36 >= 0, s36 <= 4 * z'' + z' * z'', s37 >= 0, s37 <= s36 + z'', z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s39 :|: s38 >= 0, s38 <= 4 * z'' + z' * z'', s39 >= 0, s39 <= s38 + 0, z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s41 :|: s40 >= 0, s40 <= 4 * z'' + 0 * z'', s41 >= 0, s41 <= s40 + z'', z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s43 :|: s42 >= 0, s42 <= 4 * z'' + 0 * z'', s43 >= 0, s43 <= s42 + 0, z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s45 :|: s44 >= 0, s44 <= 4 * 0 + z' * 0, s45 >= 0, s45 <= s44 + z'', z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s47 :|: s46 >= 0, s46 <= 4 * 0 + z' * 0, s47 >= 0, s47 <= s46 + 0, z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s49 :|: s48 >= 0, s48 <= 4 * 0 + 0 * 0, s49 >= 0, s49 <= s48 + z'', z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s51 :|: s50 >= 0, s50 <= 4 * 0 + 0 * 0, s51 >= 0, s51 <= s50 + 0, z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s17 :|: s16 >= 0, s16 <= 4 * z'' + z' * z'', s17 >= 0, s17 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s19 :|: s18 >= 0, s18 <= 4 * z'' + z' * z'', s19 >= 0, s19 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s21 :|: s20 >= 0, s20 <= 4 * z'' + 0 * z'', s21 >= 0, s21 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s23 :|: s22 >= 0, s22 <= 4 * z'' + 0 * z'', s23 >= 0, s23 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s25 :|: s24 >= 0, s24 <= 4 * 0 + z' * 0, s25 >= 0, s25 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s27 :|: s26 >= 0, s26 <= 4 * 0 + z' * 0, s27 >= 0, s27 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s29 :|: s28 >= 0, s28 <= 4 * 0 + 0 * 0, s29 >= 0, s29 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s31 :|: s30 >= 0, s30 <= 4 * 0 + 0 * 0, s31 >= 0, s31 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s100 + X(z'', z') :|: s99 >= 0, s99 <= 4 * z'' + 0 * z'', s100 >= 0, s100 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s102 + X(0, 0) :|: s101 >= 0, s101 <= 4 * z'' + z' * z'', s102 >= 0, s102 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s104 + X(0, z') :|: s103 >= 0, s103 <= 4 * z'' + z' * z'', s104 >= 0, s104 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s106 + X(z'', 0) :|: s105 >= 0, s105 <= 4 * z'' + z' * z'', s106 >= 0, s106 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s108 + X(z'', z') :|: s107 >= 0, s107 <= 4 * z'' + z' * z'', s108 >= 0, s108 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s110 + X(0, 0) :|: s109 >= 0, s109 <= 4 * z'' + z' * z'', s110 >= 0, s110 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s112 + X(0, z') :|: s111 >= 0, s111 <= 4 * z'' + z' * z'', s112 >= 0, s112 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s114 + X(z'', 0) :|: s113 >= 0, s113 <= 4 * z'' + z' * z'', s114 >= 0, s114 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s116 + X(z'', z') :|: s115 >= 0, s115 <= 4 * z'' + z' * z'', s116 >= 0, s116 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s54 + X(0, 0) :|: s53 >= 0, s53 <= 4 * 0 + 0 * 0, s54 >= 0, s54 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s56 + X(0, z') :|: s55 >= 0, s55 <= 4 * 0 + 0 * 0, s56 >= 0, s56 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s58 + X(z'', 0) :|: s57 >= 0, s57 <= 4 * 0 + 0 * 0, s58 >= 0, s58 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s60 + X(z'', z') :|: s59 >= 0, s59 <= 4 * 0 + 0 * 0, s60 >= 0, s60 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s62 + X(0, 0) :|: s61 >= 0, s61 <= 4 * 0 + 0 * 0, s62 >= 0, s62 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s64 + X(0, z') :|: s63 >= 0, s63 <= 4 * 0 + 0 * 0, s64 >= 0, s64 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s66 + X(z'', 0) :|: s65 >= 0, s65 <= 4 * 0 + 0 * 0, s66 >= 0, s66 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s68 + X(z'', z') :|: s67 >= 0, s67 <= 4 * 0 + 0 * 0, s68 >= 0, s68 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s70 + X(0, 0) :|: s69 >= 0, s69 <= 4 * 0 + z' * 0, s70 >= 0, s70 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s72 + X(0, z') :|: s71 >= 0, s71 <= 4 * 0 + z' * 0, s72 >= 0, s72 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s74 + X(z'', 0) :|: s73 >= 0, s73 <= 4 * 0 + z' * 0, s74 >= 0, s74 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s76 + X(z'', z') :|: s75 >= 0, s75 <= 4 * 0 + z' * 0, s76 >= 0, s76 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s78 + X(0, 0) :|: s77 >= 0, s77 <= 4 * 0 + z' * 0, s78 >= 0, s78 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s80 + X(0, z') :|: s79 >= 0, s79 <= 4 * 0 + z' * 0, s80 >= 0, s80 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s82 + X(z'', 0) :|: s81 >= 0, s81 <= 4 * 0 + z' * 0, s82 >= 0, s82 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s84 + X(z'', z') :|: s83 >= 0, s83 <= 4 * 0 + z' * 0, s84 >= 0, s84 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s86 + X(0, 0) :|: s85 >= 0, s85 <= 4 * z'' + 0 * z'', s86 >= 0, s86 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s88 + X(0, z') :|: s87 >= 0, s87 <= 4 * z'' + 0 * z'', s88 >= 0, s88 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s90 + X(z'', 0) :|: s89 >= 0, s89 <= 4 * z'' + 0 * z'', s90 >= 0, s90 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s92 + X(z'', z') :|: s91 >= 0, s91 <= 4 * z'' + 0 * z'', s92 >= 0, s92 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s94 + X(0, 0) :|: s93 >= 0, s93 <= 4 * z'' + 0 * z'', s94 >= 0, s94 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s96 + X(0, z') :|: s95 >= 0, s95 <= 4 * z'' + 0 * z'', s96 >= 0, s96 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s98 + X(z'', 0) :|: s97 >= 0, s97 <= 4 * z'' + 0 * z'', s98 >= 0, s98 <= 1, z'' >= 0, z' >= 0, z = 0
   X(z, z') -{ 1 }-> 1 + U21'(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   x(z, z') -{ 0 }-> s52 :|: s52 >= 0, s52 <= 4 * z + z * (z' - 1), z' - 1 >= 0, z >= 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {U21',U22',X}
Previous analysis results are:
  U11': runtime: O(n^1) [2 + 3*z'], size: O(1) [0]
  PLUS: runtime: O(n^1) [3*z'], size: O(1) [1]
  U12': runtime: O(n^1) [1 + 3*z'], size: O(1) [2]
  activate: runtime: O(1) [0], size: O(n^1) [z]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  U12: runtime: O(1) [0], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(1) [0], size: O(n^1) [1 + z' + z'']
  U22: runtime: O(1) [0], size: O(n^2) [z'*z'' + 2*z'']
  x: runtime: O(1) [0], size: O(n^2) [4*z + z*z']
  U21: runtime: O(1) [0], size: O(n^2) [z'*z'' + 4*z'']

