WORST_CASE(Omega(n^3),O(n^3))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^3, n^3).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 360 ms]
(2) CpxRelTRS
(3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 180 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 87 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 151 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 83 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 160 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 25 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 273 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 5 ms]
    (40) CpxRNTS
    (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 754 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 58 ms]
    (46) CpxRNTS
    (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 2116 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 418 ms]
    (52) CpxRNTS
    (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (54) CpxRNTS
    (55) IntTrsBoundProof [UPPER BOUND(ID), 529 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (58) CpxRNTS
    (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (60) CpxRNTS
    (61) IntTrsBoundProof [UPPER BOUND(ID), 480 ms]
    (62) CpxRNTS
    (63) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (64) CpxRNTS
    (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (66) CpxRNTS
    (67) IntTrsBoundProof [UPPER BOUND(ID), 1951 ms]
    (68) CpxRNTS
    (69) IntTrsBoundProof [UPPER BOUND(ID), 109 ms]
    (70) CpxRNTS
    (71) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (72) CpxRNTS
    (73) IntTrsBoundProof [UPPER BOUND(ID), 1624 ms]
    (74) CpxRNTS
    (75) IntTrsBoundProof [UPPER BOUND(ID), 8 ms]
    (76) CpxRNTS
    (77) FinalProof [FINISHED, 0 ms]
    (78) BOUNDS(1, n^3)
(79) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(80) CpxRelTRS
    (81) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) typed CpxTrs
    (83) OrderProof [LOWER BOUND(ID), 0 ms]
    (84) typed CpxTrs
    (85) RewriteLemmaProof [LOWER BOUND(ID), 312 ms]
    (86) BEST
        (87) proven lower bound
            (88) LowerBoundPropagationProof [FINISHED, 0 ms]
            (89) BOUNDS(n^1, INF)
        (90) typed CpxTrs
            (91) RewriteLemmaProof [LOWER BOUND(ID), 93 ms]
            (92) typed CpxTrs
            (93) RewriteLemmaProof [LOWER BOUND(ID), 35 ms]
            (94) typed CpxTrs
            (95) RewriteLemmaProof [LOWER BOUND(ID), 69 ms]
            (96) typed CpxTrs
            (97) RewriteLemmaProof [LOWER BOUND(ID), 68 ms]
            (98) typed CpxTrs
            (99) RewriteLemmaProof [LOWER BOUND(ID), 537 ms]
            (100) BEST
                (101) proven lower bound
                    (102) LowerBoundPropagationProof [FINISHED, 0 ms]
                    (103) BOUNDS(n^3, INF)
                (104) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^3, n^3).


The TRS R consists of the following rules:

   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1
   SQR(0) -> c2
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
   SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0))
   DBL(0) -> c5
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(0, z0) -> c7
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(0, z0) -> c9
   FIRST(s(z0), cons(z1, z2)) -> c10(ACTIVATE(z2))
   FIRST(z0, z1) -> c11
   ACTIVATE(n__terms(z0)) -> c12(TERMS(z0))
   ACTIVATE(n__first(z0, z1)) -> c13(FIRST(z0, z1))
   ACTIVATE(z0) -> c14

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

   terms(z0) -> cons(recip(sqr(z0)), n__terms(s(z0)))
   terms(z0) -> n__terms(z0)
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   first(0, z0) -> nil
   first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2)))
   first(z0, z1) -> n__first(z0, z1)
   activate(n__terms(z0)) -> terms(z0)
   activate(n__first(z0, z1)) -> first(z0, z1)
   activate(z0) -> z0

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

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

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^3, n^3).


The TRS R consists of the following rules:

   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1
   SQR(0) -> c2
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
   SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0))
   DBL(0) -> c5
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(0, z0) -> c7
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(0, z0) -> c9
   FIRST(s(z0), cons(z1, z2)) -> c10(ACTIVATE(z2))
   FIRST(z0, z1) -> c11
   ACTIVATE(n__terms(z0)) -> c12(TERMS(z0))
   ACTIVATE(n__first(z0, z1)) -> c13(FIRST(z0, z1))
   ACTIVATE(z0) -> c14

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

   terms(z0) -> cons(recip(sqr(z0)), n__terms(s(z0)))
   terms(z0) -> n__terms(z0)
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   first(0, z0) -> nil
   first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2)))
   first(z0, z1) -> n__first(z0, z1)
   activate(n__terms(z0)) -> terms(z0)
   activate(n__first(z0, z1)) -> first(z0, z1)
   activate(z0) -> z0

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

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   TERMS(z0) -> c(SQR(z0)) [1]
   TERMS(z0) -> c1 [1]
   SQR(0) -> c2 [1]
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0)) [1]
   SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0)) [1]
   DBL(0) -> c5 [1]
   DBL(s(z0)) -> c6(DBL(z0)) [1]
   ADD(0, z0) -> c7 [1]
   ADD(s(z0), z1) -> c8(ADD(z0, z1)) [1]
   FIRST(0, z0) -> c9 [1]
   FIRST(s(z0), cons(z1, z2)) -> c10(ACTIVATE(z2)) [1]
   FIRST(z0, z1) -> c11 [1]
   ACTIVATE(n__terms(z0)) -> c12(TERMS(z0)) [1]
   ACTIVATE(n__first(z0, z1)) -> c13(FIRST(z0, z1)) [1]
   ACTIVATE(z0) -> c14 [1]
   terms(z0) -> cons(recip(sqr(z0)), n__terms(s(z0))) [0]
   terms(z0) -> n__terms(z0) [0]
   sqr(0) -> 0 [0]
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) [0]
   dbl(0) -> 0 [0]
   dbl(s(z0)) -> s(s(dbl(z0))) [0]
   add(0, z0) -> z0 [0]
   add(s(z0), z1) -> s(add(z0, z1)) [0]
   first(0, z0) -> nil [0]
   first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2))) [0]
   first(z0, z1) -> n__first(z0, z1) [0]
   activate(n__terms(z0)) -> terms(z0) [0]
   activate(n__first(z0, z1)) -> first(z0, z1) [0]
   activate(z0) -> z0 [0]

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

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

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   TERMS(z0) -> c(SQR(z0)) [1]
   TERMS(z0) -> c1 [1]
   SQR(0) -> c2 [1]
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0)) [1]
   SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0)) [1]
   DBL(0) -> c5 [1]
   DBL(s(z0)) -> c6(DBL(z0)) [1]
   ADD(0, z0) -> c7 [1]
   ADD(s(z0), z1) -> c8(ADD(z0, z1)) [1]
   FIRST(0, z0) -> c9 [1]
   FIRST(s(z0), cons(z1, z2)) -> c10(ACTIVATE(z2)) [1]
   FIRST(z0, z1) -> c11 [1]
   ACTIVATE(n__terms(z0)) -> c12(TERMS(z0)) [1]
   ACTIVATE(n__first(z0, z1)) -> c13(FIRST(z0, z1)) [1]
   ACTIVATE(z0) -> c14 [1]
   terms(z0) -> cons(recip(sqr(z0)), n__terms(s(z0))) [0]
   terms(z0) -> n__terms(z0) [0]
   sqr(0) -> 0 [0]
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) [0]
   dbl(0) -> 0 [0]
   dbl(s(z0)) -> s(s(dbl(z0))) [0]
   add(0, z0) -> z0 [0]
   add(s(z0), z1) -> s(add(z0, z1)) [0]
   first(0, z0) -> nil [0]
   first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2))) [0]
   first(z0, z1) -> n__first(z0, z1) [0]
   activate(n__terms(z0)) -> terms(z0) [0]
   activate(n__first(z0, z1)) -> first(z0, z1) [0]
   activate(z0) -> z0 [0]

The TRS has the following type information:
TERMS :: 0:s -> c:c1
c :: c2:c3:c4 -> c:c1
SQR :: 0:s -> c2:c3:c4
c1 :: c:c1
0 :: 0:s
c2 :: c2:c3:c4
s :: 0:s -> 0:s
c3 :: c7:c8 -> c2:c3:c4 -> c2:c3:c4
ADD :: 0:s -> 0:s -> c7:c8
sqr :: 0:s -> 0:s
dbl :: 0:s -> 0:s
c4 :: c7:c8 -> c5:c6 -> c2:c3:c4
DBL :: 0:s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8
c8 :: c7:c8 -> c7:c8
FIRST :: 0:s -> cons:n__terms:n__first:nil -> c9:c10:c11
c9 :: c9:c10:c11
cons :: recip -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c10 :: c12:c13:c14 -> c9:c10:c11
ACTIVATE :: cons:n__terms:n__first:nil -> c12:c13:c14
c11 :: c9:c10:c11
n__terms :: 0:s -> cons:n__terms:n__first:nil
c12 :: c:c1 -> c12:c13:c14
n__first :: 0:s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c13 :: c9:c10:c11 -> c12:c13:c14
c14 :: c12:c13:c14
terms :: 0:s -> cons:n__terms:n__first:nil
recip :: 0:s -> recip
add :: 0:s -> 0:s -> 0:s
first :: 0:s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
nil :: cons:n__terms:n__first:nil
activate :: cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil

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

(7) 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:

   TERMS_1
   SQR_1
   DBL_1
   ADD_2
   FIRST_2
   ACTIVATE_1

(c) The following functions are completely defined:

   terms_1
   sqr_1
   dbl_1
   add_2
   first_2
   activate_1

Due to the following rules being added:

   terms(v0) -> nil [0]
   sqr(v0) -> 0 [0]
   dbl(v0) -> 0 [0]
   add(v0, v1) -> 0 [0]
   first(v0, v1) -> nil [0]
   activate(v0) -> nil [0]

And the following fresh constants:    const

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

(8)
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:

   TERMS(z0) -> c(SQR(z0)) [1]
   TERMS(z0) -> c1 [1]
   SQR(0) -> c2 [1]
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0)) [1]
   SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0)) [1]
   DBL(0) -> c5 [1]
   DBL(s(z0)) -> c6(DBL(z0)) [1]
   ADD(0, z0) -> c7 [1]
   ADD(s(z0), z1) -> c8(ADD(z0, z1)) [1]
   FIRST(0, z0) -> c9 [1]
   FIRST(s(z0), cons(z1, z2)) -> c10(ACTIVATE(z2)) [1]
   FIRST(z0, z1) -> c11 [1]
   ACTIVATE(n__terms(z0)) -> c12(TERMS(z0)) [1]
   ACTIVATE(n__first(z0, z1)) -> c13(FIRST(z0, z1)) [1]
   ACTIVATE(z0) -> c14 [1]
   terms(z0) -> cons(recip(sqr(z0)), n__terms(s(z0))) [0]
   terms(z0) -> n__terms(z0) [0]
   sqr(0) -> 0 [0]
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) [0]
   dbl(0) -> 0 [0]
   dbl(s(z0)) -> s(s(dbl(z0))) [0]
   add(0, z0) -> z0 [0]
   add(s(z0), z1) -> s(add(z0, z1)) [0]
   first(0, z0) -> nil [0]
   first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2))) [0]
   first(z0, z1) -> n__first(z0, z1) [0]
   activate(n__terms(z0)) -> terms(z0) [0]
   activate(n__first(z0, z1)) -> first(z0, z1) [0]
   activate(z0) -> z0 [0]
   terms(v0) -> nil [0]
   sqr(v0) -> 0 [0]
   dbl(v0) -> 0 [0]
   add(v0, v1) -> 0 [0]
   first(v0, v1) -> nil [0]
   activate(v0) -> nil [0]

The TRS has the following type information:
TERMS :: 0:s -> c:c1
c :: c2:c3:c4 -> c:c1
SQR :: 0:s -> c2:c3:c4
c1 :: c:c1
0 :: 0:s
c2 :: c2:c3:c4
s :: 0:s -> 0:s
c3 :: c7:c8 -> c2:c3:c4 -> c2:c3:c4
ADD :: 0:s -> 0:s -> c7:c8
sqr :: 0:s -> 0:s
dbl :: 0:s -> 0:s
c4 :: c7:c8 -> c5:c6 -> c2:c3:c4
DBL :: 0:s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8
c8 :: c7:c8 -> c7:c8
FIRST :: 0:s -> cons:n__terms:n__first:nil -> c9:c10:c11
c9 :: c9:c10:c11
cons :: recip -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c10 :: c12:c13:c14 -> c9:c10:c11
ACTIVATE :: cons:n__terms:n__first:nil -> c12:c13:c14
c11 :: c9:c10:c11
n__terms :: 0:s -> cons:n__terms:n__first:nil
c12 :: c:c1 -> c12:c13:c14
n__first :: 0:s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c13 :: c9:c10:c11 -> c12:c13:c14
c14 :: c12:c13:c14
terms :: 0:s -> cons:n__terms:n__first:nil
recip :: 0:s -> recip
add :: 0:s -> 0:s -> 0:s
first :: 0:s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
nil :: cons:n__terms:n__first:nil
activate :: cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
const :: recip

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

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

(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   TERMS(z0) -> c(SQR(z0)) [1]
   TERMS(z0) -> c1 [1]
   SQR(0) -> c2 [1]
   SQR(s(0)) -> c3(ADD(0, 0), SQR(0)) [1]
   SQR(s(0)) -> c3(ADD(0, 0), SQR(0)) [1]
   SQR(s(s(z0'))) -> c3(ADD(s(add(sqr(z0'), dbl(z0'))), s(s(dbl(z0')))), SQR(s(z0'))) [1]
   SQR(s(s(z0'))) -> c3(ADD(s(add(sqr(z0'), dbl(z0'))), 0), SQR(s(z0'))) [1]
   SQR(s(0)) -> c3(ADD(0, 0), SQR(0)) [1]
   SQR(s(s(z0''))) -> c3(ADD(0, s(s(dbl(z0'')))), SQR(s(z0''))) [1]
   SQR(s(z0)) -> c3(ADD(0, 0), SQR(z0)) [1]
   SQR(s(0)) -> c4(ADD(0, 0), DBL(0)) [1]
   SQR(s(0)) -> c4(ADD(0, 0), DBL(0)) [1]
   SQR(s(s(z01))) -> c4(ADD(s(add(sqr(z01), dbl(z01))), s(s(dbl(z01)))), DBL(s(z01))) [1]
   SQR(s(s(z01))) -> c4(ADD(s(add(sqr(z01), dbl(z01))), 0), DBL(s(z01))) [1]
   SQR(s(0)) -> c4(ADD(0, 0), DBL(0)) [1]
   SQR(s(s(z02))) -> c4(ADD(0, s(s(dbl(z02)))), DBL(s(z02))) [1]
   SQR(s(z0)) -> c4(ADD(0, 0), DBL(z0)) [1]
   DBL(0) -> c5 [1]
   DBL(s(z0)) -> c6(DBL(z0)) [1]
   ADD(0, z0) -> c7 [1]
   ADD(s(z0), z1) -> c8(ADD(z0, z1)) [1]
   FIRST(0, z0) -> c9 [1]
   FIRST(s(z0), cons(z1, z2)) -> c10(ACTIVATE(z2)) [1]
   FIRST(z0, z1) -> c11 [1]
   ACTIVATE(n__terms(z0)) -> c12(TERMS(z0)) [1]
   ACTIVATE(n__first(z0, z1)) -> c13(FIRST(z0, z1)) [1]
   ACTIVATE(z0) -> c14 [1]
   terms(z0) -> cons(recip(sqr(z0)), n__terms(s(z0))) [0]
   terms(z0) -> n__terms(z0) [0]
   sqr(0) -> 0 [0]
   sqr(s(0)) -> s(add(0, 0)) [0]
   sqr(s(0)) -> s(add(0, 0)) [0]
   sqr(s(s(z03))) -> s(add(s(add(sqr(z03), dbl(z03))), s(s(dbl(z03))))) [0]
   sqr(s(s(z03))) -> s(add(s(add(sqr(z03), dbl(z03))), 0)) [0]
   sqr(s(0)) -> s(add(0, 0)) [0]
   sqr(s(s(z04))) -> s(add(0, s(s(dbl(z04))))) [0]
   sqr(s(z0)) -> s(add(0, 0)) [0]
   dbl(0) -> 0 [0]
   dbl(s(z0)) -> s(s(dbl(z0))) [0]
   add(0, z0) -> z0 [0]
   add(s(z0), z1) -> s(add(z0, z1)) [0]
   first(0, z0) -> nil [0]
   first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2))) [0]
   first(z0, z1) -> n__first(z0, z1) [0]
   activate(n__terms(z0)) -> terms(z0) [0]
   activate(n__first(z0, z1)) -> first(z0, z1) [0]
   activate(z0) -> z0 [0]
   terms(v0) -> nil [0]
   sqr(v0) -> 0 [0]
   dbl(v0) -> 0 [0]
   add(v0, v1) -> 0 [0]
   first(v0, v1) -> nil [0]
   activate(v0) -> nil [0]

The TRS has the following type information:
TERMS :: 0:s -> c:c1
c :: c2:c3:c4 -> c:c1
SQR :: 0:s -> c2:c3:c4
c1 :: c:c1
0 :: 0:s
c2 :: c2:c3:c4
s :: 0:s -> 0:s
c3 :: c7:c8 -> c2:c3:c4 -> c2:c3:c4
ADD :: 0:s -> 0:s -> c7:c8
sqr :: 0:s -> 0:s
dbl :: 0:s -> 0:s
c4 :: c7:c8 -> c5:c6 -> c2:c3:c4
DBL :: 0:s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8
c8 :: c7:c8 -> c7:c8
FIRST :: 0:s -> cons:n__terms:n__first:nil -> c9:c10:c11
c9 :: c9:c10:c11
cons :: recip -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c10 :: c12:c13:c14 -> c9:c10:c11
ACTIVATE :: cons:n__terms:n__first:nil -> c12:c13:c14
c11 :: c9:c10:c11
n__terms :: 0:s -> cons:n__terms:n__first:nil
c12 :: c:c1 -> c12:c13:c14
n__first :: 0:s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c13 :: c9:c10:c11 -> c12:c13:c14
c14 :: c12:c13:c14
terms :: 0:s -> cons:n__terms:n__first:nil
recip :: 0:s -> recip
add :: 0:s -> 0:s -> 0:s
first :: 0:s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
nil :: cons:n__terms:n__first:nil
activate :: cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
const :: recip