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: U21'
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + 4*z'

Computed SIZE bound using CoFloCo for: U22'
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + 4*z'

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

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

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

   PLUS(z, z') -{ 3*z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z >= 0
   U11(z, z', z'') -{ 0 }-> s10 :|: s10 >= 0, s10 <= 0 + 0 + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> s7 :|: s7 >= 0, s7 <= z' + z'' + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> s8 :|: s8 >= 0, s8 <= z' + 0 + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 0 + z'' + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s' :|: s' >= 0, s' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s'' :|: s'' >= 0, s'' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s1 :|: s1 >= 0, s1 <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s2 :|: s2 >= 0, s2 <= 2, z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 1 + s11 :|: s11 >= 0, s11 <= z'' + z', z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= z'' + 0, z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= 0 + z', z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + s14 :|: s14 >= 0, s14 <= 0 + 0, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s6 :|: s6 >= 0, s6 <= 1, z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> s32 :|: s32 >= 0, s32 <= 2 * z'' + z' * z'', z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 2 * 0 + z' * 0, z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> s34 :|: s34 >= 0, s34 <= 2 * z'' + 0 * z'', z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 2 * 0 + 0 * 0, z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s37 :|: s36 >= 0, s36 <= 4 * z'' + z' * z'', s37 >= 0, s37 <= s36 + z'', z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s39 :|: s38 >= 0, s38 <= 4 * z'' + z' * z'', s39 >= 0, s39 <= s38 + 0, z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s41 :|: s40 >= 0, s40 <= 4 * z'' + 0 * z'', s41 >= 0, s41 <= s40 + z'', z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s43 :|: s42 >= 0, s42 <= 4 * z'' + 0 * z'', s43 >= 0, s43 <= s42 + 0, z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s45 :|: s44 >= 0, s44 <= 4 * 0 + z' * 0, s45 >= 0, s45 <= s44 + z'', z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s47 :|: s46 >= 0, s46 <= 4 * 0 + z' * 0, s47 >= 0, s47 <= s46 + 0, z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s49 :|: s48 >= 0, s48 <= 4 * 0 + 0 * 0, s49 >= 0, s49 <= s48 + z'', z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s51 :|: s50 >= 0, s50 <= 4 * 0 + 0 * 0, s51 >= 0, s51 <= s50 + 0, z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s17 :|: s16 >= 0, s16 <= 4 * z'' + z' * z'', s17 >= 0, s17 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s19 :|: s18 >= 0, s18 <= 4 * z'' + z' * z'', s19 >= 0, s19 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s21 :|: s20 >= 0, s20 <= 4 * z'' + 0 * z'', s21 >= 0, s21 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s23 :|: s22 >= 0, s22 <= 4 * z'' + 0 * z'', s23 >= 0, s23 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s25 :|: s24 >= 0, s24 <= 4 * 0 + z' * 0, s25 >= 0, s25 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s27 :|: s26 >= 0, s26 <= 4 * 0 + z' * 0, s27 >= 0, s27 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s29 :|: s28 >= 0, s28 <= 4 * 0 + 0 * 0, s29 >= 0, s29 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s31 :|: s30 >= 0, s30 <= 4 * 0 + 0 * 0, s31 >= 0, s31 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s100 + X(z'', z') :|: s99 >= 0, s99 <= 4 * z'' + 0 * z'', s100 >= 0, s100 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s102 + X(0, 0) :|: s101 >= 0, s101 <= 4 * z'' + z' * z'', s102 >= 0, s102 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s104 + X(0, z') :|: s103 >= 0, s103 <= 4 * z'' + z' * z'', s104 >= 0, s104 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s106 + X(z'', 0) :|: s105 >= 0, s105 <= 4 * z'' + z' * z'', s106 >= 0, s106 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s108 + X(z'', z') :|: s107 >= 0, s107 <= 4 * z'' + z' * z'', s108 >= 0, s108 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s110 + X(0, 0) :|: s109 >= 0, s109 <= 4 * z'' + z' * z'', s110 >= 0, s110 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s112 + X(0, z') :|: s111 >= 0, s111 <= 4 * z'' + z' * z'', s112 >= 0, s112 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s114 + X(z'', 0) :|: s113 >= 0, s113 <= 4 * z'' + z' * z'', s114 >= 0, s114 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s116 + X(z'', z') :|: s115 >= 0, s115 <= 4 * z'' + z' * z'', s116 >= 0, s116 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s54 + X(0, 0) :|: s53 >= 0, s53 <= 4 * 0 + 0 * 0, s54 >= 0, s54 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s56 + X(0, z') :|: s55 >= 0, s55 <= 4 * 0 + 0 * 0, s56 >= 0, s56 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s58 + X(z'', 0) :|: s57 >= 0, s57 <= 4 * 0 + 0 * 0, s58 >= 0, s58 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s60 + X(z'', z') :|: s59 >= 0, s59 <= 4 * 0 + 0 * 0, s60 >= 0, s60 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s62 + X(0, 0) :|: s61 >= 0, s61 <= 4 * 0 + 0 * 0, s62 >= 0, s62 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s64 + X(0, z') :|: s63 >= 0, s63 <= 4 * 0 + 0 * 0, s64 >= 0, s64 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s66 + X(z'', 0) :|: s65 >= 0, s65 <= 4 * 0 + 0 * 0, s66 >= 0, s66 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s68 + X(z'', z') :|: s67 >= 0, s67 <= 4 * 0 + 0 * 0, s68 >= 0, s68 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s70 + X(0, 0) :|: s69 >= 0, s69 <= 4 * 0 + z' * 0, s70 >= 0, s70 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s72 + X(0, z') :|: s71 >= 0, s71 <= 4 * 0 + z' * 0, s72 >= 0, s72 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s74 + X(z'', 0) :|: s73 >= 0, s73 <= 4 * 0 + z' * 0, s74 >= 0, s74 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s76 + X(z'', z') :|: s75 >= 0, s75 <= 4 * 0 + z' * 0, s76 >= 0, s76 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s78 + X(0, 0) :|: s77 >= 0, s77 <= 4 * 0 + z' * 0, s78 >= 0, s78 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s80 + X(0, z') :|: s79 >= 0, s79 <= 4 * 0 + z' * 0, s80 >= 0, s80 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s82 + X(z'', 0) :|: s81 >= 0, s81 <= 4 * 0 + z' * 0, s82 >= 0, s82 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s84 + X(z'', z') :|: s83 >= 0, s83 <= 4 * 0 + z' * 0, s84 >= 0, s84 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s86 + X(0, 0) :|: s85 >= 0, s85 <= 4 * z'' + 0 * z'', s86 >= 0, s86 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s88 + X(0, z') :|: s87 >= 0, s87 <= 4 * z'' + 0 * z'', s88 >= 0, s88 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s90 + X(z'', 0) :|: s89 >= 0, s89 <= 4 * z'' + 0 * z'', s90 >= 0, s90 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s92 + X(z'', z') :|: s91 >= 0, s91 <= 4 * z'' + 0 * z'', s92 >= 0, s92 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s94 + X(0, 0) :|: s93 >= 0, s93 <= 4 * z'' + 0 * z'', s94 >= 0, s94 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s96 + X(0, z') :|: s95 >= 0, s95 <= 4 * z'' + 0 * z'', s96 >= 0, s96 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s98 + X(z'', 0) :|: s97 >= 0, s97 <= 4 * z'' + 0 * z'', s98 >= 0, s98 <= 1, z'' >= 0, z' >= 0, z = 0
   X(z, z') -{ 1 }-> 1 + U21'(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   x(z, z') -{ 0 }-> s52 :|: s52 >= 0, s52 <= 4 * z + z * (z' - 1), z' - 1 >= 0, z >= 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {U21',U22',X}
Previous analysis results are:
  U11': runtime: O(n^1) [2 + 3*z'], size: O(1) [0]
  PLUS: runtime: O(n^1) [3*z'], size: O(1) [1]
  U12': runtime: O(n^1) [1 + 3*z'], size: O(1) [2]
  activate: runtime: O(1) [0], size: O(n^1) [z]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  U12: runtime: O(1) [0], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(1) [0], size: O(n^1) [1 + z' + z'']
  U22: runtime: O(1) [0], size: O(n^2) [z'*z'' + 2*z'']
  x: runtime: O(1) [0], size: O(n^2) [4*z + z*z']
  U21: runtime: O(1) [0], size: O(n^2) [z'*z'' + 4*z'']
  U21': runtime: ?, size: O(n^1) [3 + 4*z']
  U22': runtime: ?, size: O(n^1) [2 + 4*z']
  X: runtime: ?, size: O(n^1) [4*z']

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: U21'
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 82 + 111*z' + 144*z'*z'' + 108*z''

Computed RUNTIME bound using KoAT for: U22'
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 40 + 1776*z' + 2304*z'*z'' + 60*z''

Computed RUNTIME bound using KoAT for: X
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 144*z*z' + 111*z'

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

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

   PLUS(z, z') -{ 3*z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z >= 0
   U11(z, z', z'') -{ 0 }-> s10 :|: s10 >= 0, s10 <= 0 + 0 + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> s7 :|: s7 >= 0, s7 <= z' + z'' + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> s8 :|: s8 >= 0, s8 <= z' + 0 + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 0 + z'' + 1, z'' >= 0, z' >= 0, z = 0
   U11(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s' :|: s' >= 0, s' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 + 3*z' }-> 1 + s'' :|: s'' >= 0, s'' <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s1 :|: s1 >= 0, s1 <= 2, z'' >= 0, z' >= 0, z = 0
   U11'(z, z', z'') -{ 2 }-> 1 + s2 :|: s2 >= 0, s2 <= 2, z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U12(z, z', z'') -{ 0 }-> 1 + s11 :|: s11 >= 0, s11 <= z'' + z', z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= z'' + 0, z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= 0 + z', z'' >= 0, z' >= 0, z = 0
   U12(z, z', z'') -{ 0 }-> 1 + s14 :|: s14 >= 0, s14 <= 0 + 0, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 + 3*z' }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z'' >= 0, z' >= 0, z = 0
   U12'(z, z', z'') -{ 1 }-> 1 + s6 :|: s6 >= 0, s6 <= 1, z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> s32 :|: s32 >= 0, s32 <= 2 * z'' + z' * z'', z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 2 * 0 + z' * 0, z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> s34 :|: s34 >= 0, s34 <= 2 * z'' + 0 * z'', z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 2 * 0 + 0 * 0, z'' >= 0, z' >= 0, z = 0
   U21(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, z', 0) :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, z'') :|: z'' >= 0, z' >= 0, z = 0
   U21'(z, z', z'') -{ 1 }-> 1 + U22'(0, 0, 0) :|: z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s37 :|: s36 >= 0, s36 <= 4 * z'' + z' * z'', s37 >= 0, s37 <= s36 + z'', z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s39 :|: s38 >= 0, s38 <= 4 * z'' + z' * z'', s39 >= 0, s39 <= s38 + 0, z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s41 :|: s40 >= 0, s40 <= 4 * z'' + 0 * z'', s41 >= 0, s41 <= s40 + z'', z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s43 :|: s42 >= 0, s42 <= 4 * z'' + 0 * z'', s43 >= 0, s43 <= s42 + 0, z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s45 :|: s44 >= 0, s44 <= 4 * 0 + z' * 0, s45 >= 0, s45 <= s44 + z'', z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s47 :|: s46 >= 0, s46 <= 4 * 0 + z' * 0, s47 >= 0, s47 <= s46 + 0, z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s49 :|: s48 >= 0, s48 <= 4 * 0 + 0 * 0, s49 >= 0, s49 <= s48 + z'', z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> s51 :|: s50 >= 0, s50 <= 4 * 0 + 0 * 0, s51 >= 0, s51 <= s50 + 0, z'' >= 0, z' >= 0, z = 0
   U22(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s17 :|: s16 >= 0, s16 <= 4 * z'' + z' * z'', s17 >= 0, s17 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s19 :|: s18 >= 0, s18 <= 4 * z'' + z' * z'', s19 >= 0, s19 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s21 :|: s20 >= 0, s20 <= 4 * z'' + 0 * z'', s21 >= 0, s21 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s23 :|: s22 >= 0, s22 <= 4 * z'' + 0 * z'', s23 >= 0, s23 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s25 :|: s24 >= 0, s24 <= 4 * 0 + z' * 0, s25 >= 0, s25 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s27 :|: s26 >= 0, s26 <= 4 * 0 + z' * 0, s27 >= 0, s27 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s29 :|: s28 >= 0, s28 <= 4 * 0 + 0 * 0, s29 >= 0, s29 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s31 :|: s30 >= 0, s30 <= 4 * 0 + 0 * 0, s31 >= 0, s31 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s100 + X(z'', z') :|: s99 >= 0, s99 <= 4 * z'' + 0 * z'', s100 >= 0, s100 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s102 + X(0, 0) :|: s101 >= 0, s101 <= 4 * z'' + z' * z'', s102 >= 0, s102 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s104 + X(0, z') :|: s103 >= 0, s103 <= 4 * z'' + z' * z'', s104 >= 0, s104 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s106 + X(z'', 0) :|: s105 >= 0, s105 <= 4 * z'' + z' * z'', s106 >= 0, s106 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s108 + X(z'', z') :|: s107 >= 0, s107 <= 4 * z'' + z' * z'', s108 >= 0, s108 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s110 + X(0, 0) :|: s109 >= 0, s109 <= 4 * z'' + z' * z'', s110 >= 0, s110 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s112 + X(0, z') :|: s111 >= 0, s111 <= 4 * z'' + z' * z'', s112 >= 0, s112 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s114 + X(z'', 0) :|: s113 >= 0, s113 <= 4 * z'' + z' * z'', s114 >= 0, s114 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s116 + X(z'', z') :|: s115 >= 0, s115 <= 4 * z'' + z' * z'', s116 >= 0, s116 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s54 + X(0, 0) :|: s53 >= 0, s53 <= 4 * 0 + 0 * 0, s54 >= 0, s54 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s56 + X(0, z') :|: s55 >= 0, s55 <= 4 * 0 + 0 * 0, s56 >= 0, s56 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s58 + X(z'', 0) :|: s57 >= 0, s57 <= 4 * 0 + 0 * 0, s58 >= 0, s58 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s60 + X(z'', z') :|: s59 >= 0, s59 <= 4 * 0 + 0 * 0, s60 >= 0, s60 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s62 + X(0, 0) :|: s61 >= 0, s61 <= 4 * 0 + 0 * 0, s62 >= 0, s62 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s64 + X(0, z') :|: s63 >= 0, s63 <= 4 * 0 + 0 * 0, s64 >= 0, s64 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s66 + X(z'', 0) :|: s65 >= 0, s65 <= 4 * 0 + 0 * 0, s66 >= 0, s66 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s68 + X(z'', z') :|: s67 >= 0, s67 <= 4 * 0 + 0 * 0, s68 >= 0, s68 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s70 + X(0, 0) :|: s69 >= 0, s69 <= 4 * 0 + z' * 0, s70 >= 0, s70 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s72 + X(0, z') :|: s71 >= 0, s71 <= 4 * 0 + z' * 0, s72 >= 0, s72 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s74 + X(z'', 0) :|: s73 >= 0, s73 <= 4 * 0 + z' * 0, s74 >= 0, s74 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s76 + X(z'', z') :|: s75 >= 0, s75 <= 4 * 0 + z' * 0, s76 >= 0, s76 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s78 + X(0, 0) :|: s77 >= 0, s77 <= 4 * 0 + z' * 0, s78 >= 0, s78 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s80 + X(0, z') :|: s79 >= 0, s79 <= 4 * 0 + z' * 0, s80 >= 0, s80 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s82 + X(z'', 0) :|: s81 >= 0, s81 <= 4 * 0 + z' * 0, s82 >= 0, s82 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s84 + X(z'', z') :|: s83 >= 0, s83 <= 4 * 0 + z' * 0, s84 >= 0, s84 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s86 + X(0, 0) :|: s85 >= 0, s85 <= 4 * z'' + 0 * z'', s86 >= 0, s86 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s88 + X(0, z') :|: s87 >= 0, s87 <= 4 * z'' + 0 * z'', s88 >= 0, s88 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s90 + X(z'', 0) :|: s89 >= 0, s89 <= 4 * z'' + 0 * z'', s90 >= 0, s90 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 }-> 1 + s92 + X(z'', z') :|: s91 >= 0, s91 <= 4 * z'' + 0 * z'', s92 >= 0, s92 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s94 + X(0, 0) :|: s93 >= 0, s93 <= 4 * z'' + 0 * z'', s94 >= 0, s94 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s96 + X(0, z') :|: s95 >= 0, s95 <= 4 * z'' + 0 * z'', s96 >= 0, s96 <= 1, z'' >= 0, z' >= 0, z = 0
   U22'(z, z', z'') -{ 1 + 3*z'' }-> 1 + s98 + X(z'', 0) :|: s97 >= 0, s97 <= 4 * z'' + 0 * z'', s98 >= 0, s98 <= 1, z'' >= 0, z' >= 0, z = 0
   X(z, z') -{ 1 }-> 1 + U21'(0, z' - 1, z) :|: z' - 1 >= 0, z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= z' - 1 + z + 1, z' - 1 >= 0, z >= 0
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   x(z, z') -{ 0 }-> s52 :|: s52 >= 0, s52 <= 4 * z + z * (z' - 1), z' - 1 >= 0, z >= 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   x(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  U11': runtime: O(n^1) [2 + 3*z'], size: O(1) [0]
  PLUS: runtime: O(n^1) [3*z'], size: O(1) [1]
  U12': runtime: O(n^1) [1 + 3*z'], size: O(1) [2]
  activate: runtime: O(1) [0], size: O(n^1) [z]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  U12: runtime: O(1) [0], size: O(n^1) [1 + z' + z'']
  U11: runtime: O(1) [0], size: O(n^1) [1 + z' + z'']
  U22: runtime: O(1) [0], size: O(n^2) [z'*z'' + 2*z'']
  x: runtime: O(1) [0], size: O(n^2) [4*z + z*z']
  U21: runtime: O(1) [0], size: O(n^2) [z'*z'' + 4*z'']
  U21': runtime: O(n^2) [82 + 111*z' + 144*z'*z'' + 108*z''], size: O(n^1) [3 + 4*z']
  U22': runtime: O(n^2) [40 + 1776*z' + 2304*z'*z'' + 60*z''], size: O(n^1) [2 + 4*z']
  X: runtime: O(n^2) [144*z*z' + 111*z'], size: O(n^1) [4*z']

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

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

(56)
BOUNDS(1, n^2)

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

(57) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1))
   U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0)))
   U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1))
   U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1))
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> U11(tt, z1, z0)
   x(z0, 0) -> 0
   x(z0, s(z1)) -> U21(tt, z1, z0)
   activate(z0) -> z0
Tuples:
   U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
   U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
   PLUS(z0, 0) -> c9
   PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
   X(z0, 0) -> c11
   X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
   ACTIVATE(z0) -> c13
S tuples:
   U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
   U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
   PLUS(z0, 0) -> c9
   PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
   X(z0, 0) -> c11
   X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
   ACTIVATE(z0) -> c13
K tuples:none
Defined Rule Symbols:   U11_3, U12_3, U21_3, U22_3, plus_2, x_2, activate_1