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

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

   c1 => 0
   0 => 0
   c2 => 0
   c5 => 0
   c7 => 0
   c9 => 1
   c11 => 0
   c14 => 0
   nil => 0
   const => 0

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   ACTIVATE(z) -{ 1 }-> 0 :|: z = z0, z0 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z0) :|: z = 1 + z0, z0 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   ADD(z, z') -{ 1 }-> 1 + ADD(z0, z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 }-> 1 + DBL(z0) :|: z = 1 + z0, z0 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z0 >= 0, z = 0, z' = z0
   FIRST(z, z') -{ 1 }-> 0 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 1 + z0, z0 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + SQR(z0) :|: z = 1 + z0, z0 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + SQR(0) :|: z = 1 + 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + DBL(z0) :|: z = 1 + z0, z0 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + DBL(0) :|: z = 1 + 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 1 + (1 + dbl(z0''))) + SQR(1 + z0'') :|: z = 1 + (1 + z0''), z0'' >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 1 + (1 + dbl(z02))) + DBL(1 + z02) :|: z = 1 + (1 + z02), z02 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z0'), dbl(z0')), 0) + SQR(1 + z0') :|: z = 1 + (1 + z0'), z0' >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z0'), dbl(z0')), 1 + (1 + dbl(z0'))) + SQR(1 + z0') :|: z = 1 + (1 + z0'), z0' >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z01), dbl(z01)), 0) + DBL(1 + z01) :|: z = 1 + (1 + z01), z01 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z01), dbl(z01)), 1 + (1 + dbl(z01))) + DBL(1 + z01) :|: z = 1 + (1 + z01), z01 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z = z0, z0 >= 0
   TERMS(z) -{ 1 }-> 1 + SQR(z0) :|: z = z0, z0 >= 0
   activate(z) -{ 0 }-> z0 :|: z = z0, z0 >= 0
   activate(z) -{ 0 }-> terms(z0) :|: z = 1 + z0, z0 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   add(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0
   add(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   add(z, z') -{ 0 }-> 1 + add(z0, z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   dbl(z) -{ 0 }-> 1 + (1 + dbl(z0)) :|: z = 1 + z0, z0 >= 0
   first(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   first(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   first(z, z') -{ 0 }-> 1 + z0 + z1 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + z0 + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z = 1 + z0, z0 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z = 1 + z0, z0 >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 1 + (1 + dbl(z04))) :|: z04 >= 0, z = 1 + (1 + z04)
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z03), dbl(z03)), 0) :|: z = 1 + (1 + z03), z03 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z03), dbl(z03)), 1 + (1 + dbl(z03))) :|: z = 1 + (1 + z03), z03 >= 0
   terms(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   terms(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0
   terms(z) -{ 0 }-> 1 + (1 + sqr(z0)) + (1 + (1 + z0)) :|: z = z0, z0 >= 0


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

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 }-> 1 + ADD(z - 1, z') :|: z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 }-> 1 + DBL(z - 1) :|: z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + SQR(0) :|: z = 1 + 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + SQR(z - 1) :|: z - 1 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + DBL(0) :|: z = 1 + 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + DBL(z - 1) :|: z - 1 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 1 + (1 + dbl(z - 2))) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 1 + (1 + dbl(z - 2))) + DBL(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 0) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 0) + DBL(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) + DBL(1 + (z - 2)) :|: z - 2 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 1 }-> 1 + SQR(z) :|: z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 0) :|: z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0


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

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

   { DBL }
   { ADD }
   { dbl }
   { add }
   { sqr }
   { SQR }
   { terms }
   { TERMS }
   { first, activate }
   { FIRST, ACTIVATE }

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 }-> 1 + ADD(z - 1, z') :|: z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 }-> 1 + DBL(z - 1) :|: z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + SQR(0) :|: z = 1 + 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + SQR(z - 1) :|: z - 1 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + DBL(0) :|: z = 1 + 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + DBL(z - 1) :|: z - 1 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 1 + (1 + dbl(z - 2))) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 1 + (1 + dbl(z - 2))) + DBL(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 0) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 0) + DBL(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) + DBL(1 + (z - 2)) :|: z - 2 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 1 }-> 1 + SQR(z) :|: z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 0) :|: z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {DBL}, {ADD}, {dbl}, {add}, {sqr}, {SQR}, {terms}, {TERMS}, {first,activate}, {FIRST,ACTIVATE}

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

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 }-> 1 + ADD(z - 1, z') :|: z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 }-> 1 + DBL(z - 1) :|: z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + SQR(0) :|: z = 1 + 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + SQR(z - 1) :|: z - 1 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + DBL(0) :|: z = 1 + 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + DBL(z - 1) :|: z - 1 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 1 + (1 + dbl(z - 2))) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 1 + (1 + dbl(z - 2))) + DBL(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 0) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 0) + DBL(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) + DBL(1 + (z - 2)) :|: z - 2 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 1 }-> 1 + SQR(z) :|: z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 0) :|: z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {DBL}, {ADD}, {dbl}, {add}, {sqr}, {SQR}, {terms}, {TERMS}, {first,activate}, {FIRST,ACTIVATE}

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 }-> 1 + ADD(z - 1, z') :|: z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 }-> 1 + DBL(z - 1) :|: z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + SQR(0) :|: z = 1 + 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + SQR(z - 1) :|: z - 1 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + DBL(0) :|: z = 1 + 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + DBL(z - 1) :|: z - 1 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 1 + (1 + dbl(z - 2))) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 1 + (1 + dbl(z - 2))) + DBL(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 0) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 0) + DBL(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) + DBL(1 + (z - 2)) :|: z - 2 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 1 }-> 1 + SQR(z) :|: z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 0) :|: z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {DBL}, {ADD}, {dbl}, {add}, {sqr}, {SQR}, {terms}, {TERMS}, {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: ?, size: O(n^1) [z]

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 }-> 1 + ADD(z - 1, z') :|: z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 }-> 1 + DBL(z - 1) :|: z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + SQR(0) :|: z = 1 + 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + SQR(z - 1) :|: z - 1 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + DBL(0) :|: z = 1 + 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + DBL(z - 1) :|: z - 1 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 1 + (1 + dbl(z - 2))) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 1 + (1 + dbl(z - 2))) + DBL(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 0) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 0) + DBL(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) + DBL(1 + (z - 2)) :|: z - 2 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 1 }-> 1 + SQR(z) :|: z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 0) :|: z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {ADD}, {dbl}, {add}, {sqr}, {SQR}, {terms}, {TERMS}, {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]

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

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 }-> 1 + ADD(z - 1, z') :|: z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 2 }-> 1 + ADD(0, 0) + s :|: s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 1 + z }-> 1 + ADD(0, 0) + s2 :|: s2 >= 0, s2 <= z - 1, z - 1 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + SQR(0) :|: z = 1 + 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + SQR(z - 1) :|: z - 1 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(0, 1 + (1 + dbl(z - 2))) + s1 :|: s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 1 + (1 + dbl(z - 2))) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 0) + s'' :|: s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 0) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) + s' :|: s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 1 }-> 1 + SQR(z) :|: z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 0) :|: z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {ADD}, {dbl}, {add}, {sqr}, {SQR}, {terms}, {TERMS}, {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 }-> 1 + ADD(z - 1, z') :|: z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 2 }-> 1 + ADD(0, 0) + s :|: s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 1 + z }-> 1 + ADD(0, 0) + s2 :|: s2 >= 0, s2 <= z - 1, z - 1 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + SQR(0) :|: z = 1 + 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + SQR(z - 1) :|: z - 1 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(0, 1 + (1 + dbl(z - 2))) + s1 :|: s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 1 + (1 + dbl(z - 2))) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 0) + s'' :|: s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 0) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) + s' :|: s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 1 }-> 1 + SQR(z) :|: z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 0) :|: z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {ADD}, {dbl}, {add}, {sqr}, {SQR}, {terms}, {TERMS}, {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: ?, size: O(n^1) [z]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 }-> 1 + ADD(z - 1, z') :|: z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 2 }-> 1 + ADD(0, 0) + s :|: s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 1 + z }-> 1 + ADD(0, 0) + s2 :|: s2 >= 0, s2 <= z - 1, z - 1 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + SQR(0) :|: z = 1 + 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 0) + SQR(z - 1) :|: z - 1 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(0, 1 + (1 + dbl(z - 2))) + s1 :|: s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 1 + (1 + dbl(z - 2))) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 0) + s'' :|: s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 0) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) + s' :|: s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 1 }-> 1 + SQR(z) :|: z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 0) :|: z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {dbl}, {add}, {sqr}, {SQR}, {terms}, {TERMS}, {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]

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

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 2 }-> 1 + s4 + SQR(0) :|: s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 2 }-> 1 + s5 + SQR(z - 1) :|: s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(0, 1 + (1 + dbl(z - 2))) + s1 :|: s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 1 + (1 + dbl(z - 2))) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 0) + s'' :|: s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 0) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) + s' :|: s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 1 }-> 1 + SQR(z) :|: z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 0) :|: z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {dbl}, {add}, {sqr}, {SQR}, {terms}, {TERMS}, {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 2 }-> 1 + s4 + SQR(0) :|: s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 2 }-> 1 + s5 + SQR(z - 1) :|: s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(0, 1 + (1 + dbl(z - 2))) + s1 :|: s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 1 + (1 + dbl(z - 2))) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 0) + s'' :|: s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 0) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) + s' :|: s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 1 }-> 1 + SQR(z) :|: z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 0) :|: z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {dbl}, {add}, {sqr}, {SQR}, {terms}, {TERMS}, {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: ?, size: O(n^1) [2*z]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 2 }-> 1 + s4 + SQR(0) :|: s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 2 }-> 1 + s5 + SQR(z - 1) :|: s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(0, 1 + (1 + dbl(z - 2))) + s1 :|: s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(0, 1 + (1 + dbl(z - 2))) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 0) + s'' :|: s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 0) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) + s' :|: s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) + SQR(1 + (z - 2)) :|: z - 2 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 1 }-> 1 + SQR(z) :|: z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 0) :|: z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {add}, {sqr}, {SQR}, {terms}, {TERMS}, {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: O(1) [0], size: O(n^1) [2*z]

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

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 2 }-> 1 + s13 + SQR(1 + (z - 2)) :|: s12 >= 0, s12 <= 2 * (z - 2), s13 >= 0, s13 <= 0, z - 2 >= 0
   SQR(z) -{ 2 + z }-> 1 + s18 + s1 :|: s17 >= 0, s17 <= 2 * (z - 2), s18 >= 0, s18 <= 0, s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 2 }-> 1 + s4 + SQR(0) :|: s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 2 }-> 1 + s5 + SQR(z - 1) :|: s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), s11), 0) + SQR(1 + (z - 2)) :|: s11 >= 0, s11 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), s14), 1 + (1 + s15)) + s' :|: s14 >= 0, s14 <= 2 * (z - 2), s15 >= 0, s15 <= 2 * (z - 2), s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), s16), 0) + s'' :|: s16 >= 0, s16 <= 2 * (z - 2), s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), s9), 1 + (1 + s10)) + SQR(1 + (z - 2)) :|: s9 >= 0, s9 <= 2 * (z - 2), s10 >= 0, s10 <= 2 * (z - 2), z - 2 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 1 }-> 1 + SQR(z) :|: z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + s23) :|: s23 >= 0, s23 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 1 + (1 + s22)) :|: s22 >= 0, s22 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), s19), 1 + (1 + s20)) :|: s19 >= 0, s19 <= 2 * (z - 2), s20 >= 0, s20 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), s21), 0) :|: s21 >= 0, s21 <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {add}, {sqr}, {SQR}, {terms}, {TERMS}, {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: O(1) [0], size: O(n^1) [2*z]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 2 }-> 1 + s13 + SQR(1 + (z - 2)) :|: s12 >= 0, s12 <= 2 * (z - 2), s13 >= 0, s13 <= 0, z - 2 >= 0
   SQR(z) -{ 2 + z }-> 1 + s18 + s1 :|: s17 >= 0, s17 <= 2 * (z - 2), s18 >= 0, s18 <= 0, s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 2 }-> 1 + s4 + SQR(0) :|: s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 2 }-> 1 + s5 + SQR(z - 1) :|: s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), s11), 0) + SQR(1 + (z - 2)) :|: s11 >= 0, s11 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), s14), 1 + (1 + s15)) + s' :|: s14 >= 0, s14 <= 2 * (z - 2), s15 >= 0, s15 <= 2 * (z - 2), s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), s16), 0) + s'' :|: s16 >= 0, s16 <= 2 * (z - 2), s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), s9), 1 + (1 + s10)) + SQR(1 + (z - 2)) :|: s9 >= 0, s9 <= 2 * (z - 2), s10 >= 0, s10 <= 2 * (z - 2), z - 2 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 1 }-> 1 + SQR(z) :|: z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + s23) :|: s23 >= 0, s23 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 1 + (1 + s22)) :|: s22 >= 0, s22 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), s19), 1 + (1 + s20)) :|: s19 >= 0, s19 <= 2 * (z - 2), s20 >= 0, s20 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), s21), 0) :|: s21 >= 0, s21 <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {add}, {sqr}, {SQR}, {terms}, {TERMS}, {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: O(1) [0], size: O(n^1) [2*z]
  add: runtime: ?, size: O(n^1) [z + z']

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 2 }-> 1 + s13 + SQR(1 + (z - 2)) :|: s12 >= 0, s12 <= 2 * (z - 2), s13 >= 0, s13 <= 0, z - 2 >= 0
   SQR(z) -{ 2 + z }-> 1 + s18 + s1 :|: s17 >= 0, s17 <= 2 * (z - 2), s18 >= 0, s18 <= 0, s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 2 }-> 1 + s4 + SQR(0) :|: s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 2 }-> 1 + s5 + SQR(z - 1) :|: s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), s11), 0) + SQR(1 + (z - 2)) :|: s11 >= 0, s11 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), s14), 1 + (1 + s15)) + s' :|: s14 >= 0, s14 <= 2 * (z - 2), s15 >= 0, s15 <= 2 * (z - 2), s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), s16), 0) + s'' :|: s16 >= 0, s16 <= 2 * (z - 2), s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), s9), 1 + (1 + s10)) + SQR(1 + (z - 2)) :|: s9 >= 0, s9 <= 2 * (z - 2), s10 >= 0, s10 <= 2 * (z - 2), z - 2 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 1 }-> 1 + SQR(z) :|: z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + s23) :|: s23 >= 0, s23 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 0 }-> 1 + add(0, 0) :|: z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + add(0, 1 + (1 + s22)) :|: s22 >= 0, s22 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), s19), 1 + (1 + s20)) :|: s19 >= 0, s19 <= 2 * (z - 2), s20 >= 0, s20 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), s21), 0) :|: s21 >= 0, s21 <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {sqr}, {SQR}, {terms}, {TERMS}, {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: O(1) [0], size: O(n^1) [2*z]
  add: runtime: O(1) [0], size: O(n^1) [z + z']

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

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 2 }-> 1 + s13 + SQR(1 + (z - 2)) :|: s12 >= 0, s12 <= 2 * (z - 2), s13 >= 0, s13 <= 0, z - 2 >= 0
   SQR(z) -{ 2 + z }-> 1 + s18 + s1 :|: s17 >= 0, s17 <= 2 * (z - 2), s18 >= 0, s18 <= 0, s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 2 }-> 1 + s4 + SQR(0) :|: s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 2 }-> 1 + s5 + SQR(z - 1) :|: s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), s11), 0) + SQR(1 + (z - 2)) :|: s11 >= 0, s11 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), s14), 1 + (1 + s15)) + s' :|: s14 >= 0, s14 <= 2 * (z - 2), s15 >= 0, s15 <= 2 * (z - 2), s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), s16), 0) + s'' :|: s16 >= 0, s16 <= 2 * (z - 2), s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), s9), 1 + (1 + s10)) + SQR(1 + (z - 2)) :|: s9 >= 0, s9 <= 2 * (z - 2), s10 >= 0, s10 <= 2 * (z - 2), z - 2 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 1 }-> 1 + SQR(z) :|: z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + s27 :|: s27 >= 0, s27 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + s23) :|: s23 >= 0, s23 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + s24 :|: s24 >= 0, s24 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 0 }-> 1 + s25 :|: s25 >= 0, s25 <= 0 + (1 + (1 + s22)), s22 >= 0, s22 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s26 :|: s26 >= 0, s26 <= 0 + 0, z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), s19), 1 + (1 + s20)) :|: s19 >= 0, s19 <= 2 * (z - 2), s20 >= 0, s20 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), s21), 0) :|: s21 >= 0, s21 <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {sqr}, {SQR}, {terms}, {TERMS}, {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: O(1) [0], size: O(n^1) [2*z]
  add: runtime: O(1) [0], size: O(n^1) [z + z']

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: sqr
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 3 + 8*z + 6*z^2