Defined Pair Symbols:   U11'_3, U12'_3, U21'_3, U22'_3, PLUS_2, X_2, ACTIVATE_1

Compound Symbols:   c_2, c1_2, c2_2, c3_2, c4_2, c5_2, c6_3, c7_3, c8_2, c9, c10_1, c11, c12_1, c13


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

(59) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(60)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
   U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
   PLUS(z0, 0) -> c9
   PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
   X(z0, 0) -> c11
   X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
   ACTIVATE(z0) -> c13

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

   U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1))
   U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0)))
   U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1))
   U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1))
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> U11(tt, z1, z0)
   x(z0, 0) -> 0
   x(z0, s(z1)) -> U21(tt, z1, z0)
   activate(z0) -> z0

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

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

(62)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
   U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
   U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
   U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
   U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
   PLUS(z0, 0') -> c9
   PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
   X(z0, 0') -> c11
   X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
   ACTIVATE(z0) -> c13

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

   U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1))
   U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0)))
   U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1))
   U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1))
   plus(z0, 0') -> z0
   plus(z0, s(z1)) -> U11(tt, z1, z0)
   x(z0, 0') -> 0'
   x(z0, s(z1)) -> U21(tt, z1, z0)
   activate(z0) -> z0

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

(63) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(64)
Obligation:
Innermost TRS:
Rules:
U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
PLUS(z0, 0') -> c9
PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
X(z0, 0') -> c11
X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
ACTIVATE(z0) -> c13
U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0') -> z0
plus(z0, s(z1)) -> U11(tt, z1, z0)
x(z0, 0') -> 0'
x(z0, s(z1)) -> U21(tt, z1, z0)
activate(z0) -> z0

Types:
U11' :: tt -> 0':s -> 0':s -> c:c1
tt :: tt
c :: c2:c3 -> c13 -> c:c1
U12' :: tt -> 0':s -> 0':s -> c2:c3
activate :: 0':s -> 0':s
ACTIVATE :: 0':s -> c13
c1 :: c2:c3 -> c13 -> c:c1
c2 :: c9:c10 -> c13 -> c2:c3
PLUS :: 0':s -> 0':s -> c9:c10
c3 :: c9:c10 -> c13 -> c2:c3
U21' :: tt -> 0':s -> 0':s -> c4:c5
c4 :: c6:c7:c8 -> c13 -> c4:c5
U22' :: tt -> 0':s -> 0':s -> c6:c7:c8
c5 :: c6:c7:c8 -> c13 -> c4:c5
c6 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
x :: 0':s -> 0':s -> 0':s
X :: 0':s -> 0':s -> c11:c12
c7 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
c8 :: c9:c10 -> c13 -> c6:c7:c8
0' :: 0':s
c9 :: c9:c10
s :: 0':s -> 0':s
c10 :: c:c1 -> c9:c10
c11 :: c11:c12
c12 :: c4:c5 -> c11:c12
c13 :: c13
U11 :: tt -> 0':s -> 0':s -> 0':s
U12 :: tt -> 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
U21 :: tt -> 0':s -> 0':s -> 0':s
U22 :: tt -> 0':s -> 0':s -> 0':s
hole_c:c11_14 :: c:c1
hole_tt2_14 :: tt
hole_0':s3_14 :: 0':s
hole_c2:c34_14 :: c2:c3
hole_c135_14 :: c13
hole_c9:c106_14 :: c9:c10
hole_c4:c57_14 :: c4:c5
hole_c6:c7:c88_14 :: c6:c7:c8
hole_c11:c129_14 :: c11:c12
gen_0':s10_14 :: Nat -> 0':s

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

(65) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
PLUS, x, X, plus

They will be analysed ascendingly in the following order:
PLUS < X
x < X
plus < x

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

(66)
Obligation:
Innermost TRS:
Rules:
U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
PLUS(z0, 0') -> c9
PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
X(z0, 0') -> c11
X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
ACTIVATE(z0) -> c13
U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0') -> z0
plus(z0, s(z1)) -> U11(tt, z1, z0)
x(z0, 0') -> 0'
x(z0, s(z1)) -> U21(tt, z1, z0)
activate(z0) -> z0

Types:
U11' :: tt -> 0':s -> 0':s -> c:c1
tt :: tt
c :: c2:c3 -> c13 -> c:c1
U12' :: tt -> 0':s -> 0':s -> c2:c3
activate :: 0':s -> 0':s
ACTIVATE :: 0':s -> c13
c1 :: c2:c3 -> c13 -> c:c1
c2 :: c9:c10 -> c13 -> c2:c3
PLUS :: 0':s -> 0':s -> c9:c10
c3 :: c9:c10 -> c13 -> c2:c3
U21' :: tt -> 0':s -> 0':s -> c4:c5
c4 :: c6:c7:c8 -> c13 -> c4:c5
U22' :: tt -> 0':s -> 0':s -> c6:c7:c8
c5 :: c6:c7:c8 -> c13 -> c4:c5
c6 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
x :: 0':s -> 0':s -> 0':s
X :: 0':s -> 0':s -> c11:c12
c7 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
c8 :: c9:c10 -> c13 -> c6:c7:c8
0' :: 0':s
c9 :: c9:c10
s :: 0':s -> 0':s
c10 :: c:c1 -> c9:c10
c11 :: c11:c12
c12 :: c4:c5 -> c11:c12
c13 :: c13
U11 :: tt -> 0':s -> 0':s -> 0':s
U12 :: tt -> 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
U21 :: tt -> 0':s -> 0':s -> 0':s
U22 :: tt -> 0':s -> 0':s -> 0':s
hole_c:c11_14 :: c:c1
hole_tt2_14 :: tt
hole_0':s3_14 :: 0':s
hole_c2:c34_14 :: c2:c3
hole_c135_14 :: c13
hole_c9:c106_14 :: c9:c10
hole_c4:c57_14 :: c4:c5
hole_c6:c7:c88_14 :: c6:c7:c8
hole_c11:c129_14 :: c11:c12
gen_0':s10_14 :: Nat -> 0':s


Generator Equations:
gen_0':s10_14(0) <=> 0'
gen_0':s10_14(+(x, 1)) <=> s(gen_0':s10_14(x))


The following defined symbols remain to be analysed:
PLUS, x, X, plus

They will be analysed ascendingly in the following order:
PLUS < X
x < X
plus < x

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

(67) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
PLUS(gen_0':s10_14(a), gen_0':s10_14(n12_14)) -> *11_14, rt in Omega(n12_14)

Induction Base:
PLUS(gen_0':s10_14(a), gen_0':s10_14(0))

Induction Step:
PLUS(gen_0':s10_14(a), gen_0':s10_14(+(n12_14, 1))) ->_R^Omega(1)
c10(U11'(tt, gen_0':s10_14(n12_14), gen_0':s10_14(a))) ->_R^Omega(1)
c10(c(U12'(tt, activate(gen_0':s10_14(n12_14)), activate(gen_0':s10_14(a))), ACTIVATE(gen_0':s10_14(n12_14)))) ->_R^Omega(0)
c10(c(U12'(tt, gen_0':s10_14(n12_14), activate(gen_0':s10_14(a))), ACTIVATE(gen_0':s10_14(n12_14)))) ->_R^Omega(0)
c10(c(U12'(tt, gen_0':s10_14(n12_14), gen_0':s10_14(a)), ACTIVATE(gen_0':s10_14(n12_14)))) ->_R^Omega(1)
c10(c(c2(PLUS(activate(gen_0':s10_14(a)), activate(gen_0':s10_14(n12_14))), ACTIVATE(gen_0':s10_14(a))), ACTIVATE(gen_0':s10_14(n12_14)))) ->_R^Omega(0)
c10(c(c2(PLUS(gen_0':s10_14(a), activate(gen_0':s10_14(n12_14))), ACTIVATE(gen_0':s10_14(a))), ACTIVATE(gen_0':s10_14(n12_14)))) ->_R^Omega(0)
c10(c(c2(PLUS(gen_0':s10_14(a), gen_0':s10_14(n12_14)), ACTIVATE(gen_0':s10_14(a))), ACTIVATE(gen_0':s10_14(n12_14)))) ->_IH
c10(c(c2(*11_14, ACTIVATE(gen_0':s10_14(a))), ACTIVATE(gen_0':s10_14(n12_14)))) ->_R^Omega(1)
c10(c(c2(*11_14, c13), ACTIVATE(gen_0':s10_14(n12_14)))) ->_R^Omega(1)
c10(c(c2(*11_14, c13), c13))

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

(68)
Complex Obligation (BEST)

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

(69)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
PLUS(z0, 0') -> c9
PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
X(z0, 0') -> c11
X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
ACTIVATE(z0) -> c13
U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0') -> z0
plus(z0, s(z1)) -> U11(tt, z1, z0)
x(z0, 0') -> 0'
x(z0, s(z1)) -> U21(tt, z1, z0)
activate(z0) -> z0

Types:
U11' :: tt -> 0':s -> 0':s -> c:c1
tt :: tt
c :: c2:c3 -> c13 -> c:c1
U12' :: tt -> 0':s -> 0':s -> c2:c3
activate :: 0':s -> 0':s
ACTIVATE :: 0':s -> c13
c1 :: c2:c3 -> c13 -> c:c1
c2 :: c9:c10 -> c13 -> c2:c3
PLUS :: 0':s -> 0':s -> c9:c10
c3 :: c9:c10 -> c13 -> c2:c3
U21' :: tt -> 0':s -> 0':s -> c4:c5
c4 :: c6:c7:c8 -> c13 -> c4:c5
U22' :: tt -> 0':s -> 0':s -> c6:c7:c8
c5 :: c6:c7:c8 -> c13 -> c4:c5
c6 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
x :: 0':s -> 0':s -> 0':s
X :: 0':s -> 0':s -> c11:c12
c7 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
c8 :: c9:c10 -> c13 -> c6:c7:c8
0' :: 0':s
c9 :: c9:c10
s :: 0':s -> 0':s
c10 :: c:c1 -> c9:c10
c11 :: c11:c12
c12 :: c4:c5 -> c11:c12
c13 :: c13
U11 :: tt -> 0':s -> 0':s -> 0':s
U12 :: tt -> 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
U21 :: tt -> 0':s -> 0':s -> 0':s
U22 :: tt -> 0':s -> 0':s -> 0':s
hole_c:c11_14 :: c:c1
hole_tt2_14 :: tt
hole_0':s3_14 :: 0':s
hole_c2:c34_14 :: c2:c3
hole_c135_14 :: c13
hole_c9:c106_14 :: c9:c10
hole_c4:c57_14 :: c4:c5
hole_c6:c7:c88_14 :: c6:c7:c8
hole_c11:c129_14 :: c11:c12
gen_0':s10_14 :: Nat -> 0':s


Generator Equations:
gen_0':s10_14(0) <=> 0'
gen_0':s10_14(+(x, 1)) <=> s(gen_0':s10_14(x))


The following defined symbols remain to be analysed:
PLUS, x, X, plus

They will be analysed ascendingly in the following order:
PLUS < X
x < X
plus < x

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

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

(71)
BOUNDS(n^1, INF)

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

(72)
Obligation:
Innermost TRS:
Rules:
U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
PLUS(z0, 0') -> c9
PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
X(z0, 0') -> c11
X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
ACTIVATE(z0) -> c13
U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0') -> z0
plus(z0, s(z1)) -> U11(tt, z1, z0)
x(z0, 0') -> 0'
x(z0, s(z1)) -> U21(tt, z1, z0)
activate(z0) -> z0

Types:
U11' :: tt -> 0':s -> 0':s -> c:c1
tt :: tt
c :: c2:c3 -> c13 -> c:c1
U12' :: tt -> 0':s -> 0':s -> c2:c3
activate :: 0':s -> 0':s
ACTIVATE :: 0':s -> c13
c1 :: c2:c3 -> c13 -> c:c1
c2 :: c9:c10 -> c13 -> c2:c3
PLUS :: 0':s -> 0':s -> c9:c10
c3 :: c9:c10 -> c13 -> c2:c3
U21' :: tt -> 0':s -> 0':s -> c4:c5
c4 :: c6:c7:c8 -> c13 -> c4:c5
U22' :: tt -> 0':s -> 0':s -> c6:c7:c8
c5 :: c6:c7:c8 -> c13 -> c4:c5
c6 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
x :: 0':s -> 0':s -> 0':s
X :: 0':s -> 0':s -> c11:c12
c7 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
c8 :: c9:c10 -> c13 -> c6:c7:c8
0' :: 0':s
c9 :: c9:c10
s :: 0':s -> 0':s
c10 :: c:c1 -> c9:c10
c11 :: c11:c12
c12 :: c4:c5 -> c11:c12
c13 :: c13
U11 :: tt -> 0':s -> 0':s -> 0':s
U12 :: tt -> 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
U21 :: tt -> 0':s -> 0':s -> 0':s
U22 :: tt -> 0':s -> 0':s -> 0':s
hole_c:c11_14 :: c:c1
hole_tt2_14 :: tt
hole_0':s3_14 :: 0':s
hole_c2:c34_14 :: c2:c3
hole_c135_14 :: c13
hole_c9:c106_14 :: c9:c10
hole_c4:c57_14 :: c4:c5
hole_c6:c7:c88_14 :: c6:c7:c8
hole_c11:c129_14 :: c11:c12
gen_0':s10_14 :: Nat -> 0':s


Lemmas:
PLUS(gen_0':s10_14(a), gen_0':s10_14(n12_14)) -> *11_14, rt in Omega(n12_14)