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 2 }-> 1 + s13 + SQR(1 + (z - 2)) :|: s12 >= 0, s12 <= 2 * (z - 2), s13 >= 0, s13 <= 0, z - 2 >= 0
   SQR(z) -{ 2 + z }-> 1 + s18 + s1 :|: s17 >= 0, s17 <= 2 * (z - 2), s18 >= 0, s18 <= 0, s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 2 }-> 1 + s4 + SQR(0) :|: s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 2 }-> 1 + s5 + SQR(z - 1) :|: s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), s11), 0) + SQR(1 + (z - 2)) :|: s11 >= 0, s11 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), s14), 1 + (1 + s15)) + s' :|: s14 >= 0, s14 <= 2 * (z - 2), s15 >= 0, s15 <= 2 * (z - 2), s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), s16), 0) + s'' :|: s16 >= 0, s16 <= 2 * (z - 2), s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), s9), 1 + (1 + s10)) + SQR(1 + (z - 2)) :|: s9 >= 0, s9 <= 2 * (z - 2), s10 >= 0, s10 <= 2 * (z - 2), z - 2 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 1 }-> 1 + SQR(z) :|: z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + s27 :|: s27 >= 0, s27 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + s23) :|: s23 >= 0, s23 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + s24 :|: s24 >= 0, s24 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 0 }-> 1 + s25 :|: s25 >= 0, s25 <= 0 + (1 + (1 + s22)), s22 >= 0, s22 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s26 :|: s26 >= 0, s26 <= 0 + 0, z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), s19), 1 + (1 + s20)) :|: s19 >= 0, s19 <= 2 * (z - 2), s20 >= 0, s20 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), s21), 0) :|: s21 >= 0, s21 <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {sqr}, {SQR}, {terms}, {TERMS}, {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: O(1) [0], size: O(n^1) [2*z]
  add: runtime: O(1) [0], size: O(n^1) [z + z']
  sqr: runtime: ?, size: O(n^2) [3 + 8*z + 6*z^2]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 2 }-> 1 + s13 + SQR(1 + (z - 2)) :|: s12 >= 0, s12 <= 2 * (z - 2), s13 >= 0, s13 <= 0, z - 2 >= 0
   SQR(z) -{ 2 + z }-> 1 + s18 + s1 :|: s17 >= 0, s17 <= 2 * (z - 2), s18 >= 0, s18 <= 0, s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 2 }-> 1 + s4 + SQR(0) :|: s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 2 }-> 1 + s5 + SQR(z - 1) :|: s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), s11), 0) + SQR(1 + (z - 2)) :|: s11 >= 0, s11 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), s14), 1 + (1 + s15)) + s' :|: s14 >= 0, s14 <= 2 * (z - 2), s15 >= 0, s15 <= 2 * (z - 2), s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + z }-> 1 + ADD(1 + add(sqr(z - 2), s16), 0) + s'' :|: s16 >= 0, s16 <= 2 * (z - 2), s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 }-> 1 + ADD(1 + add(sqr(z - 2), s9), 1 + (1 + s10)) + SQR(1 + (z - 2)) :|: s9 >= 0, s9 <= 2 * (z - 2), s10 >= 0, s10 <= 2 * (z - 2), z - 2 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 1 }-> 1 + SQR(z) :|: z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + s27 :|: s27 >= 0, s27 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + s23) :|: s23 >= 0, s23 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + s24 :|: s24 >= 0, s24 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 0 }-> 1 + s25 :|: s25 >= 0, s25 <= 0 + (1 + (1 + s22)), s22 >= 0, s22 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s26 :|: s26 >= 0, s26 <= 0 + 0, z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), s19), 1 + (1 + s20)) :|: s19 >= 0, s19 <= 2 * (z - 2), s20 >= 0, s20 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + add(1 + add(sqr(z - 2), s21), 0) :|: s21 >= 0, s21 <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {SQR}, {terms}, {TERMS}, {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: O(1) [0], size: O(n^1) [2*z]
  add: runtime: O(1) [0], size: O(n^1) [z + z']
  sqr: runtime: O(1) [0], size: O(n^2) [3 + 8*z + 6*z^2]

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

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 2 }-> 1 + s13 + SQR(1 + (z - 2)) :|: s12 >= 0, s12 <= 2 * (z - 2), s13 >= 0, s13 <= 0, z - 2 >= 0
   SQR(z) -{ 2 + z }-> 1 + s18 + s1 :|: s17 >= 0, s17 <= 2 * (z - 2), s18 >= 0, s18 <= 0, s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s29 }-> 1 + s30 + SQR(1 + (z - 2)) :|: s28 >= 0, s28 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s29 >= 0, s29 <= s28 + s9, s30 >= 0, s30 <= 1 + s29, s9 >= 0, s9 <= 2 * (z - 2), s10 >= 0, s10 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s32 }-> 1 + s33 + SQR(1 + (z - 2)) :|: s31 >= 0, s31 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s32 >= 0, s32 <= s31 + s11, s33 >= 0, s33 <= 1 + s32, s11 >= 0, s11 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s35 + z }-> 1 + s36 + s' :|: s34 >= 0, s34 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s35 >= 0, s35 <= s34 + s14, s36 >= 0, s36 <= 1 + s35, s14 >= 0, s14 <= 2 * (z - 2), s15 >= 0, s15 <= 2 * (z - 2), s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s38 + z }-> 1 + s39 + s'' :|: s37 >= 0, s37 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s38 >= 0, s38 <= s37 + s16, s39 >= 0, s39 <= 1 + s38, s16 >= 0, s16 <= 2 * (z - 2), s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 2 }-> 1 + s4 + SQR(0) :|: s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 2 }-> 1 + s5 + SQR(z - 1) :|: s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 1 }-> 1 + SQR(z) :|: z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + s27 :|: s27 >= 0, s27 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + s23) :|: s23 >= 0, s23 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + s24 :|: s24 >= 0, s24 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 0 }-> 1 + s25 :|: s25 >= 0, s25 <= 0 + (1 + (1 + s22)), s22 >= 0, s22 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s26 :|: s26 >= 0, s26 <= 0 + 0, z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + s43 :|: s41 >= 0, s41 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s42 >= 0, s42 <= s41 + s19, s43 >= 0, s43 <= 1 + s42 + (1 + (1 + s20)), s19 >= 0, s19 <= 2 * (z - 2), s20 >= 0, s20 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s46 :|: s44 >= 0, s44 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s45 >= 0, s45 <= s44 + s21, s46 >= 0, s46 <= 1 + s45 + 0, s21 >= 0, s21 <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + s40) + (1 + (1 + z)) :|: s40 >= 0, s40 <= 6 * (z * z) + 8 * z + 3, z >= 0

Function symbols to be analyzed: {SQR}, {terms}, {TERMS}, {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: O(1) [0], size: O(n^1) [2*z]
  add: runtime: O(1) [0], size: O(n^1) [z + z']
  sqr: runtime: O(1) [0], size: O(n^2) [3 + 8*z + 6*z^2]

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: SQR
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 2 }-> 1 + s13 + SQR(1 + (z - 2)) :|: s12 >= 0, s12 <= 2 * (z - 2), s13 >= 0, s13 <= 0, z - 2 >= 0
   SQR(z) -{ 2 + z }-> 1 + s18 + s1 :|: s17 >= 0, s17 <= 2 * (z - 2), s18 >= 0, s18 <= 0, s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s29 }-> 1 + s30 + SQR(1 + (z - 2)) :|: s28 >= 0, s28 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s29 >= 0, s29 <= s28 + s9, s30 >= 0, s30 <= 1 + s29, s9 >= 0, s9 <= 2 * (z - 2), s10 >= 0, s10 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s32 }-> 1 + s33 + SQR(1 + (z - 2)) :|: s31 >= 0, s31 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s32 >= 0, s32 <= s31 + s11, s33 >= 0, s33 <= 1 + s32, s11 >= 0, s11 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s35 + z }-> 1 + s36 + s' :|: s34 >= 0, s34 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s35 >= 0, s35 <= s34 + s14, s36 >= 0, s36 <= 1 + s35, s14 >= 0, s14 <= 2 * (z - 2), s15 >= 0, s15 <= 2 * (z - 2), s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s38 + z }-> 1 + s39 + s'' :|: s37 >= 0, s37 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s38 >= 0, s38 <= s37 + s16, s39 >= 0, s39 <= 1 + s38, s16 >= 0, s16 <= 2 * (z - 2), s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 2 }-> 1 + s4 + SQR(0) :|: s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 2 }-> 1 + s5 + SQR(z - 1) :|: s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 1 }-> 1 + SQR(z) :|: z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + s27 :|: s27 >= 0, s27 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + s23) :|: s23 >= 0, s23 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + s24 :|: s24 >= 0, s24 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 0 }-> 1 + s25 :|: s25 >= 0, s25 <= 0 + (1 + (1 + s22)), s22 >= 0, s22 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s26 :|: s26 >= 0, s26 <= 0 + 0, z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + s43 :|: s41 >= 0, s41 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s42 >= 0, s42 <= s41 + s19, s43 >= 0, s43 <= 1 + s42 + (1 + (1 + s20)), s19 >= 0, s19 <= 2 * (z - 2), s20 >= 0, s20 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s46 :|: s44 >= 0, s44 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s45 >= 0, s45 <= s44 + s21, s46 >= 0, s46 <= 1 + s45 + 0, s21 >= 0, s21 <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + s40) + (1 + (1 + z)) :|: s40 >= 0, s40 <= 6 * (z * z) + 8 * z + 3, z >= 0

Function symbols to be analyzed: {SQR}, {terms}, {TERMS}, {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: O(1) [0], size: O(n^1) [2*z]
  add: runtime: O(1) [0], size: O(n^1) [z + z']
  sqr: runtime: O(1) [0], size: O(n^2) [3 + 8*z + 6*z^2]
  SQR: runtime: ?, size: INF

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: SQR
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 36 + 74*z + 48*z^2 + 12*z^3

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 2 }-> 1 + s13 + SQR(1 + (z - 2)) :|: s12 >= 0, s12 <= 2 * (z - 2), s13 >= 0, s13 <= 0, z - 2 >= 0
   SQR(z) -{ 2 + z }-> 1 + s18 + s1 :|: s17 >= 0, s17 <= 2 * (z - 2), s18 >= 0, s18 <= 0, s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s29 }-> 1 + s30 + SQR(1 + (z - 2)) :|: s28 >= 0, s28 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s29 >= 0, s29 <= s28 + s9, s30 >= 0, s30 <= 1 + s29, s9 >= 0, s9 <= 2 * (z - 2), s10 >= 0, s10 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s32 }-> 1 + s33 + SQR(1 + (z - 2)) :|: s31 >= 0, s31 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s32 >= 0, s32 <= s31 + s11, s33 >= 0, s33 <= 1 + s32, s11 >= 0, s11 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s35 + z }-> 1 + s36 + s' :|: s34 >= 0, s34 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s35 >= 0, s35 <= s34 + s14, s36 >= 0, s36 <= 1 + s35, s14 >= 0, s14 <= 2 * (z - 2), s15 >= 0, s15 <= 2 * (z - 2), s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s38 + z }-> 1 + s39 + s'' :|: s37 >= 0, s37 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s38 >= 0, s38 <= s37 + s16, s39 >= 0, s39 <= 1 + s38, s16 >= 0, s16 <= 2 * (z - 2), s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 2 }-> 1 + s4 + SQR(0) :|: s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 2 }-> 1 + s5 + SQR(z - 1) :|: s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 1 }-> 1 + SQR(z) :|: z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + s27 :|: s27 >= 0, s27 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + s23) :|: s23 >= 0, s23 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + s24 :|: s24 >= 0, s24 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 0 }-> 1 + s25 :|: s25 >= 0, s25 <= 0 + (1 + (1 + s22)), s22 >= 0, s22 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s26 :|: s26 >= 0, s26 <= 0 + 0, z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + s43 :|: s41 >= 0, s41 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s42 >= 0, s42 <= s41 + s19, s43 >= 0, s43 <= 1 + s42 + (1 + (1 + s20)), s19 >= 0, s19 <= 2 * (z - 2), s20 >= 0, s20 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s46 :|: s44 >= 0, s44 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s45 >= 0, s45 <= s44 + s21, s46 >= 0, s46 <= 1 + s45 + 0, s21 >= 0, s21 <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + s40) + (1 + (1 + z)) :|: s40 >= 0, s40 <= 6 * (z * z) + 8 * z + 3, z >= 0

Function symbols to be analyzed: {terms}, {TERMS}, {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: O(1) [0], size: O(n^1) [2*z]
  add: runtime: O(1) [0], size: O(n^1) [z + z']
  sqr: runtime: O(1) [0], size: O(n^2) [3 + 8*z + 6*z^2]
  SQR: runtime: O(n^3) [36 + 74*z + 48*z^2 + 12*z^3], size: INF

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

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s13 + s51 :|: s51 >= 0, s51 <= inf2, s12 >= 0, s12 <= 2 * (z - 2), s13 >= 0, s13 <= 0, z - 2 >= 0
   SQR(z) -{ 2 + z }-> 1 + s18 + s1 :|: s17 >= 0, s17 <= 2 * (z - 2), s18 >= 0, s18 <= 0, s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s29 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s30 + s49 :|: s49 >= 0, s49 <= inf'', s28 >= 0, s28 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s29 >= 0, s29 <= s28 + s9, s30 >= 0, s30 <= 1 + s29, s9 >= 0, s9 <= 2 * (z - 2), s10 >= 0, s10 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s32 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s33 + s50 :|: s50 >= 0, s50 <= inf1, s31 >= 0, s31 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s32 >= 0, s32 <= s31 + s11, s33 >= 0, s33 <= 1 + s32, s11 >= 0, s11 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s35 + z }-> 1 + s36 + s' :|: s34 >= 0, s34 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s35 >= 0, s35 <= s34 + s14, s36 >= 0, s36 <= 1 + s35, s14 >= 0, s14 <= 2 * (z - 2), s15 >= 0, s15 <= 2 * (z - 2), s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s38 + z }-> 1 + s39 + s'' :|: s37 >= 0, s37 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s38 >= 0, s38 <= s37 + s16, s39 >= 0, s39 <= 1 + s38, s16 >= 0, s16 <= 2 * (z - 2), s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 38 }-> 1 + s4 + s48 :|: s48 >= 0, s48 <= inf', s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s5 + s52 :|: s52 >= 0, s52 <= inf3, s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 37 + 74*z + 48*z^2 + 12*z^3 }-> 1 + s47 :|: s47 >= 0, s47 <= inf, z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + s27 :|: s27 >= 0, s27 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + s23) :|: s23 >= 0, s23 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + s24 :|: s24 >= 0, s24 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 0 }-> 1 + s25 :|: s25 >= 0, s25 <= 0 + (1 + (1 + s22)), s22 >= 0, s22 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s26 :|: s26 >= 0, s26 <= 0 + 0, z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + s43 :|: s41 >= 0, s41 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s42 >= 0, s42 <= s41 + s19, s43 >= 0, s43 <= 1 + s42 + (1 + (1 + s20)), s19 >= 0, s19 <= 2 * (z - 2), s20 >= 0, s20 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s46 :|: s44 >= 0, s44 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s45 >= 0, s45 <= s44 + s21, s46 >= 0, s46 <= 1 + s45 + 0, s21 >= 0, s21 <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + s40) + (1 + (1 + z)) :|: s40 >= 0, s40 <= 6 * (z * z) + 8 * z + 3, z >= 0

Function symbols to be analyzed: {terms}, {TERMS}, {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: O(1) [0], size: O(n^1) [2*z]
  add: runtime: O(1) [0], size: O(n^1) [z + z']
  sqr: runtime: O(1) [0], size: O(n^2) [3 + 8*z + 6*z^2]
  SQR: runtime: O(n^3) [36 + 74*z + 48*z^2 + 12*z^3], size: INF

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: terms
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 8 + 10*z + 6*z^2

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s13 + s51 :|: s51 >= 0, s51 <= inf2, s12 >= 0, s12 <= 2 * (z - 2), s13 >= 0, s13 <= 0, z - 2 >= 0
   SQR(z) -{ 2 + z }-> 1 + s18 + s1 :|: s17 >= 0, s17 <= 2 * (z - 2), s18 >= 0, s18 <= 0, s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s29 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s30 + s49 :|: s49 >= 0, s49 <= inf'', s28 >= 0, s28 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s29 >= 0, s29 <= s28 + s9, s30 >= 0, s30 <= 1 + s29, s9 >= 0, s9 <= 2 * (z - 2), s10 >= 0, s10 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s32 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s33 + s50 :|: s50 >= 0, s50 <= inf1, s31 >= 0, s31 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s32 >= 0, s32 <= s31 + s11, s33 >= 0, s33 <= 1 + s32, s11 >= 0, s11 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s35 + z }-> 1 + s36 + s' :|: s34 >= 0, s34 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s35 >= 0, s35 <= s34 + s14, s36 >= 0, s36 <= 1 + s35, s14 >= 0, s14 <= 2 * (z - 2), s15 >= 0, s15 <= 2 * (z - 2), s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s38 + z }-> 1 + s39 + s'' :|: s37 >= 0, s37 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s38 >= 0, s38 <= s37 + s16, s39 >= 0, s39 <= 1 + s38, s16 >= 0, s16 <= 2 * (z - 2), s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 38 }-> 1 + s4 + s48 :|: s48 >= 0, s48 <= inf', s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s5 + s52 :|: s52 >= 0, s52 <= inf3, s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 37 + 74*z + 48*z^2 + 12*z^3 }-> 1 + s47 :|: s47 >= 0, s47 <= inf, z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + s27 :|: s27 >= 0, s27 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + s23) :|: s23 >= 0, s23 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + s24 :|: s24 >= 0, s24 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 0 }-> 1 + s25 :|: s25 >= 0, s25 <= 0 + (1 + (1 + s22)), s22 >= 0, s22 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s26 :|: s26 >= 0, s26 <= 0 + 0, z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + s43 :|: s41 >= 0, s41 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s42 >= 0, s42 <= s41 + s19, s43 >= 0, s43 <= 1 + s42 + (1 + (1 + s20)), s19 >= 0, s19 <= 2 * (z - 2), s20 >= 0, s20 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s46 :|: s44 >= 0, s44 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s45 >= 0, s45 <= s44 + s21, s46 >= 0, s46 <= 1 + s45 + 0, s21 >= 0, s21 <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + s40) + (1 + (1 + z)) :|: s40 >= 0, s40 <= 6 * (z * z) + 8 * z + 3, z >= 0

Function symbols to be analyzed: {terms}, {TERMS}, {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: O(1) [0], size: O(n^1) [2*z]
  add: runtime: O(1) [0], size: O(n^1) [z + z']
  sqr: runtime: O(1) [0], size: O(n^2) [3 + 8*z + 6*z^2]
  SQR: runtime: O(n^3) [36 + 74*z + 48*z^2 + 12*z^3], size: INF
  terms: runtime: ?, size: O(n^2) [8 + 10*z + 6*z^2]

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s13 + s51 :|: s51 >= 0, s51 <= inf2, s12 >= 0, s12 <= 2 * (z - 2), s13 >= 0, s13 <= 0, z - 2 >= 0
   SQR(z) -{ 2 + z }-> 1 + s18 + s1 :|: s17 >= 0, s17 <= 2 * (z - 2), s18 >= 0, s18 <= 0, s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s29 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s30 + s49 :|: s49 >= 0, s49 <= inf'', s28 >= 0, s28 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s29 >= 0, s29 <= s28 + s9, s30 >= 0, s30 <= 1 + s29, s9 >= 0, s9 <= 2 * (z - 2), s10 >= 0, s10 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s32 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s33 + s50 :|: s50 >= 0, s50 <= inf1, s31 >= 0, s31 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s32 >= 0, s32 <= s31 + s11, s33 >= 0, s33 <= 1 + s32, s11 >= 0, s11 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s35 + z }-> 1 + s36 + s' :|: s34 >= 0, s34 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s35 >= 0, s35 <= s34 + s14, s36 >= 0, s36 <= 1 + s35, s14 >= 0, s14 <= 2 * (z - 2), s15 >= 0, s15 <= 2 * (z - 2), s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s38 + z }-> 1 + s39 + s'' :|: s37 >= 0, s37 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s38 >= 0, s38 <= s37 + s16, s39 >= 0, s39 <= 1 + s38, s16 >= 0, s16 <= 2 * (z - 2), s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 38 }-> 1 + s4 + s48 :|: s48 >= 0, s48 <= inf', s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s5 + s52 :|: s52 >= 0, s52 <= inf3, s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 37 + 74*z + 48*z^2 + 12*z^3 }-> 1 + s47 :|: s47 >= 0, s47 <= inf, z >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + s27 :|: s27 >= 0, s27 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + s23) :|: s23 >= 0, s23 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + s24 :|: s24 >= 0, s24 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 0 }-> 1 + s25 :|: s25 >= 0, s25 <= 0 + (1 + (1 + s22)), s22 >= 0, s22 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s26 :|: s26 >= 0, s26 <= 0 + 0, z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + s43 :|: s41 >= 0, s41 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s42 >= 0, s42 <= s41 + s19, s43 >= 0, s43 <= 1 + s42 + (1 + (1 + s20)), s19 >= 0, s19 <= 2 * (z - 2), s20 >= 0, s20 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s46 :|: s44 >= 0, s44 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s45 >= 0, s45 <= s44 + s21, s46 >= 0, s46 <= 1 + s45 + 0, s21 >= 0, s21 <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + s40) + (1 + (1 + z)) :|: s40 >= 0, s40 <= 6 * (z * z) + 8 * z + 3, z >= 0

Function symbols to be analyzed: {TERMS}, {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: O(1) [0], size: O(n^1) [2*z]
  add: runtime: O(1) [0], size: O(n^1) [z + z']
  sqr: runtime: O(1) [0], size: O(n^2) [3 + 8*z + 6*z^2]
  SQR: runtime: O(n^3) [36 + 74*z + 48*z^2 + 12*z^3], size: INF
  terms: runtime: O(1) [0], size: O(n^2) [8 + 10*z + 6*z^2]

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

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s13 + s51 :|: s51 >= 0, s51 <= inf2, s12 >= 0, s12 <= 2 * (z - 2), s13 >= 0, s13 <= 0, z - 2 >= 0
   SQR(z) -{ 2 + z }-> 1 + s18 + s1 :|: s17 >= 0, s17 <= 2 * (z - 2), s18 >= 0, s18 <= 0, s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s29 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s30 + s49 :|: s49 >= 0, s49 <= inf'', s28 >= 0, s28 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s29 >= 0, s29 <= s28 + s9, s30 >= 0, s30 <= 1 + s29, s9 >= 0, s9 <= 2 * (z - 2), s10 >= 0, s10 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s32 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s33 + s50 :|: s50 >= 0, s50 <= inf1, s31 >= 0, s31 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s32 >= 0, s32 <= s31 + s11, s33 >= 0, s33 <= 1 + s32, s11 >= 0, s11 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s35 + z }-> 1 + s36 + s' :|: s34 >= 0, s34 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s35 >= 0, s35 <= s34 + s14, s36 >= 0, s36 <= 1 + s35, s14 >= 0, s14 <= 2 * (z - 2), s15 >= 0, s15 <= 2 * (z - 2), s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s38 + z }-> 1 + s39 + s'' :|: s37 >= 0, s37 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s38 >= 0, s38 <= s37 + s16, s39 >= 0, s39 <= 1 + s38, s16 >= 0, s16 <= 2 * (z - 2), s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 38 }-> 1 + s4 + s48 :|: s48 >= 0, s48 <= inf', s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s5 + s52 :|: s52 >= 0, s52 <= inf3, s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 37 + 74*z + 48*z^2 + 12*z^3 }-> 1 + s47 :|: s47 >= 0, s47 <= inf, z >= 0
   activate(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 10 * (z - 1) + 6 * ((z - 1) * (z - 1)) + 8, z - 1 >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + s27 :|: s27 >= 0, s27 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + s23) :|: s23 >= 0, s23 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + s24 :|: s24 >= 0, s24 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 0 }-> 1 + s25 :|: s25 >= 0, s25 <= 0 + (1 + (1 + s22)), s22 >= 0, s22 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s26 :|: s26 >= 0, s26 <= 0 + 0, z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + s43 :|: s41 >= 0, s41 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s42 >= 0, s42 <= s41 + s19, s43 >= 0, s43 <= 1 + s42 + (1 + (1 + s20)), s19 >= 0, s19 <= 2 * (z - 2), s20 >= 0, s20 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s46 :|: s44 >= 0, s44 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s45 >= 0, s45 <= s44 + s21, s46 >= 0, s46 <= 1 + s45 + 0, s21 >= 0, s21 <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + s40) + (1 + (1 + z)) :|: s40 >= 0, s40 <= 6 * (z * z) + 8 * z + 3, z >= 0

Function symbols to be analyzed: {TERMS}, {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: O(1) [0], size: O(n^1) [2*z]
  add: runtime: O(1) [0], size: O(n^1) [z + z']
  sqr: runtime: O(1) [0], size: O(n^2) [3 + 8*z + 6*z^2]
  SQR: runtime: O(n^3) [36 + 74*z + 48*z^2 + 12*z^3], size: INF
  terms: runtime: O(1) [0], size: O(n^2) [8 + 10*z + 6*z^2]

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

(61) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: TERMS
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s13 + s51 :|: s51 >= 0, s51 <= inf2, s12 >= 0, s12 <= 2 * (z - 2), s13 >= 0, s13 <= 0, z - 2 >= 0
   SQR(z) -{ 2 + z }-> 1 + s18 + s1 :|: s17 >= 0, s17 <= 2 * (z - 2), s18 >= 0, s18 <= 0, s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s29 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s30 + s49 :|: s49 >= 0, s49 <= inf'', s28 >= 0, s28 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s29 >= 0, s29 <= s28 + s9, s30 >= 0, s30 <= 1 + s29, s9 >= 0, s9 <= 2 * (z - 2), s10 >= 0, s10 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s32 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s33 + s50 :|: s50 >= 0, s50 <= inf1, s31 >= 0, s31 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s32 >= 0, s32 <= s31 + s11, s33 >= 0, s33 <= 1 + s32, s11 >= 0, s11 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s35 + z }-> 1 + s36 + s' :|: s34 >= 0, s34 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s35 >= 0, s35 <= s34 + s14, s36 >= 0, s36 <= 1 + s35, s14 >= 0, s14 <= 2 * (z - 2), s15 >= 0, s15 <= 2 * (z - 2), s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s38 + z }-> 1 + s39 + s'' :|: s37 >= 0, s37 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s38 >= 0, s38 <= s37 + s16, s39 >= 0, s39 <= 1 + s38, s16 >= 0, s16 <= 2 * (z - 2), s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 38 }-> 1 + s4 + s48 :|: s48 >= 0, s48 <= inf', s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s5 + s52 :|: s52 >= 0, s52 <= inf3, s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 37 + 74*z + 48*z^2 + 12*z^3 }-> 1 + s47 :|: s47 >= 0, s47 <= inf, z >= 0
   activate(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 10 * (z - 1) + 6 * ((z - 1) * (z - 1)) + 8, z - 1 >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + s27 :|: s27 >= 0, s27 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + s23) :|: s23 >= 0, s23 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + s24 :|: s24 >= 0, s24 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 0 }-> 1 + s25 :|: s25 >= 0, s25 <= 0 + (1 + (1 + s22)), s22 >= 0, s22 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s26 :|: s26 >= 0, s26 <= 0 + 0, z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + s43 :|: s41 >= 0, s41 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s42 >= 0, s42 <= s41 + s19, s43 >= 0, s43 <= 1 + s42 + (1 + (1 + s20)), s19 >= 0, s19 <= 2 * (z - 2), s20 >= 0, s20 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s46 :|: s44 >= 0, s44 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s45 >= 0, s45 <= s44 + s21, s46 >= 0, s46 <= 1 + s45 + 0, s21 >= 0, s21 <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + s40) + (1 + (1 + z)) :|: s40 >= 0, s40 <= 6 * (z * z) + 8 * z + 3, z >= 0

Function symbols to be analyzed: {TERMS}, {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: O(1) [0], size: O(n^1) [2*z]
  add: runtime: O(1) [0], size: O(n^1) [z + z']
  sqr: runtime: O(1) [0], size: O(n^2) [3 + 8*z + 6*z^2]
  SQR: runtime: O(n^3) [36 + 74*z + 48*z^2 + 12*z^3], size: INF
  terms: runtime: O(1) [0], size: O(n^2) [8 + 10*z + 6*z^2]
  TERMS: runtime: ?, size: INF

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: TERMS
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 38 + 74*z + 48*z^2 + 12*z^3

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 }-> 1 + TERMS(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s13 + s51 :|: s51 >= 0, s51 <= inf2, s12 >= 0, s12 <= 2 * (z - 2), s13 >= 0, s13 <= 0, z - 2 >= 0
   SQR(z) -{ 2 + z }-> 1 + s18 + s1 :|: s17 >= 0, s17 <= 2 * (z - 2), s18 >= 0, s18 <= 0, s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s29 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s30 + s49 :|: s49 >= 0, s49 <= inf'', s28 >= 0, s28 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s29 >= 0, s29 <= s28 + s9, s30 >= 0, s30 <= 1 + s29, s9 >= 0, s9 <= 2 * (z - 2), s10 >= 0, s10 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s32 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s33 + s50 :|: s50 >= 0, s50 <= inf1, s31 >= 0, s31 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s32 >= 0, s32 <= s31 + s11, s33 >= 0, s33 <= 1 + s32, s11 >= 0, s11 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s35 + z }-> 1 + s36 + s' :|: s34 >= 0, s34 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s35 >= 0, s35 <= s34 + s14, s36 >= 0, s36 <= 1 + s35, s14 >= 0, s14 <= 2 * (z - 2), s15 >= 0, s15 <= 2 * (z - 2), s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s38 + z }-> 1 + s39 + s'' :|: s37 >= 0, s37 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s38 >= 0, s38 <= s37 + s16, s39 >= 0, s39 <= 1 + s38, s16 >= 0, s16 <= 2 * (z - 2), s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 38 }-> 1 + s4 + s48 :|: s48 >= 0, s48 <= inf', s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s5 + s52 :|: s52 >= 0, s52 <= inf3, s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 37 + 74*z + 48*z^2 + 12*z^3 }-> 1 + s47 :|: s47 >= 0, s47 <= inf, z >= 0
   activate(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 10 * (z - 1) + 6 * ((z - 1) * (z - 1)) + 8, z - 1 >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + s27 :|: s27 >= 0, s27 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + s23) :|: s23 >= 0, s23 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + s24 :|: s24 >= 0, s24 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 0 }-> 1 + s25 :|: s25 >= 0, s25 <= 0 + (1 + (1 + s22)), s22 >= 0, s22 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s26 :|: s26 >= 0, s26 <= 0 + 0, z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + s43 :|: s41 >= 0, s41 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s42 >= 0, s42 <= s41 + s19, s43 >= 0, s43 <= 1 + s42 + (1 + (1 + s20)), s19 >= 0, s19 <= 2 * (z - 2), s20 >= 0, s20 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s46 :|: s44 >= 0, s44 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s45 >= 0, s45 <= s44 + s21, s46 >= 0, s46 <= 1 + s45 + 0, s21 >= 0, s21 <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + s40) + (1 + (1 + z)) :|: s40 >= 0, s40 <= 6 * (z * z) + 8 * z + 3, z >= 0

Function symbols to be analyzed: {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: O(1) [0], size: O(n^1) [2*z]
  add: runtime: O(1) [0], size: O(n^1) [z + z']
  sqr: runtime: O(1) [0], size: O(n^2) [3 + 8*z + 6*z^2]
  SQR: runtime: O(n^3) [36 + 74*z + 48*z^2 + 12*z^3], size: INF
  terms: runtime: O(1) [0], size: O(n^2) [8 + 10*z + 6*z^2]
  TERMS: runtime: O(n^3) [38 + 74*z + 48*z^2 + 12*z^3], size: INF

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

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s54 :|: s54 >= 0, s54 <= inf4, z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s13 + s51 :|: s51 >= 0, s51 <= inf2, s12 >= 0, s12 <= 2 * (z - 2), s13 >= 0, s13 <= 0, z - 2 >= 0
   SQR(z) -{ 2 + z }-> 1 + s18 + s1 :|: s17 >= 0, s17 <= 2 * (z - 2), s18 >= 0, s18 <= 0, s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s29 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s30 + s49 :|: s49 >= 0, s49 <= inf'', s28 >= 0, s28 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s29 >= 0, s29 <= s28 + s9, s30 >= 0, s30 <= 1 + s29, s9 >= 0, s9 <= 2 * (z - 2), s10 >= 0, s10 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s32 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s33 + s50 :|: s50 >= 0, s50 <= inf1, s31 >= 0, s31 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s32 >= 0, s32 <= s31 + s11, s33 >= 0, s33 <= 1 + s32, s11 >= 0, s11 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s35 + z }-> 1 + s36 + s' :|: s34 >= 0, s34 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s35 >= 0, s35 <= s34 + s14, s36 >= 0, s36 <= 1 + s35, s14 >= 0, s14 <= 2 * (z - 2), s15 >= 0, s15 <= 2 * (z - 2), s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s38 + z }-> 1 + s39 + s'' :|: s37 >= 0, s37 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s38 >= 0, s38 <= s37 + s16, s39 >= 0, s39 <= 1 + s38, s16 >= 0, s16 <= 2 * (z - 2), s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 38 }-> 1 + s4 + s48 :|: s48 >= 0, s48 <= inf', s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s5 + s52 :|: s52 >= 0, s52 <= inf3, s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 37 + 74*z + 48*z^2 + 12*z^3 }-> 1 + s47 :|: s47 >= 0, s47 <= inf, z >= 0
   activate(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 10 * (z - 1) + 6 * ((z - 1) * (z - 1)) + 8, z - 1 >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + s27 :|: s27 >= 0, s27 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + s23) :|: s23 >= 0, s23 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + s24 :|: s24 >= 0, s24 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 0 }-> 1 + s25 :|: s25 >= 0, s25 <= 0 + (1 + (1 + s22)), s22 >= 0, s22 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s26 :|: s26 >= 0, s26 <= 0 + 0, z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + s43 :|: s41 >= 0, s41 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s42 >= 0, s42 <= s41 + s19, s43 >= 0, s43 <= 1 + s42 + (1 + (1 + s20)), s19 >= 0, s19 <= 2 * (z - 2), s20 >= 0, s20 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s46 :|: s44 >= 0, s44 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s45 >= 0, s45 <= s44 + s21, s46 >= 0, s46 <= 1 + s45 + 0, s21 >= 0, s21 <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + s40) + (1 + (1 + z)) :|: s40 >= 0, s40 <= 6 * (z * z) + 8 * z + 3, z >= 0