Generator Equations:
gen_0':s10_14(0) <=> 0'
gen_0':s10_14(+(x, 1)) <=> s(gen_0':s10_14(x))


The following defined symbols remain to be analysed:
plus, x, X

They will be analysed ascendingly in the following order:
x < X
plus < x

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

(73) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s10_14(a), gen_0':s10_14(n2625_14)) -> gen_0':s10_14(+(n2625_14, a)), rt in Omega(0)

Induction Base:
plus(gen_0':s10_14(a), gen_0':s10_14(0)) ->_R^Omega(0)
gen_0':s10_14(a)

Induction Step:
plus(gen_0':s10_14(a), gen_0':s10_14(+(n2625_14, 1))) ->_R^Omega(0)
U11(tt, gen_0':s10_14(n2625_14), gen_0':s10_14(a)) ->_R^Omega(0)
U12(tt, activate(gen_0':s10_14(n2625_14)), activate(gen_0':s10_14(a))) ->_R^Omega(0)
U12(tt, gen_0':s10_14(n2625_14), activate(gen_0':s10_14(a))) ->_R^Omega(0)
U12(tt, gen_0':s10_14(n2625_14), gen_0':s10_14(a)) ->_R^Omega(0)
s(plus(activate(gen_0':s10_14(a)), activate(gen_0':s10_14(n2625_14)))) ->_R^Omega(0)
s(plus(gen_0':s10_14(a), activate(gen_0':s10_14(n2625_14)))) ->_R^Omega(0)
s(plus(gen_0':s10_14(a), gen_0':s10_14(n2625_14))) ->_IH
s(gen_0':s10_14(+(a, c2626_14)))

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

(74)
Obligation:
Innermost TRS:
Rules:
U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
PLUS(z0, 0') -> c9
PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
X(z0, 0') -> c11
X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
ACTIVATE(z0) -> c13
U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0') -> z0
plus(z0, s(z1)) -> U11(tt, z1, z0)
x(z0, 0') -> 0'
x(z0, s(z1)) -> U21(tt, z1, z0)
activate(z0) -> z0

Types:
U11' :: tt -> 0':s -> 0':s -> c:c1
tt :: tt
c :: c2:c3 -> c13 -> c:c1
U12' :: tt -> 0':s -> 0':s -> c2:c3
activate :: 0':s -> 0':s
ACTIVATE :: 0':s -> c13
c1 :: c2:c3 -> c13 -> c:c1
c2 :: c9:c10 -> c13 -> c2:c3
PLUS :: 0':s -> 0':s -> c9:c10
c3 :: c9:c10 -> c13 -> c2:c3
U21' :: tt -> 0':s -> 0':s -> c4:c5
c4 :: c6:c7:c8 -> c13 -> c4:c5
U22' :: tt -> 0':s -> 0':s -> c6:c7:c8
c5 :: c6:c7:c8 -> c13 -> c4:c5
c6 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
x :: 0':s -> 0':s -> 0':s
X :: 0':s -> 0':s -> c11:c12
c7 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
c8 :: c9:c10 -> c13 -> c6:c7:c8
0' :: 0':s
c9 :: c9:c10
s :: 0':s -> 0':s
c10 :: c:c1 -> c9:c10
c11 :: c11:c12
c12 :: c4:c5 -> c11:c12
c13 :: c13
U11 :: tt -> 0':s -> 0':s -> 0':s
U12 :: tt -> 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
U21 :: tt -> 0':s -> 0':s -> 0':s
U22 :: tt -> 0':s -> 0':s -> 0':s
hole_c:c11_14 :: c:c1
hole_tt2_14 :: tt
hole_0':s3_14 :: 0':s
hole_c2:c34_14 :: c2:c3
hole_c135_14 :: c13
hole_c9:c106_14 :: c9:c10
hole_c4:c57_14 :: c4:c5
hole_c6:c7:c88_14 :: c6:c7:c8
hole_c11:c129_14 :: c11:c12
gen_0':s10_14 :: Nat -> 0':s


Lemmas:
PLUS(gen_0':s10_14(a), gen_0':s10_14(n12_14)) -> *11_14, rt in Omega(n12_14)
plus(gen_0':s10_14(a), gen_0':s10_14(n2625_14)) -> gen_0':s10_14(+(n2625_14, a)), rt in Omega(0)


Generator Equations:
gen_0':s10_14(0) <=> 0'
gen_0':s10_14(+(x, 1)) <=> s(gen_0':s10_14(x))


The following defined symbols remain to be analysed:
x, X

They will be analysed ascendingly in the following order:
x < X

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

(75) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
x(gen_0':s10_14(a), gen_0':s10_14(n4085_14)) -> gen_0':s10_14(*(n4085_14, a)), rt in Omega(0)

Induction Base:
x(gen_0':s10_14(a), gen_0':s10_14(0)) ->_R^Omega(0)
0'

Induction Step:
x(gen_0':s10_14(a), gen_0':s10_14(+(n4085_14, 1))) ->_R^Omega(0)
U21(tt, gen_0':s10_14(n4085_14), gen_0':s10_14(a)) ->_R^Omega(0)
U22(tt, activate(gen_0':s10_14(n4085_14)), activate(gen_0':s10_14(a))) ->_R^Omega(0)
U22(tt, gen_0':s10_14(n4085_14), activate(gen_0':s10_14(a))) ->_R^Omega(0)
U22(tt, gen_0':s10_14(n4085_14), gen_0':s10_14(a)) ->_R^Omega(0)
plus(x(activate(gen_0':s10_14(a)), activate(gen_0':s10_14(n4085_14))), activate(gen_0':s10_14(a))) ->_R^Omega(0)
plus(x(gen_0':s10_14(a), activate(gen_0':s10_14(n4085_14))), activate(gen_0':s10_14(a))) ->_R^Omega(0)
plus(x(gen_0':s10_14(a), gen_0':s10_14(n4085_14)), activate(gen_0':s10_14(a))) ->_IH
plus(gen_0':s10_14(*(c4086_14, a)), activate(gen_0':s10_14(a))) ->_R^Omega(0)
plus(gen_0':s10_14(*(n4085_14, a)), gen_0':s10_14(a)) ->_L^Omega(0)
gen_0':s10_14(+(a, *(n4085_14, a)))