Function symbols to be analyzed: {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: O(1) [0], size: O(n^1) [2*z]
  add: runtime: O(1) [0], size: O(n^1) [z + z']
  sqr: runtime: O(1) [0], size: O(n^2) [3 + 8*z + 6*z^2]
  SQR: runtime: O(n^3) [36 + 74*z + 48*z^2 + 12*z^3], size: INF
  terms: runtime: O(1) [0], size: O(n^2) [8 + 10*z + 6*z^2]
  TERMS: runtime: O(n^3) [38 + 74*z + 48*z^2 + 12*z^3], size: INF

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

(67) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: first
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

Computed SIZE bound using CoFloCo for: activate
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s54 :|: s54 >= 0, s54 <= inf4, z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s13 + s51 :|: s51 >= 0, s51 <= inf2, s12 >= 0, s12 <= 2 * (z - 2), s13 >= 0, s13 <= 0, z - 2 >= 0
   SQR(z) -{ 2 + z }-> 1 + s18 + s1 :|: s17 >= 0, s17 <= 2 * (z - 2), s18 >= 0, s18 <= 0, s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s29 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s30 + s49 :|: s49 >= 0, s49 <= inf'', s28 >= 0, s28 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s29 >= 0, s29 <= s28 + s9, s30 >= 0, s30 <= 1 + s29, s9 >= 0, s9 <= 2 * (z - 2), s10 >= 0, s10 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s32 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s33 + s50 :|: s50 >= 0, s50 <= inf1, s31 >= 0, s31 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s32 >= 0, s32 <= s31 + s11, s33 >= 0, s33 <= 1 + s32, s11 >= 0, s11 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s35 + z }-> 1 + s36 + s' :|: s34 >= 0, s34 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s35 >= 0, s35 <= s34 + s14, s36 >= 0, s36 <= 1 + s35, s14 >= 0, s14 <= 2 * (z - 2), s15 >= 0, s15 <= 2 * (z - 2), s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s38 + z }-> 1 + s39 + s'' :|: s37 >= 0, s37 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s38 >= 0, s38 <= s37 + s16, s39 >= 0, s39 <= 1 + s38, s16 >= 0, s16 <= 2 * (z - 2), s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 38 }-> 1 + s4 + s48 :|: s48 >= 0, s48 <= inf', s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s5 + s52 :|: s52 >= 0, s52 <= inf3, s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 37 + 74*z + 48*z^2 + 12*z^3 }-> 1 + s47 :|: s47 >= 0, s47 <= inf, z >= 0
   activate(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 10 * (z - 1) + 6 * ((z - 1) * (z - 1)) + 8, z - 1 >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + s27 :|: s27 >= 0, s27 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + s23) :|: s23 >= 0, s23 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + s24 :|: s24 >= 0, s24 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 0 }-> 1 + s25 :|: s25 >= 0, s25 <= 0 + (1 + (1 + s22)), s22 >= 0, s22 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s26 :|: s26 >= 0, s26 <= 0 + 0, z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + s43 :|: s41 >= 0, s41 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s42 >= 0, s42 <= s41 + s19, s43 >= 0, s43 <= 1 + s42 + (1 + (1 + s20)), s19 >= 0, s19 <= 2 * (z - 2), s20 >= 0, s20 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s46 :|: s44 >= 0, s44 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s45 >= 0, s45 <= s44 + s21, s46 >= 0, s46 <= 1 + s45 + 0, s21 >= 0, s21 <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + s40) + (1 + (1 + z)) :|: s40 >= 0, s40 <= 6 * (z * z) + 8 * z + 3, z >= 0

Function symbols to be analyzed: {first,activate}, {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: O(1) [0], size: O(n^1) [2*z]
  add: runtime: O(1) [0], size: O(n^1) [z + z']
  sqr: runtime: O(1) [0], size: O(n^2) [3 + 8*z + 6*z^2]
  SQR: runtime: O(n^3) [36 + 74*z + 48*z^2 + 12*z^3], size: INF
  terms: runtime: O(1) [0], size: O(n^2) [8 + 10*z + 6*z^2]
  TERMS: runtime: O(n^3) [38 + 74*z + 48*z^2 + 12*z^3], size: INF
  first: runtime: ?, size: INF
  activate: runtime: ?, size: INF

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

(69) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s54 :|: s54 >= 0, s54 <= inf4, z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s13 + s51 :|: s51 >= 0, s51 <= inf2, s12 >= 0, s12 <= 2 * (z - 2), s13 >= 0, s13 <= 0, z - 2 >= 0
   SQR(z) -{ 2 + z }-> 1 + s18 + s1 :|: s17 >= 0, s17 <= 2 * (z - 2), s18 >= 0, s18 <= 0, s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s29 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s30 + s49 :|: s49 >= 0, s49 <= inf'', s28 >= 0, s28 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s29 >= 0, s29 <= s28 + s9, s30 >= 0, s30 <= 1 + s29, s9 >= 0, s9 <= 2 * (z - 2), s10 >= 0, s10 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s32 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s33 + s50 :|: s50 >= 0, s50 <= inf1, s31 >= 0, s31 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s32 >= 0, s32 <= s31 + s11, s33 >= 0, s33 <= 1 + s32, s11 >= 0, s11 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s35 + z }-> 1 + s36 + s' :|: s34 >= 0, s34 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s35 >= 0, s35 <= s34 + s14, s36 >= 0, s36 <= 1 + s35, s14 >= 0, s14 <= 2 * (z - 2), s15 >= 0, s15 <= 2 * (z - 2), s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s38 + z }-> 1 + s39 + s'' :|: s37 >= 0, s37 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s38 >= 0, s38 <= s37 + s16, s39 >= 0, s39 <= 1 + s38, s16 >= 0, s16 <= 2 * (z - 2), s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 38 }-> 1 + s4 + s48 :|: s48 >= 0, s48 <= inf', s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s5 + s52 :|: s52 >= 0, s52 <= inf3, s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 37 + 74*z + 48*z^2 + 12*z^3 }-> 1 + s47 :|: s47 >= 0, s47 <= inf, z >= 0
   activate(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 10 * (z - 1) + 6 * ((z - 1) * (z - 1)) + 8, z - 1 >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> first(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + s27 :|: s27 >= 0, s27 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + s23) :|: s23 >= 0, s23 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + activate(z2)) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + s24 :|: s24 >= 0, s24 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 0 }-> 1 + s25 :|: s25 >= 0, s25 <= 0 + (1 + (1 + s22)), s22 >= 0, s22 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s26 :|: s26 >= 0, s26 <= 0 + 0, z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + s43 :|: s41 >= 0, s41 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s42 >= 0, s42 <= s41 + s19, s43 >= 0, s43 <= 1 + s42 + (1 + (1 + s20)), s19 >= 0, s19 <= 2 * (z - 2), s20 >= 0, s20 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s46 :|: s44 >= 0, s44 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s45 >= 0, s45 <= s44 + s21, s46 >= 0, s46 <= 1 + s45 + 0, s21 >= 0, s21 <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + s40) + (1 + (1 + z)) :|: s40 >= 0, s40 <= 6 * (z * z) + 8 * z + 3, z >= 0

Function symbols to be analyzed: {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: O(1) [0], size: O(n^1) [2*z]
  add: runtime: O(1) [0], size: O(n^1) [z + z']
  sqr: runtime: O(1) [0], size: O(n^2) [3 + 8*z + 6*z^2]
  SQR: runtime: O(n^3) [36 + 74*z + 48*z^2 + 12*z^3], size: INF
  terms: runtime: O(1) [0], size: O(n^2) [8 + 10*z + 6*z^2]
  TERMS: runtime: O(n^3) [38 + 74*z + 48*z^2 + 12*z^3], size: INF
  first: runtime: O(1) [0], size: INF
  activate: runtime: O(1) [0], size: INF

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

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s54 :|: s54 >= 0, s54 <= inf4, z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s13 + s51 :|: s51 >= 0, s51 <= inf2, s12 >= 0, s12 <= 2 * (z - 2), s13 >= 0, s13 <= 0, z - 2 >= 0
   SQR(z) -{ 2 + z }-> 1 + s18 + s1 :|: s17 >= 0, s17 <= 2 * (z - 2), s18 >= 0, s18 <= 0, s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s29 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s30 + s49 :|: s49 >= 0, s49 <= inf'', s28 >= 0, s28 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s29 >= 0, s29 <= s28 + s9, s30 >= 0, s30 <= 1 + s29, s9 >= 0, s9 <= 2 * (z - 2), s10 >= 0, s10 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s32 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s33 + s50 :|: s50 >= 0, s50 <= inf1, s31 >= 0, s31 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s32 >= 0, s32 <= s31 + s11, s33 >= 0, s33 <= 1 + s32, s11 >= 0, s11 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s35 + z }-> 1 + s36 + s' :|: s34 >= 0, s34 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s35 >= 0, s35 <= s34 + s14, s36 >= 0, s36 <= 1 + s35, s14 >= 0, s14 <= 2 * (z - 2), s15 >= 0, s15 <= 2 * (z - 2), s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s38 + z }-> 1 + s39 + s'' :|: s37 >= 0, s37 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s38 >= 0, s38 <= s37 + s16, s39 >= 0, s39 <= 1 + s38, s16 >= 0, s16 <= 2 * (z - 2), s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 38 }-> 1 + s4 + s48 :|: s48 >= 0, s48 <= inf', s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s5 + s52 :|: s52 >= 0, s52 <= inf3, s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 37 + 74*z + 48*z^2 + 12*z^3 }-> 1 + s47 :|: s47 >= 0, s47 <= inf, z >= 0
   activate(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 10 * (z - 1) + 6 * ((z - 1) * (z - 1)) + 8, z - 1 >= 0
   activate(z) -{ 0 }-> s56 :|: s56 >= 0, s56 <= inf6, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + s27 :|: s27 >= 0, s27 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + s23) :|: s23 >= 0, s23 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + s55) :|: s55 >= 0, s55 <= inf5, z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + s24 :|: s24 >= 0, s24 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 0 }-> 1 + s25 :|: s25 >= 0, s25 <= 0 + (1 + (1 + s22)), s22 >= 0, s22 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s26 :|: s26 >= 0, s26 <= 0 + 0, z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + s43 :|: s41 >= 0, s41 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s42 >= 0, s42 <= s41 + s19, s43 >= 0, s43 <= 1 + s42 + (1 + (1 + s20)), s19 >= 0, s19 <= 2 * (z - 2), s20 >= 0, s20 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s46 :|: s44 >= 0, s44 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s45 >= 0, s45 <= s44 + s21, s46 >= 0, s46 <= 1 + s45 + 0, s21 >= 0, s21 <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + s40) + (1 + (1 + z)) :|: s40 >= 0, s40 <= 6 * (z * z) + 8 * z + 3, z >= 0

Function symbols to be analyzed: {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: O(1) [0], size: O(n^1) [2*z]
  add: runtime: O(1) [0], size: O(n^1) [z + z']
  sqr: runtime: O(1) [0], size: O(n^2) [3 + 8*z + 6*z^2]
  SQR: runtime: O(n^3) [36 + 74*z + 48*z^2 + 12*z^3], size: INF
  terms: runtime: O(1) [0], size: O(n^2) [8 + 10*z + 6*z^2]
  TERMS: runtime: O(n^3) [38 + 74*z + 48*z^2 + 12*z^3], size: INF
  first: runtime: O(1) [0], size: INF
  activate: runtime: O(1) [0], size: INF

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

(73) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: FIRST
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

Computed SIZE bound using CoFloCo for: ACTIVATE
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s54 :|: s54 >= 0, s54 <= inf4, z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s13 + s51 :|: s51 >= 0, s51 <= inf2, s12 >= 0, s12 <= 2 * (z - 2), s13 >= 0, s13 <= 0, z - 2 >= 0
   SQR(z) -{ 2 + z }-> 1 + s18 + s1 :|: s17 >= 0, s17 <= 2 * (z - 2), s18 >= 0, s18 <= 0, s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s29 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s30 + s49 :|: s49 >= 0, s49 <= inf'', s28 >= 0, s28 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s29 >= 0, s29 <= s28 + s9, s30 >= 0, s30 <= 1 + s29, s9 >= 0, s9 <= 2 * (z - 2), s10 >= 0, s10 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s32 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s33 + s50 :|: s50 >= 0, s50 <= inf1, s31 >= 0, s31 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s32 >= 0, s32 <= s31 + s11, s33 >= 0, s33 <= 1 + s32, s11 >= 0, s11 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s35 + z }-> 1 + s36 + s' :|: s34 >= 0, s34 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s35 >= 0, s35 <= s34 + s14, s36 >= 0, s36 <= 1 + s35, s14 >= 0, s14 <= 2 * (z - 2), s15 >= 0, s15 <= 2 * (z - 2), s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s38 + z }-> 1 + s39 + s'' :|: s37 >= 0, s37 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s38 >= 0, s38 <= s37 + s16, s39 >= 0, s39 <= 1 + s38, s16 >= 0, s16 <= 2 * (z - 2), s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 38 }-> 1 + s4 + s48 :|: s48 >= 0, s48 <= inf', s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s5 + s52 :|: s52 >= 0, s52 <= inf3, s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 37 + 74*z + 48*z^2 + 12*z^3 }-> 1 + s47 :|: s47 >= 0, s47 <= inf, z >= 0
   activate(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 10 * (z - 1) + 6 * ((z - 1) * (z - 1)) + 8, z - 1 >= 0
   activate(z) -{ 0 }-> s56 :|: s56 >= 0, s56 <= inf6, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + s27 :|: s27 >= 0, s27 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + s23) :|: s23 >= 0, s23 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + s55) :|: s55 >= 0, s55 <= inf5, z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + s24 :|: s24 >= 0, s24 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 0 }-> 1 + s25 :|: s25 >= 0, s25 <= 0 + (1 + (1 + s22)), s22 >= 0, s22 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s26 :|: s26 >= 0, s26 <= 0 + 0, z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + s43 :|: s41 >= 0, s41 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s42 >= 0, s42 <= s41 + s19, s43 >= 0, s43 <= 1 + s42 + (1 + (1 + s20)), s19 >= 0, s19 <= 2 * (z - 2), s20 >= 0, s20 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s46 :|: s44 >= 0, s44 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s45 >= 0, s45 <= s44 + s21, s46 >= 0, s46 <= 1 + s45 + 0, s21 >= 0, s21 <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + s40) + (1 + (1 + z)) :|: s40 >= 0, s40 <= 6 * (z * z) + 8 * z + 3, z >= 0

Function symbols to be analyzed: {FIRST,ACTIVATE}
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: O(1) [0], size: O(n^1) [2*z]
  add: runtime: O(1) [0], size: O(n^1) [z + z']
  sqr: runtime: O(1) [0], size: O(n^2) [3 + 8*z + 6*z^2]
  SQR: runtime: O(n^3) [36 + 74*z + 48*z^2 + 12*z^3], size: INF
  terms: runtime: O(1) [0], size: O(n^2) [8 + 10*z + 6*z^2]
  TERMS: runtime: O(n^3) [38 + 74*z + 48*z^2 + 12*z^3], size: INF
  first: runtime: O(1) [0], size: INF
  activate: runtime: O(1) [0], size: INF
  FIRST: runtime: ?, size: INF
  ACTIVATE: runtime: ?, size: INF

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

(75) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: FIRST
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 4 + 16*z' + 12*z'^2 + 12*z'^3

Computed RUNTIME bound using KoAT for: ACTIVATE
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 7 + 30*z + 24*z^2 + 24*z^3

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

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

   ACTIVATE(z) -{ 1 }-> 0 :|: z >= 0
   ACTIVATE(z) -{ 1 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s54 :|: s54 >= 0, s54 <= inf4, z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + FIRST(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ADD(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   ADD(z, z') -{ 1 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 1, z' >= 0, z - 1 >= 0
   DBL(z) -{ 1 }-> 0 :|: z = 0
   DBL(z) -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1, z - 1 >= 0
   FIRST(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   FIRST(z, z') -{ 1 }-> 0 :|: z' >= 0, z >= 0
   FIRST(z, z') -{ 1 }-> 1 + ACTIVATE(z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   SQR(z) -{ 1 }-> 0 :|: z = 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s13 + s51 :|: s51 >= 0, s51 <= inf2, s12 >= 0, s12 <= 2 * (z - 2), s13 >= 0, s13 <= 0, z - 2 >= 0
   SQR(z) -{ 2 + z }-> 1 + s18 + s1 :|: s17 >= 0, s17 <= 2 * (z - 2), s18 >= 0, s18 <= 0, s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s29 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s30 + s49 :|: s49 >= 0, s49 <= inf'', s28 >= 0, s28 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s29 >= 0, s29 <= s28 + s9, s30 >= 0, s30 <= 1 + s29, s9 >= 0, s9 <= 2 * (z - 2), s10 >= 0, s10 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 1 + s32 + 14*z + 12*z^2 + 12*z^3 }-> 1 + s33 + s50 :|: s50 >= 0, s50 <= inf1, s31 >= 0, s31 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s32 >= 0, s32 <= s31 + s11, s33 >= 0, s33 <= 1 + s32, s11 >= 0, s11 <= 2 * (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s35 + z }-> 1 + s36 + s' :|: s34 >= 0, s34 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s35 >= 0, s35 <= s34 + s14, s36 >= 0, s36 <= 1 + s35, s14 >= 0, s14 <= 2 * (z - 2), s15 >= 0, s15 <= 2 * (z - 2), s' >= 0, s' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 3 + s38 + z }-> 1 + s39 + s'' :|: s37 >= 0, s37 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s38 >= 0, s38 <= s37 + s16, s39 >= 0, s39 <= 1 + s38, s16 >= 0, s16 <= 2 * (z - 2), s'' >= 0, s'' <= 1 + (z - 2), z - 2 >= 0
   SQR(z) -{ 38 }-> 1 + s4 + s48 :|: s48 >= 0, s48 <= inf', s4 >= 0, s4 <= 0, z = 1 + 0
   SQR(z) -{ 14*z + 12*z^2 + 12*z^3 }-> 1 + s5 + s52 :|: s52 >= 0, s52 <= inf3, s5 >= 0, s5 <= 0, z - 1 >= 0
   SQR(z) -{ 3 }-> 1 + s6 + s :|: s6 >= 0, s6 <= 0, s >= 0, s <= 0, z = 1 + 0
   SQR(z) -{ 2 + z }-> 1 + s7 + s2 :|: s7 >= 0, s7 <= 0, s2 >= 0, s2 <= z - 1, z - 1 >= 0
   TERMS(z) -{ 1 }-> 0 :|: z >= 0
   TERMS(z) -{ 37 + 74*z + 48*z^2 + 12*z^3 }-> 1 + s47 :|: s47 >= 0, s47 <= inf, z >= 0
   activate(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 10 * (z - 1) + 6 * ((z - 1) * (z - 1)) + 8, z - 1 >= 0
   activate(z) -{ 0 }-> s56 :|: s56 >= 0, s56 <= inf6, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   add(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   add(z, z') -{ 0 }-> 1 + s27 :|: s27 >= 0, s27 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 0 }-> 0 :|: z = 0
   dbl(z) -{ 0 }-> 0 :|: z >= 0
   dbl(z) -{ 0 }-> 1 + (1 + s23) :|: s23 >= 0, s23 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   first(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0
   first(z, z') -{ 0 }-> 1 + z1 + (1 + (z - 1) + s55) :|: s55 >= 0, s55 <= inf5, z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   sqr(z) -{ 0 }-> 0 :|: z = 0
   sqr(z) -{ 0 }-> 0 :|: z >= 0
   sqr(z) -{ 0 }-> 1 + s24 :|: s24 >= 0, s24 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 0 }-> 1 + s25 :|: s25 >= 0, s25 <= 0 + (1 + (1 + s22)), s22 >= 0, s22 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s26 :|: s26 >= 0, s26 <= 0 + 0, z - 1 >= 0
   sqr(z) -{ 0 }-> 1 + s43 :|: s41 >= 0, s41 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s42 >= 0, s42 <= s41 + s19, s43 >= 0, s43 <= 1 + s42 + (1 + (1 + s20)), s19 >= 0, s19 <= 2 * (z - 2), s20 >= 0, s20 <= 2 * (z - 2), z - 2 >= 0
   sqr(z) -{ 0 }-> 1 + s46 :|: s44 >= 0, s44 <= 6 * ((z - 2) * (z - 2)) + 8 * (z - 2) + 3, s45 >= 0, s45 <= s44 + s21, s46 >= 0, s46 <= 1 + s45 + 0, s21 >= 0, s21 <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 0 }-> 0 :|: z >= 0
   terms(z) -{ 0 }-> 1 + z :|: z >= 0
   terms(z) -{ 0 }-> 1 + (1 + s40) + (1 + (1 + z)) :|: s40 >= 0, s40 <= 6 * (z * z) + 8 * z + 3, z >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  DBL: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  ADD: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  dbl: runtime: O(1) [0], size: O(n^1) [2*z]
  add: runtime: O(1) [0], size: O(n^1) [z + z']
  sqr: runtime: O(1) [0], size: O(n^2) [3 + 8*z + 6*z^2]
  SQR: runtime: O(n^3) [36 + 74*z + 48*z^2 + 12*z^3], size: INF
  terms: runtime: O(1) [0], size: O(n^2) [8 + 10*z + 6*z^2]
  TERMS: runtime: O(n^3) [38 + 74*z + 48*z^2 + 12*z^3], size: INF
  first: runtime: O(1) [0], size: INF
  activate: runtime: O(1) [0], size: INF
  FIRST: runtime: O(n^3) [4 + 16*z' + 12*z'^2 + 12*z'^3], size: INF
  ACTIVATE: runtime: O(n^3) [7 + 30*z + 24*z^2 + 24*z^3], size: INF

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

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

(78)
BOUNDS(1, n^3)

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

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

(80)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1
   SQR(0') -> c2
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
   SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0))
   DBL(0') -> c5
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(0', z0) -> c7
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(0', z0) -> c9
   FIRST(s(z0), cons(z1, z2)) -> c10(ACTIVATE(z2))
   FIRST(z0, z1) -> c11
   ACTIVATE(n__terms(z0)) -> c12(TERMS(z0))
   ACTIVATE(n__first(z0, z1)) -> c13(FIRST(z0, z1))
   ACTIVATE(z0) -> c14

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

   terms(z0) -> cons(recip(sqr(z0)), n__terms(s(z0)))
   terms(z0) -> n__terms(z0)
   sqr(0') -> 0'
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   dbl(0') -> 0'
   dbl(s(z0)) -> s(s(dbl(z0)))
   add(0', z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   first(0', z0) -> nil
   first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2)))
   first(z0, z1) -> n__first(z0, z1)
   activate(n__terms(z0)) -> terms(z0)
   activate(n__first(z0, z1)) -> first(z0, z1)
   activate(z0) -> z0

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

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

(82)
Obligation:
Innermost TRS:
Rules:
TERMS(z0) -> c(SQR(z0))
TERMS(z0) -> c1
SQR(0') -> c2
SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0))
DBL(0') -> c5
DBL(s(z0)) -> c6(DBL(z0))
ADD(0', z0) -> c7
ADD(s(z0), z1) -> c8(ADD(z0, z1))
FIRST(0', z0) -> c9
FIRST(s(z0), cons(z1, z2)) -> c10(ACTIVATE(z2))
FIRST(z0, z1) -> c11
ACTIVATE(n__terms(z0)) -> c12(TERMS(z0))
ACTIVATE(n__first(z0, z1)) -> c13(FIRST(z0, z1))
ACTIVATE(z0) -> c14
terms(z0) -> cons(recip(sqr(z0)), n__terms(s(z0)))
terms(z0) -> n__terms(z0)
sqr(0') -> 0'
sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
dbl(0') -> 0'
dbl(s(z0)) -> s(s(dbl(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
first(0', z0) -> nil
first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2)))
first(z0, z1) -> n__first(z0, z1)
activate(n__terms(z0)) -> terms(z0)
activate(n__first(z0, z1)) -> first(z0, z1)
activate(z0) -> z0

Types:
TERMS :: 0':s -> c:c1
c :: c2:c3:c4 -> c:c1
SQR :: 0':s -> c2:c3:c4
c1 :: c:c1
0' :: 0':s
c2 :: c2:c3:c4
s :: 0':s -> 0':s
c3 :: c7:c8 -> c2:c3:c4 -> c2:c3:c4
ADD :: 0':s -> 0':s -> c7:c8
sqr :: 0':s -> 0':s
dbl :: 0':s -> 0':s
c4 :: c7:c8 -> c5:c6 -> c2:c3:c4
DBL :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8
c8 :: c7:c8 -> c7:c8
FIRST :: 0':s -> cons:n__terms:n__first:nil -> c9:c10:c11
c9 :: c9:c10:c11
cons :: recip -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c10 :: c12:c13:c14 -> c9:c10:c11
ACTIVATE :: cons:n__terms:n__first:nil -> c12:c13:c14
c11 :: c9:c10:c11
n__terms :: 0':s -> cons:n__terms:n__first:nil
c12 :: c:c1 -> c12:c13:c14
n__first :: 0':s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c13 :: c9:c10:c11 -> c12:c13:c14
c14 :: c12:c13:c14
terms :: 0':s -> cons:n__terms:n__first:nil
recip :: 0':s -> recip
add :: 0':s -> 0':s -> 0':s
first :: 0':s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
nil :: cons:n__terms:n__first:nil
activate :: cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
hole_c:c11_15 :: c:c1
hole_0':s2_15 :: 0':s
hole_c2:c3:c43_15 :: c2:c3:c4
hole_c7:c84_15 :: c7:c8
hole_c5:c65_15 :: c5:c6
hole_c9:c10:c116_15 :: c9:c10:c11
hole_cons:n__terms:n__first:nil7_15 :: cons:n__terms:n__first:nil
hole_recip8_15 :: recip
hole_c12:c13:c149_15 :: c12:c13:c14
gen_0':s10_15 :: Nat -> 0':s
gen_c2:c3:c411_15 :: Nat -> c2:c3:c4
gen_c7:c812_15 :: Nat -> c7:c8
gen_c5:c613_15 :: Nat -> c5:c6
gen_cons:n__terms:n__first:nil14_15 :: Nat -> cons:n__terms:n__first:nil

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

(83) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
SQR, ADD, sqr, dbl, DBL, ACTIVATE, add, activate

They will be analysed ascendingly in the following order:
ADD < SQR
sqr < SQR
dbl < SQR
DBL < SQR
dbl < sqr
add < sqr

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

(84)
Obligation:
Innermost TRS:
Rules:
TERMS(z0) -> c(SQR(z0))
TERMS(z0) -> c1
SQR(0') -> c2
SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0))
DBL(0') -> c5
DBL(s(z0)) -> c6(DBL(z0))
ADD(0', z0) -> c7
ADD(s(z0), z1) -> c8(ADD(z0, z1))
FIRST(0', z0) -> c9
FIRST(s(z0), cons(z1, z2)) -> c10(ACTIVATE(z2))
FIRST(z0, z1) -> c11
ACTIVATE(n__terms(z0)) -> c12(TERMS(z0))
ACTIVATE(n__first(z0, z1)) -> c13(FIRST(z0, z1))
ACTIVATE(z0) -> c14
terms(z0) -> cons(recip(sqr(z0)), n__terms(s(z0)))
terms(z0) -> n__terms(z0)
sqr(0') -> 0'
sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
dbl(0') -> 0'
dbl(s(z0)) -> s(s(dbl(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
first(0', z0) -> nil
first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2)))
first(z0, z1) -> n__first(z0, z1)
activate(n__terms(z0)) -> terms(z0)
activate(n__first(z0, z1)) -> first(z0, z1)
activate(z0) -> z0

Types:
TERMS :: 0':s -> c:c1
c :: c2:c3:c4 -> c:c1
SQR :: 0':s -> c2:c3:c4
c1 :: c:c1
0' :: 0':s
c2 :: c2:c3:c4
s :: 0':s -> 0':s
c3 :: c7:c8 -> c2:c3:c4 -> c2:c3:c4
ADD :: 0':s -> 0':s -> c7:c8
sqr :: 0':s -> 0':s
dbl :: 0':s -> 0':s
c4 :: c7:c8 -> c5:c6 -> c2:c3:c4
DBL :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8
c8 :: c7:c8 -> c7:c8
FIRST :: 0':s -> cons:n__terms:n__first:nil -> c9:c10:c11
c9 :: c9:c10:c11
cons :: recip -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c10 :: c12:c13:c14 -> c9:c10:c11
ACTIVATE :: cons:n__terms:n__first:nil -> c12:c13:c14
c11 :: c9:c10:c11
n__terms :: 0':s -> cons:n__terms:n__first:nil
c12 :: c:c1 -> c12:c13:c14
n__first :: 0':s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c13 :: c9:c10:c11 -> c12:c13:c14
c14 :: c12:c13:c14
terms :: 0':s -> cons:n__terms:n__first:nil
recip :: 0':s -> recip
add :: 0':s -> 0':s -> 0':s
first :: 0':s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
nil :: cons:n__terms:n__first:nil
activate :: cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
hole_c:c11_15 :: c:c1
hole_0':s2_15 :: 0':s
hole_c2:c3:c43_15 :: c2:c3:c4
hole_c7:c84_15 :: c7:c8
hole_c5:c65_15 :: c5:c6
hole_c9:c10:c116_15 :: c9:c10:c11
hole_cons:n__terms:n__first:nil7_15 :: cons:n__terms:n__first:nil
hole_recip8_15 :: recip
hole_c12:c13:c149_15 :: c12:c13:c14
gen_0':s10_15 :: Nat -> 0':s
gen_c2:c3:c411_15 :: Nat -> c2:c3:c4
gen_c7:c812_15 :: Nat -> c7:c8
gen_c5:c613_15 :: Nat -> c5:c6
gen_cons:n__terms:n__first:nil14_15 :: Nat -> cons:n__terms:n__first:nil


Generator Equations:
gen_0':s10_15(0) <=> 0'
gen_0':s10_15(+(x, 1)) <=> s(gen_0':s10_15(x))
gen_c2:c3:c411_15(0) <=> c2
gen_c2:c3:c411_15(+(x, 1)) <=> c3(c7, gen_c2:c3:c411_15(x))
gen_c7:c812_15(0) <=> c7
gen_c7:c812_15(+(x, 1)) <=> c8(gen_c7:c812_15(x))
gen_c5:c613_15(0) <=> c5
gen_c5:c613_15(+(x, 1)) <=> c6(gen_c5:c613_15(x))
gen_cons:n__terms:n__first:nil14_15(0) <=> n__terms(0')
gen_cons:n__terms:n__first:nil14_15(+(x, 1)) <=> cons(recip(0'), gen_cons:n__terms:n__first:nil14_15(x))


The following defined symbols remain to be analysed:
ADD, SQR, sqr, dbl, DBL, ACTIVATE, add, activate

They will be analysed ascendingly in the following order:
ADD < SQR
sqr < SQR
dbl < SQR
DBL < SQR
dbl < sqr
add < sqr

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

(85) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ADD(gen_0':s10_15(n16_15), gen_0':s10_15(b)) -> gen_c7:c812_15(n16_15), rt in Omega(1 + n16_15)

Induction Base:
ADD(gen_0':s10_15(0), gen_0':s10_15(b)) ->_R^Omega(1)
c7

Induction Step:
ADD(gen_0':s10_15(+(n16_15, 1)), gen_0':s10_15(b)) ->_R^Omega(1)
c8(ADD(gen_0':s10_15(n16_15), gen_0':s10_15(b))) ->_IH
c8(gen_c7:c812_15(c17_15))

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

(86)
Complex Obligation (BEST)

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

(87)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
TERMS(z0) -> c(SQR(z0))
TERMS(z0) -> c1
SQR(0') -> c2
SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0))
DBL(0') -> c5
DBL(s(z0)) -> c6(DBL(z0))
ADD(0', z0) -> c7
ADD(s(z0), z1) -> c8(ADD(z0, z1))
FIRST(0', z0) -> c9
FIRST(s(z0), cons(z1, z2)) -> c10(ACTIVATE(z2))
FIRST(z0, z1) -> c11
ACTIVATE(n__terms(z0)) -> c12(TERMS(z0))
ACTIVATE(n__first(z0, z1)) -> c13(FIRST(z0, z1))
ACTIVATE(z0) -> c14
terms(z0) -> cons(recip(sqr(z0)), n__terms(s(z0)))
terms(z0) -> n__terms(z0)
sqr(0') -> 0'
sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
dbl(0') -> 0'
dbl(s(z0)) -> s(s(dbl(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
first(0', z0) -> nil
first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2)))
first(z0, z1) -> n__first(z0, z1)
activate(n__terms(z0)) -> terms(z0)
activate(n__first(z0, z1)) -> first(z0, z1)
activate(z0) -> z0

Types:
TERMS :: 0':s -> c:c1
c :: c2:c3:c4 -> c:c1
SQR :: 0':s -> c2:c3:c4
c1 :: c:c1
0' :: 0':s
c2 :: c2:c3:c4
s :: 0':s -> 0':s
c3 :: c7:c8 -> c2:c3:c4 -> c2:c3:c4
ADD :: 0':s -> 0':s -> c7:c8
sqr :: 0':s -> 0':s
dbl :: 0':s -> 0':s
c4 :: c7:c8 -> c5:c6 -> c2:c3:c4
DBL :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8
c8 :: c7:c8 -> c7:c8
FIRST :: 0':s -> cons:n__terms:n__first:nil -> c9:c10:c11
c9 :: c9:c10:c11
cons :: recip -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c10 :: c12:c13:c14 -> c9:c10:c11
ACTIVATE :: cons:n__terms:n__first:nil -> c12:c13:c14
c11 :: c9:c10:c11
n__terms :: 0':s -> cons:n__terms:n__first:nil
c12 :: c:c1 -> c12:c13:c14
n__first :: 0':s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c13 :: c9:c10:c11 -> c12:c13:c14
c14 :: c12:c13:c14
terms :: 0':s -> cons:n__terms:n__first:nil
recip :: 0':s -> recip
add :: 0':s -> 0':s -> 0':s
first :: 0':s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
nil :: cons:n__terms:n__first:nil
activate :: cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
hole_c:c11_15 :: c:c1
hole_0':s2_15 :: 0':s
hole_c2:c3:c43_15 :: c2:c3:c4
hole_c7:c84_15 :: c7:c8
hole_c5:c65_15 :: c5:c6
hole_c9:c10:c116_15 :: c9:c10:c11
hole_cons:n__terms:n__first:nil7_15 :: cons:n__terms:n__first:nil
hole_recip8_15 :: recip
hole_c12:c13:c149_15 :: c12:c13:c14
gen_0':s10_15 :: Nat -> 0':s
gen_c2:c3:c411_15 :: Nat -> c2:c3:c4
gen_c7:c812_15 :: Nat -> c7:c8
gen_c5:c613_15 :: Nat -> c5:c6
gen_cons:n__terms:n__first:nil14_15 :: Nat -> cons:n__terms:n__first:nil