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

(76)
Obligation:
Innermost TRS:
Rules:
U11'(tt, z0, z1) -> c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U11'(tt, z0, z1) -> c1(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c2(PLUS(activate(z1), activate(z0)), ACTIVATE(z1))
U12'(tt, z0, z1) -> c3(PLUS(activate(z1), activate(z0)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c4(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0))
U21'(tt, z0, z1) -> c5(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c6(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1))
U22'(tt, z0, z1) -> c7(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z0))
U22'(tt, z0, z1) -> c8(PLUS(x(activate(z1), activate(z0)), activate(z1)), ACTIVATE(z1))
PLUS(z0, 0') -> c9
PLUS(z0, s(z1)) -> c10(U11'(tt, z1, z0))
X(z0, 0') -> c11
X(z0, s(z1)) -> c12(U21'(tt, z1, z0))
ACTIVATE(z0) -> c13
U11(tt, z0, z1) -> U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) -> s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) -> U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) -> plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0') -> z0
plus(z0, s(z1)) -> U11(tt, z1, z0)
x(z0, 0') -> 0'
x(z0, s(z1)) -> U21(tt, z1, z0)
activate(z0) -> z0

Types:
U11' :: tt -> 0':s -> 0':s -> c:c1
tt :: tt
c :: c2:c3 -> c13 -> c:c1
U12' :: tt -> 0':s -> 0':s -> c2:c3
activate :: 0':s -> 0':s
ACTIVATE :: 0':s -> c13
c1 :: c2:c3 -> c13 -> c:c1
c2 :: c9:c10 -> c13 -> c2:c3
PLUS :: 0':s -> 0':s -> c9:c10
c3 :: c9:c10 -> c13 -> c2:c3
U21' :: tt -> 0':s -> 0':s -> c4:c5
c4 :: c6:c7:c8 -> c13 -> c4:c5
U22' :: tt -> 0':s -> 0':s -> c6:c7:c8
c5 :: c6:c7:c8 -> c13 -> c4:c5
c6 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
x :: 0':s -> 0':s -> 0':s
X :: 0':s -> 0':s -> c11:c12
c7 :: c9:c10 -> c11:c12 -> c13 -> c6:c7:c8
c8 :: c9:c10 -> c13 -> c6:c7:c8
0' :: 0':s
c9 :: c9:c10
s :: 0':s -> 0':s
c10 :: c:c1 -> c9:c10
c11 :: c11:c12
c12 :: c4:c5 -> c11:c12
c13 :: c13
U11 :: tt -> 0':s -> 0':s -> 0':s
U12 :: tt -> 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
U21 :: tt -> 0':s -> 0':s -> 0':s
U22 :: tt -> 0':s -> 0':s -> 0':s
hole_c:c11_14 :: c:c1
hole_tt2_14 :: tt
hole_0':s3_14 :: 0':s
hole_c2:c34_14 :: c2:c3
hole_c135_14 :: c13
hole_c9:c106_14 :: c9:c10
hole_c4:c57_14 :: c4:c5
hole_c6:c7:c88_14 :: c6:c7:c8
hole_c11:c129_14 :: c11:c12
gen_0':s10_14 :: Nat -> 0':s


Lemmas:
PLUS(gen_0':s10_14(a), gen_0':s10_14(n12_14)) -> *11_14, rt in Omega(n12_14)
plus(gen_0':s10_14(a), gen_0':s10_14(n2625_14)) -> gen_0':s10_14(+(n2625_14, a)), rt in Omega(0)
x(gen_0':s10_14(a), gen_0':s10_14(n4085_14)) -> gen_0':s10_14(*(n4085_14, a)), rt in Omega(0)


Generator Equations:
gen_0':s10_14(0) <=> 0'
gen_0':s10_14(+(x, 1)) <=> s(gen_0':s10_14(x))


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

(77) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
X(gen_0':s10_14(0), gen_0':s10_14(n6197_14)) -> *11_14, rt in Omega(n6197_14)

Induction Base:
X(gen_0':s10_14(0), gen_0':s10_14(0))

Induction Step:
X(gen_0':s10_14(0), gen_0':s10_14(+(n6197_14, 1))) ->_R^Omega(1)
c12(U21'(tt, gen_0':s10_14(n6197_14), gen_0':s10_14(0))) ->_R^Omega(1)
c12(c4(U22'(tt, activate(gen_0':s10_14(n6197_14)), activate(gen_0':s10_14(0))), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_R^Omega(0)
c12(c4(U22'(tt, gen_0':s10_14(n6197_14), activate(gen_0':s10_14(0))), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_R^Omega(0)
c12(c4(U22'(tt, gen_0':s10_14(n6197_14), gen_0':s10_14(0)), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_R^Omega(1)
c12(c4(c6(PLUS(x(activate(gen_0':s10_14(0)), activate(gen_0':s10_14(n6197_14))), activate(gen_0':s10_14(0))), X(activate(gen_0':s10_14(0)), activate(gen_0':s10_14(n6197_14))), ACTIVATE(gen_0':s10_14(0))), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_R^Omega(0)
c12(c4(c6(PLUS(x(gen_0':s10_14(0), activate(gen_0':s10_14(n6197_14))), activate(gen_0':s10_14(0))), X(activate(gen_0':s10_14(0)), activate(gen_0':s10_14(n6197_14))), ACTIVATE(gen_0':s10_14(0))), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_R^Omega(0)
c12(c4(c6(PLUS(x(gen_0':s10_14(0), gen_0':s10_14(n6197_14)), activate(gen_0':s10_14(0))), X(activate(gen_0':s10_14(0)), activate(gen_0':s10_14(n6197_14))), ACTIVATE(gen_0':s10_14(0))), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_L^Omega(0)
c12(c4(c6(PLUS(gen_0':s10_14(*(n6197_14, 0)), activate(gen_0':s10_14(0))), X(activate(gen_0':s10_14(0)), activate(gen_0':s10_14(n6197_14))), ACTIVATE(gen_0':s10_14(0))), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_R^Omega(0)
c12(c4(c6(PLUS(gen_0':s10_14(0), gen_0':s10_14(0)), X(activate(gen_0':s10_14(0)), activate(gen_0':s10_14(n6197_14))), ACTIVATE(gen_0':s10_14(0))), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_R^Omega(1)
c12(c4(c6(c9, X(activate(gen_0':s10_14(0)), activate(gen_0':s10_14(n6197_14))), ACTIVATE(gen_0':s10_14(0))), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_R^Omega(0)
c12(c4(c6(c9, X(gen_0':s10_14(0), activate(gen_0':s10_14(n6197_14))), ACTIVATE(gen_0':s10_14(0))), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_R^Omega(0)
c12(c4(c6(c9, X(gen_0':s10_14(0), gen_0':s10_14(n6197_14)), ACTIVATE(gen_0':s10_14(0))), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_IH
c12(c4(c6(c9, *11_14, ACTIVATE(gen_0':s10_14(0))), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_R^Omega(1)
c12(c4(c6(c9, *11_14, c13), ACTIVATE(gen_0':s10_14(n6197_14)))) ->_R^Omega(1)
c12(c4(c6(c9, *11_14, c13), c13))

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

(78)
BOUNDS(1, INF)