Generator Equations:
gen_0':s10_15(0) <=> 0'
gen_0':s10_15(+(x, 1)) <=> s(gen_0':s10_15(x))
gen_c2:c3:c411_15(0) <=> c2
gen_c2:c3:c411_15(+(x, 1)) <=> c3(c7, gen_c2:c3:c411_15(x))
gen_c7:c812_15(0) <=> c7
gen_c7:c812_15(+(x, 1)) <=> c8(gen_c7:c812_15(x))
gen_c5:c613_15(0) <=> c5
gen_c5:c613_15(+(x, 1)) <=> c6(gen_c5:c613_15(x))
gen_cons:n__terms:n__first:nil14_15(0) <=> n__terms(0')
gen_cons:n__terms:n__first:nil14_15(+(x, 1)) <=> cons(recip(0'), gen_cons:n__terms:n__first:nil14_15(x))


The following defined symbols remain to be analysed:
ADD, SQR, sqr, dbl, DBL, ACTIVATE, add, activate

They will be analysed ascendingly in the following order:
ADD < SQR
sqr < SQR
dbl < SQR
DBL < SQR
dbl < sqr
add < sqr

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

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

(89)
BOUNDS(n^1, INF)

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

(90)
Obligation:
Innermost TRS:
Rules:
TERMS(z0) -> c(SQR(z0))
TERMS(z0) -> c1
SQR(0') -> c2
SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0))
DBL(0') -> c5
DBL(s(z0)) -> c6(DBL(z0))
ADD(0', z0) -> c7
ADD(s(z0), z1) -> c8(ADD(z0, z1))
FIRST(0', z0) -> c9
FIRST(s(z0), cons(z1, z2)) -> c10(ACTIVATE(z2))
FIRST(z0, z1) -> c11
ACTIVATE(n__terms(z0)) -> c12(TERMS(z0))
ACTIVATE(n__first(z0, z1)) -> c13(FIRST(z0, z1))
ACTIVATE(z0) -> c14
terms(z0) -> cons(recip(sqr(z0)), n__terms(s(z0)))
terms(z0) -> n__terms(z0)
sqr(0') -> 0'
sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
dbl(0') -> 0'
dbl(s(z0)) -> s(s(dbl(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
first(0', z0) -> nil
first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2)))
first(z0, z1) -> n__first(z0, z1)
activate(n__terms(z0)) -> terms(z0)
activate(n__first(z0, z1)) -> first(z0, z1)
activate(z0) -> z0

Types:
TERMS :: 0':s -> c:c1
c :: c2:c3:c4 -> c:c1
SQR :: 0':s -> c2:c3:c4
c1 :: c:c1
0' :: 0':s
c2 :: c2:c3:c4
s :: 0':s -> 0':s
c3 :: c7:c8 -> c2:c3:c4 -> c2:c3:c4
ADD :: 0':s -> 0':s -> c7:c8
sqr :: 0':s -> 0':s
dbl :: 0':s -> 0':s
c4 :: c7:c8 -> c5:c6 -> c2:c3:c4
DBL :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8
c8 :: c7:c8 -> c7:c8
FIRST :: 0':s -> cons:n__terms:n__first:nil -> c9:c10:c11
c9 :: c9:c10:c11
cons :: recip -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c10 :: c12:c13:c14 -> c9:c10:c11
ACTIVATE :: cons:n__terms:n__first:nil -> c12:c13:c14
c11 :: c9:c10:c11
n__terms :: 0':s -> cons:n__terms:n__first:nil
c12 :: c:c1 -> c12:c13:c14
n__first :: 0':s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c13 :: c9:c10:c11 -> c12:c13:c14
c14 :: c12:c13:c14
terms :: 0':s -> cons:n__terms:n__first:nil
recip :: 0':s -> recip
add :: 0':s -> 0':s -> 0':s
first :: 0':s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
nil :: cons:n__terms:n__first:nil
activate :: cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
hole_c:c11_15 :: c:c1
hole_0':s2_15 :: 0':s
hole_c2:c3:c43_15 :: c2:c3:c4
hole_c7:c84_15 :: c7:c8
hole_c5:c65_15 :: c5:c6
hole_c9:c10:c116_15 :: c9:c10:c11
hole_cons:n__terms:n__first:nil7_15 :: cons:n__terms:n__first:nil
hole_recip8_15 :: recip
hole_c12:c13:c149_15 :: c12:c13:c14
gen_0':s10_15 :: Nat -> 0':s
gen_c2:c3:c411_15 :: Nat -> c2:c3:c4
gen_c7:c812_15 :: Nat -> c7:c8
gen_c5:c613_15 :: Nat -> c5:c6
gen_cons:n__terms:n__first:nil14_15 :: Nat -> cons:n__terms:n__first:nil


Lemmas:
ADD(gen_0':s10_15(n16_15), gen_0':s10_15(b)) -> gen_c7:c812_15(n16_15), rt in Omega(1 + n16_15)


Generator Equations:
gen_0':s10_15(0) <=> 0'
gen_0':s10_15(+(x, 1)) <=> s(gen_0':s10_15(x))
gen_c2:c3:c411_15(0) <=> c2
gen_c2:c3:c411_15(+(x, 1)) <=> c3(c7, gen_c2:c3:c411_15(x))
gen_c7:c812_15(0) <=> c7
gen_c7:c812_15(+(x, 1)) <=> c8(gen_c7:c812_15(x))
gen_c5:c613_15(0) <=> c5
gen_c5:c613_15(+(x, 1)) <=> c6(gen_c5:c613_15(x))
gen_cons:n__terms:n__first:nil14_15(0) <=> n__terms(0')
gen_cons:n__terms:n__first:nil14_15(+(x, 1)) <=> cons(recip(0'), gen_cons:n__terms:n__first:nil14_15(x))


The following defined symbols remain to be analysed:
dbl, SQR, sqr, DBL, ACTIVATE, add, activate

They will be analysed ascendingly in the following order:
sqr < SQR
dbl < SQR
DBL < SQR
dbl < sqr
add < sqr

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

(91) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
dbl(gen_0':s10_15(n750_15)) -> gen_0':s10_15(*(2, n750_15)), rt in Omega(0)

Induction Base:
dbl(gen_0':s10_15(0)) ->_R^Omega(0)
0'

Induction Step:
dbl(gen_0':s10_15(+(n750_15, 1))) ->_R^Omega(0)
s(s(dbl(gen_0':s10_15(n750_15)))) ->_IH
s(s(gen_0':s10_15(*(2, c751_15))))

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

(92)
Obligation:
Innermost TRS:
Rules:
TERMS(z0) -> c(SQR(z0))
TERMS(z0) -> c1
SQR(0') -> c2
SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0))
DBL(0') -> c5
DBL(s(z0)) -> c6(DBL(z0))
ADD(0', z0) -> c7
ADD(s(z0), z1) -> c8(ADD(z0, z1))
FIRST(0', z0) -> c9
FIRST(s(z0), cons(z1, z2)) -> c10(ACTIVATE(z2))
FIRST(z0, z1) -> c11
ACTIVATE(n__terms(z0)) -> c12(TERMS(z0))
ACTIVATE(n__first(z0, z1)) -> c13(FIRST(z0, z1))
ACTIVATE(z0) -> c14
terms(z0) -> cons(recip(sqr(z0)), n__terms(s(z0)))
terms(z0) -> n__terms(z0)
sqr(0') -> 0'
sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
dbl(0') -> 0'
dbl(s(z0)) -> s(s(dbl(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
first(0', z0) -> nil
first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2)))
first(z0, z1) -> n__first(z0, z1)
activate(n__terms(z0)) -> terms(z0)
activate(n__first(z0, z1)) -> first(z0, z1)
activate(z0) -> z0

Types:
TERMS :: 0':s -> c:c1
c :: c2:c3:c4 -> c:c1
SQR :: 0':s -> c2:c3:c4
c1 :: c:c1
0' :: 0':s
c2 :: c2:c3:c4
s :: 0':s -> 0':s
c3 :: c7:c8 -> c2:c3:c4 -> c2:c3:c4
ADD :: 0':s -> 0':s -> c7:c8
sqr :: 0':s -> 0':s
dbl :: 0':s -> 0':s
c4 :: c7:c8 -> c5:c6 -> c2:c3:c4
DBL :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8
c8 :: c7:c8 -> c7:c8
FIRST :: 0':s -> cons:n__terms:n__first:nil -> c9:c10:c11
c9 :: c9:c10:c11
cons :: recip -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c10 :: c12:c13:c14 -> c9:c10:c11
ACTIVATE :: cons:n__terms:n__first:nil -> c12:c13:c14
c11 :: c9:c10:c11
n__terms :: 0':s -> cons:n__terms:n__first:nil
c12 :: c:c1 -> c12:c13:c14
n__first :: 0':s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c13 :: c9:c10:c11 -> c12:c13:c14
c14 :: c12:c13:c14
terms :: 0':s -> cons:n__terms:n__first:nil
recip :: 0':s -> recip
add :: 0':s -> 0':s -> 0':s
first :: 0':s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
nil :: cons:n__terms:n__first:nil
activate :: cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
hole_c:c11_15 :: c:c1
hole_0':s2_15 :: 0':s
hole_c2:c3:c43_15 :: c2:c3:c4
hole_c7:c84_15 :: c7:c8
hole_c5:c65_15 :: c5:c6
hole_c9:c10:c116_15 :: c9:c10:c11
hole_cons:n__terms:n__first:nil7_15 :: cons:n__terms:n__first:nil
hole_recip8_15 :: recip
hole_c12:c13:c149_15 :: c12:c13:c14
gen_0':s10_15 :: Nat -> 0':s
gen_c2:c3:c411_15 :: Nat -> c2:c3:c4
gen_c7:c812_15 :: Nat -> c7:c8
gen_c5:c613_15 :: Nat -> c5:c6
gen_cons:n__terms:n__first:nil14_15 :: Nat -> cons:n__terms:n__first:nil


Lemmas:
ADD(gen_0':s10_15(n16_15), gen_0':s10_15(b)) -> gen_c7:c812_15(n16_15), rt in Omega(1 + n16_15)
dbl(gen_0':s10_15(n750_15)) -> gen_0':s10_15(*(2, n750_15)), rt in Omega(0)


Generator Equations:
gen_0':s10_15(0) <=> 0'
gen_0':s10_15(+(x, 1)) <=> s(gen_0':s10_15(x))
gen_c2:c3:c411_15(0) <=> c2
gen_c2:c3:c411_15(+(x, 1)) <=> c3(c7, gen_c2:c3:c411_15(x))
gen_c7:c812_15(0) <=> c7
gen_c7:c812_15(+(x, 1)) <=> c8(gen_c7:c812_15(x))
gen_c5:c613_15(0) <=> c5
gen_c5:c613_15(+(x, 1)) <=> c6(gen_c5:c613_15(x))
gen_cons:n__terms:n__first:nil14_15(0) <=> n__terms(0')
gen_cons:n__terms:n__first:nil14_15(+(x, 1)) <=> cons(recip(0'), gen_cons:n__terms:n__first:nil14_15(x))


The following defined symbols remain to be analysed:
DBL, SQR, sqr, ACTIVATE, add, activate

They will be analysed ascendingly in the following order:
sqr < SQR
DBL < SQR
add < sqr

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

(93) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
DBL(gen_0':s10_15(n1130_15)) -> gen_c5:c613_15(n1130_15), rt in Omega(1 + n1130_15)

Induction Base:
DBL(gen_0':s10_15(0)) ->_R^Omega(1)
c5

Induction Step:
DBL(gen_0':s10_15(+(n1130_15, 1))) ->_R^Omega(1)
c6(DBL(gen_0':s10_15(n1130_15))) ->_IH
c6(gen_c5:c613_15(c1131_15))

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

(94)
Obligation:
Innermost TRS:
Rules:
TERMS(z0) -> c(SQR(z0))
TERMS(z0) -> c1
SQR(0') -> c2
SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0))
DBL(0') -> c5
DBL(s(z0)) -> c6(DBL(z0))
ADD(0', z0) -> c7
ADD(s(z0), z1) -> c8(ADD(z0, z1))
FIRST(0', z0) -> c9
FIRST(s(z0), cons(z1, z2)) -> c10(ACTIVATE(z2))
FIRST(z0, z1) -> c11
ACTIVATE(n__terms(z0)) -> c12(TERMS(z0))
ACTIVATE(n__first(z0, z1)) -> c13(FIRST(z0, z1))
ACTIVATE(z0) -> c14
terms(z0) -> cons(recip(sqr(z0)), n__terms(s(z0)))
terms(z0) -> n__terms(z0)
sqr(0') -> 0'
sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
dbl(0') -> 0'
dbl(s(z0)) -> s(s(dbl(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
first(0', z0) -> nil
first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2)))
first(z0, z1) -> n__first(z0, z1)
activate(n__terms(z0)) -> terms(z0)
activate(n__first(z0, z1)) -> first(z0, z1)
activate(z0) -> z0

Types:
TERMS :: 0':s -> c:c1
c :: c2:c3:c4 -> c:c1
SQR :: 0':s -> c2:c3:c4
c1 :: c:c1
0' :: 0':s
c2 :: c2:c3:c4
s :: 0':s -> 0':s
c3 :: c7:c8 -> c2:c3:c4 -> c2:c3:c4
ADD :: 0':s -> 0':s -> c7:c8
sqr :: 0':s -> 0':s
dbl :: 0':s -> 0':s
c4 :: c7:c8 -> c5:c6 -> c2:c3:c4
DBL :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8
c8 :: c7:c8 -> c7:c8
FIRST :: 0':s -> cons:n__terms:n__first:nil -> c9:c10:c11
c9 :: c9:c10:c11
cons :: recip -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c10 :: c12:c13:c14 -> c9:c10:c11
ACTIVATE :: cons:n__terms:n__first:nil -> c12:c13:c14
c11 :: c9:c10:c11
n__terms :: 0':s -> cons:n__terms:n__first:nil
c12 :: c:c1 -> c12:c13:c14
n__first :: 0':s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c13 :: c9:c10:c11 -> c12:c13:c14
c14 :: c12:c13:c14
terms :: 0':s -> cons:n__terms:n__first:nil
recip :: 0':s -> recip
add :: 0':s -> 0':s -> 0':s
first :: 0':s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
nil :: cons:n__terms:n__first:nil
activate :: cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
hole_c:c11_15 :: c:c1
hole_0':s2_15 :: 0':s
hole_c2:c3:c43_15 :: c2:c3:c4
hole_c7:c84_15 :: c7:c8
hole_c5:c65_15 :: c5:c6
hole_c9:c10:c116_15 :: c9:c10:c11
hole_cons:n__terms:n__first:nil7_15 :: cons:n__terms:n__first:nil
hole_recip8_15 :: recip
hole_c12:c13:c149_15 :: c12:c13:c14
gen_0':s10_15 :: Nat -> 0':s
gen_c2:c3:c411_15 :: Nat -> c2:c3:c4
gen_c7:c812_15 :: Nat -> c7:c8
gen_c5:c613_15 :: Nat -> c5:c6
gen_cons:n__terms:n__first:nil14_15 :: Nat -> cons:n__terms:n__first:nil


Lemmas:
ADD(gen_0':s10_15(n16_15), gen_0':s10_15(b)) -> gen_c7:c812_15(n16_15), rt in Omega(1 + n16_15)
dbl(gen_0':s10_15(n750_15)) -> gen_0':s10_15(*(2, n750_15)), rt in Omega(0)
DBL(gen_0':s10_15(n1130_15)) -> gen_c5:c613_15(n1130_15), rt in Omega(1 + n1130_15)


Generator Equations:
gen_0':s10_15(0) <=> 0'
gen_0':s10_15(+(x, 1)) <=> s(gen_0':s10_15(x))
gen_c2:c3:c411_15(0) <=> c2
gen_c2:c3:c411_15(+(x, 1)) <=> c3(c7, gen_c2:c3:c411_15(x))
gen_c7:c812_15(0) <=> c7
gen_c7:c812_15(+(x, 1)) <=> c8(gen_c7:c812_15(x))
gen_c5:c613_15(0) <=> c5
gen_c5:c613_15(+(x, 1)) <=> c6(gen_c5:c613_15(x))
gen_cons:n__terms:n__first:nil14_15(0) <=> n__terms(0')
gen_cons:n__terms:n__first:nil14_15(+(x, 1)) <=> cons(recip(0'), gen_cons:n__terms:n__first:nil14_15(x))


The following defined symbols remain to be analysed:
ACTIVATE, SQR, sqr, add, activate

They will be analysed ascendingly in the following order:
sqr < SQR
add < sqr

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

(95) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
add(gen_0':s10_15(n1574_15), gen_0':s10_15(b)) -> gen_0':s10_15(+(n1574_15, b)), rt in Omega(0)

Induction Base:
add(gen_0':s10_15(0), gen_0':s10_15(b)) ->_R^Omega(0)
gen_0':s10_15(b)

Induction Step:
add(gen_0':s10_15(+(n1574_15, 1)), gen_0':s10_15(b)) ->_R^Omega(0)
s(add(gen_0':s10_15(n1574_15), gen_0':s10_15(b))) ->_IH
s(gen_0':s10_15(+(b, c1575_15)))

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

(96)
Obligation:
Innermost TRS:
Rules:
TERMS(z0) -> c(SQR(z0))
TERMS(z0) -> c1
SQR(0') -> c2
SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0))
DBL(0') -> c5
DBL(s(z0)) -> c6(DBL(z0))
ADD(0', z0) -> c7
ADD(s(z0), z1) -> c8(ADD(z0, z1))
FIRST(0', z0) -> c9
FIRST(s(z0), cons(z1, z2)) -> c10(ACTIVATE(z2))
FIRST(z0, z1) -> c11
ACTIVATE(n__terms(z0)) -> c12(TERMS(z0))
ACTIVATE(n__first(z0, z1)) -> c13(FIRST(z0, z1))
ACTIVATE(z0) -> c14
terms(z0) -> cons(recip(sqr(z0)), n__terms(s(z0)))
terms(z0) -> n__terms(z0)
sqr(0') -> 0'
sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
dbl(0') -> 0'
dbl(s(z0)) -> s(s(dbl(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
first(0', z0) -> nil
first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2)))
first(z0, z1) -> n__first(z0, z1)
activate(n__terms(z0)) -> terms(z0)
activate(n__first(z0, z1)) -> first(z0, z1)
activate(z0) -> z0

Types:
TERMS :: 0':s -> c:c1
c :: c2:c3:c4 -> c:c1
SQR :: 0':s -> c2:c3:c4
c1 :: c:c1
0' :: 0':s
c2 :: c2:c3:c4
s :: 0':s -> 0':s
c3 :: c7:c8 -> c2:c3:c4 -> c2:c3:c4
ADD :: 0':s -> 0':s -> c7:c8
sqr :: 0':s -> 0':s
dbl :: 0':s -> 0':s
c4 :: c7:c8 -> c5:c6 -> c2:c3:c4
DBL :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8
c8 :: c7:c8 -> c7:c8
FIRST :: 0':s -> cons:n__terms:n__first:nil -> c9:c10:c11
c9 :: c9:c10:c11
cons :: recip -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c10 :: c12:c13:c14 -> c9:c10:c11
ACTIVATE :: cons:n__terms:n__first:nil -> c12:c13:c14
c11 :: c9:c10:c11
n__terms :: 0':s -> cons:n__terms:n__first:nil
c12 :: c:c1 -> c12:c13:c14
n__first :: 0':s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c13 :: c9:c10:c11 -> c12:c13:c14
c14 :: c12:c13:c14
terms :: 0':s -> cons:n__terms:n__first:nil
recip :: 0':s -> recip
add :: 0':s -> 0':s -> 0':s
first :: 0':s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
nil :: cons:n__terms:n__first:nil
activate :: cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
hole_c:c11_15 :: c:c1
hole_0':s2_15 :: 0':s
hole_c2:c3:c43_15 :: c2:c3:c4
hole_c7:c84_15 :: c7:c8
hole_c5:c65_15 :: c5:c6
hole_c9:c10:c116_15 :: c9:c10:c11
hole_cons:n__terms:n__first:nil7_15 :: cons:n__terms:n__first:nil
hole_recip8_15 :: recip
hole_c12:c13:c149_15 :: c12:c13:c14
gen_0':s10_15 :: Nat -> 0':s
gen_c2:c3:c411_15 :: Nat -> c2:c3:c4
gen_c7:c812_15 :: Nat -> c7:c8
gen_c5:c613_15 :: Nat -> c5:c6
gen_cons:n__terms:n__first:nil14_15 :: Nat -> cons:n__terms:n__first:nil


Lemmas:
ADD(gen_0':s10_15(n16_15), gen_0':s10_15(b)) -> gen_c7:c812_15(n16_15), rt in Omega(1 + n16_15)
dbl(gen_0':s10_15(n750_15)) -> gen_0':s10_15(*(2, n750_15)), rt in Omega(0)
DBL(gen_0':s10_15(n1130_15)) -> gen_c5:c613_15(n1130_15), rt in Omega(1 + n1130_15)
add(gen_0':s10_15(n1574_15), gen_0':s10_15(b)) -> gen_0':s10_15(+(n1574_15, b)), rt in Omega(0)


Generator Equations:
gen_0':s10_15(0) <=> 0'
gen_0':s10_15(+(x, 1)) <=> s(gen_0':s10_15(x))
gen_c2:c3:c411_15(0) <=> c2
gen_c2:c3:c411_15(+(x, 1)) <=> c3(c7, gen_c2:c3:c411_15(x))
gen_c7:c812_15(0) <=> c7
gen_c7:c812_15(+(x, 1)) <=> c8(gen_c7:c812_15(x))
gen_c5:c613_15(0) <=> c5
gen_c5:c613_15(+(x, 1)) <=> c6(gen_c5:c613_15(x))
gen_cons:n__terms:n__first:nil14_15(0) <=> n__terms(0')
gen_cons:n__terms:n__first:nil14_15(+(x, 1)) <=> cons(recip(0'), gen_cons:n__terms:n__first:nil14_15(x))


The following defined symbols remain to be analysed:
sqr, SQR, activate

They will be analysed ascendingly in the following order:
sqr < SQR

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

(97) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sqr(gen_0':s10_15(n2933_15)) -> gen_0':s10_15(*(n2933_15, n2933_15)), rt in Omega(0)

Induction Base:
sqr(gen_0':s10_15(0)) ->_R^Omega(0)
0'

Induction Step:
sqr(gen_0':s10_15(+(n2933_15, 1))) ->_R^Omega(0)
s(add(sqr(gen_0':s10_15(n2933_15)), dbl(gen_0':s10_15(n2933_15)))) ->_IH
s(add(gen_0':s10_15(*(c2934_15, c2934_15)), dbl(gen_0':s10_15(n2933_15)))) ->_L^Omega(0)
s(add(gen_0':s10_15(*(n2933_15, n2933_15)), gen_0':s10_15(*(2, n2933_15)))) ->_L^Omega(0)
s(gen_0':s10_15(+(*(n2933_15, n2933_15), *(2, n2933_15))))

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

(98)
Obligation:
Innermost TRS:
Rules:
TERMS(z0) -> c(SQR(z0))
TERMS(z0) -> c1
SQR(0') -> c2
SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0))
DBL(0') -> c5
DBL(s(z0)) -> c6(DBL(z0))
ADD(0', z0) -> c7
ADD(s(z0), z1) -> c8(ADD(z0, z1))
FIRST(0', z0) -> c9
FIRST(s(z0), cons(z1, z2)) -> c10(ACTIVATE(z2))
FIRST(z0, z1) -> c11
ACTIVATE(n__terms(z0)) -> c12(TERMS(z0))
ACTIVATE(n__first(z0, z1)) -> c13(FIRST(z0, z1))
ACTIVATE(z0) -> c14
terms(z0) -> cons(recip(sqr(z0)), n__terms(s(z0)))
terms(z0) -> n__terms(z0)
sqr(0') -> 0'
sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
dbl(0') -> 0'
dbl(s(z0)) -> s(s(dbl(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
first(0', z0) -> nil
first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2)))
first(z0, z1) -> n__first(z0, z1)
activate(n__terms(z0)) -> terms(z0)
activate(n__first(z0, z1)) -> first(z0, z1)
activate(z0) -> z0

Types:
TERMS :: 0':s -> c:c1
c :: c2:c3:c4 -> c:c1
SQR :: 0':s -> c2:c3:c4
c1 :: c:c1
0' :: 0':s
c2 :: c2:c3:c4
s :: 0':s -> 0':s
c3 :: c7:c8 -> c2:c3:c4 -> c2:c3:c4
ADD :: 0':s -> 0':s -> c7:c8
sqr :: 0':s -> 0':s
dbl :: 0':s -> 0':s
c4 :: c7:c8 -> c5:c6 -> c2:c3:c4
DBL :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8
c8 :: c7:c8 -> c7:c8
FIRST :: 0':s -> cons:n__terms:n__first:nil -> c9:c10:c11
c9 :: c9:c10:c11
cons :: recip -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c10 :: c12:c13:c14 -> c9:c10:c11
ACTIVATE :: cons:n__terms:n__first:nil -> c12:c13:c14
c11 :: c9:c10:c11
n__terms :: 0':s -> cons:n__terms:n__first:nil
c12 :: c:c1 -> c12:c13:c14
n__first :: 0':s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c13 :: c9:c10:c11 -> c12:c13:c14
c14 :: c12:c13:c14
terms :: 0':s -> cons:n__terms:n__first:nil
recip :: 0':s -> recip
add :: 0':s -> 0':s -> 0':s
first :: 0':s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
nil :: cons:n__terms:n__first:nil
activate :: cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
hole_c:c11_15 :: c:c1
hole_0':s2_15 :: 0':s
hole_c2:c3:c43_15 :: c2:c3:c4
hole_c7:c84_15 :: c7:c8
hole_c5:c65_15 :: c5:c6
hole_c9:c10:c116_15 :: c9:c10:c11
hole_cons:n__terms:n__first:nil7_15 :: cons:n__terms:n__first:nil
hole_recip8_15 :: recip
hole_c12:c13:c149_15 :: c12:c13:c14
gen_0':s10_15 :: Nat -> 0':s
gen_c2:c3:c411_15 :: Nat -> c2:c3:c4
gen_c7:c812_15 :: Nat -> c7:c8
gen_c5:c613_15 :: Nat -> c5:c6
gen_cons:n__terms:n__first:nil14_15 :: Nat -> cons:n__terms:n__first:nil


Lemmas:
ADD(gen_0':s10_15(n16_15), gen_0':s10_15(b)) -> gen_c7:c812_15(n16_15), rt in Omega(1 + n16_15)
dbl(gen_0':s10_15(n750_15)) -> gen_0':s10_15(*(2, n750_15)), rt in Omega(0)
DBL(gen_0':s10_15(n1130_15)) -> gen_c5:c613_15(n1130_15), rt in Omega(1 + n1130_15)
add(gen_0':s10_15(n1574_15), gen_0':s10_15(b)) -> gen_0':s10_15(+(n1574_15, b)), rt in Omega(0)
sqr(gen_0':s10_15(n2933_15)) -> gen_0':s10_15(*(n2933_15, n2933_15)), rt in Omega(0)


Generator Equations:
gen_0':s10_15(0) <=> 0'
gen_0':s10_15(+(x, 1)) <=> s(gen_0':s10_15(x))
gen_c2:c3:c411_15(0) <=> c2
gen_c2:c3:c411_15(+(x, 1)) <=> c3(c7, gen_c2:c3:c411_15(x))
gen_c7:c812_15(0) <=> c7
gen_c7:c812_15(+(x, 1)) <=> c8(gen_c7:c812_15(x))
gen_c5:c613_15(0) <=> c5
gen_c5:c613_15(+(x, 1)) <=> c6(gen_c5:c613_15(x))
gen_cons:n__terms:n__first:nil14_15(0) <=> n__terms(0')
gen_cons:n__terms:n__first:nil14_15(+(x, 1)) <=> cons(recip(0'), gen_cons:n__terms:n__first:nil14_15(x))


The following defined symbols remain to be analysed:
SQR, activate
----------------------------------------

(99) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
SQR(gen_0':s10_15(n3490_15)) -> *15_15, rt in Omega(n3490_15 + n3490_15^3)

Induction Base:
SQR(gen_0':s10_15(0))

Induction Step:
SQR(gen_0':s10_15(+(n3490_15, 1))) ->_R^Omega(1)
c3(ADD(sqr(gen_0':s10_15(n3490_15)), dbl(gen_0':s10_15(n3490_15))), SQR(gen_0':s10_15(n3490_15))) ->_L^Omega(0)
c3(ADD(gen_0':s10_15(*(n3490_15, n3490_15)), dbl(gen_0':s10_15(n3490_15))), SQR(gen_0':s10_15(n3490_15))) ->_L^Omega(0)
c3(ADD(gen_0':s10_15(*(n3490_15, n3490_15)), gen_0':s10_15(*(2, n3490_15))), SQR(gen_0':s10_15(n3490_15))) ->_L^Omega(1 + n3490_15^2)
c3(gen_c7:c812_15(*(n3490_15, n3490_15)), SQR(gen_0':s10_15(n3490_15))) ->_IH
c3(gen_c7:c812_15(*(n3490_15, n3490_15)), *15_15)

We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3).
----------------------------------------

(100)
Complex Obligation (BEST)

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

(101)
Obligation:
Proved the lower bound n^3 for the following obligation:

Innermost TRS:
Rules:
TERMS(z0) -> c(SQR(z0))
TERMS(z0) -> c1
SQR(0') -> c2
SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0))
DBL(0') -> c5
DBL(s(z0)) -> c6(DBL(z0))
ADD(0', z0) -> c7
ADD(s(z0), z1) -> c8(ADD(z0, z1))
FIRST(0', z0) -> c9
FIRST(s(z0), cons(z1, z2)) -> c10(ACTIVATE(z2))
FIRST(z0, z1) -> c11
ACTIVATE(n__terms(z0)) -> c12(TERMS(z0))
ACTIVATE(n__first(z0, z1)) -> c13(FIRST(z0, z1))
ACTIVATE(z0) -> c14
terms(z0) -> cons(recip(sqr(z0)), n__terms(s(z0)))
terms(z0) -> n__terms(z0)
sqr(0') -> 0'
sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
dbl(0') -> 0'
dbl(s(z0)) -> s(s(dbl(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
first(0', z0) -> nil
first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2)))
first(z0, z1) -> n__first(z0, z1)
activate(n__terms(z0)) -> terms(z0)
activate(n__first(z0, z1)) -> first(z0, z1)
activate(z0) -> z0

Types:
TERMS :: 0':s -> c:c1
c :: c2:c3:c4 -> c:c1
SQR :: 0':s -> c2:c3:c4
c1 :: c:c1
0' :: 0':s
c2 :: c2:c3:c4
s :: 0':s -> 0':s
c3 :: c7:c8 -> c2:c3:c4 -> c2:c3:c4
ADD :: 0':s -> 0':s -> c7:c8
sqr :: 0':s -> 0':s
dbl :: 0':s -> 0':s
c4 :: c7:c8 -> c5:c6 -> c2:c3:c4
DBL :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8
c8 :: c7:c8 -> c7:c8
FIRST :: 0':s -> cons:n__terms:n__first:nil -> c9:c10:c11
c9 :: c9:c10:c11
cons :: recip -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c10 :: c12:c13:c14 -> c9:c10:c11
ACTIVATE :: cons:n__terms:n__first:nil -> c12:c13:c14
c11 :: c9:c10:c11
n__terms :: 0':s -> cons:n__terms:n__first:nil
c12 :: c:c1 -> c12:c13:c14
n__first :: 0':s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c13 :: c9:c10:c11 -> c12:c13:c14
c14 :: c12:c13:c14
terms :: 0':s -> cons:n__terms:n__first:nil
recip :: 0':s -> recip
add :: 0':s -> 0':s -> 0':s
first :: 0':s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
nil :: cons:n__terms:n__first:nil
activate :: cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
hole_c:c11_15 :: c:c1
hole_0':s2_15 :: 0':s
hole_c2:c3:c43_15 :: c2:c3:c4
hole_c7:c84_15 :: c7:c8
hole_c5:c65_15 :: c5:c6
hole_c9:c10:c116_15 :: c9:c10:c11
hole_cons:n__terms:n__first:nil7_15 :: cons:n__terms:n__first:nil
hole_recip8_15 :: recip
hole_c12:c13:c149_15 :: c12:c13:c14
gen_0':s10_15 :: Nat -> 0':s
gen_c2:c3:c411_15 :: Nat -> c2:c3:c4
gen_c7:c812_15 :: Nat -> c7:c8
gen_c5:c613_15 :: Nat -> c5:c6
gen_cons:n__terms:n__first:nil14_15 :: Nat -> cons:n__terms:n__first:nil


Lemmas:
ADD(gen_0':s10_15(n16_15), gen_0':s10_15(b)) -> gen_c7:c812_15(n16_15), rt in Omega(1 + n16_15)
dbl(gen_0':s10_15(n750_15)) -> gen_0':s10_15(*(2, n750_15)), rt in Omega(0)
DBL(gen_0':s10_15(n1130_15)) -> gen_c5:c613_15(n1130_15), rt in Omega(1 + n1130_15)
add(gen_0':s10_15(n1574_15), gen_0':s10_15(b)) -> gen_0':s10_15(+(n1574_15, b)), rt in Omega(0)
sqr(gen_0':s10_15(n2933_15)) -> gen_0':s10_15(*(n2933_15, n2933_15)), rt in Omega(0)


Generator Equations:
gen_0':s10_15(0) <=> 0'
gen_0':s10_15(+(x, 1)) <=> s(gen_0':s10_15(x))
gen_c2:c3:c411_15(0) <=> c2
gen_c2:c3:c411_15(+(x, 1)) <=> c3(c7, gen_c2:c3:c411_15(x))
gen_c7:c812_15(0) <=> c7
gen_c7:c812_15(+(x, 1)) <=> c8(gen_c7:c812_15(x))
gen_c5:c613_15(0) <=> c5
gen_c5:c613_15(+(x, 1)) <=> c6(gen_c5:c613_15(x))
gen_cons:n__terms:n__first:nil14_15(0) <=> n__terms(0')
gen_cons:n__terms:n__first:nil14_15(+(x, 1)) <=> cons(recip(0'), gen_cons:n__terms:n__first:nil14_15(x))


The following defined symbols remain to be analysed:
SQR, activate
----------------------------------------

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

(103)
BOUNDS(n^3, INF)

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

(104)
Obligation:
Innermost TRS:
Rules:
TERMS(z0) -> c(SQR(z0))
TERMS(z0) -> c1
SQR(0') -> c2
SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0))
DBL(0') -> c5
DBL(s(z0)) -> c6(DBL(z0))
ADD(0', z0) -> c7
ADD(s(z0), z1) -> c8(ADD(z0, z1))
FIRST(0', z0) -> c9
FIRST(s(z0), cons(z1, z2)) -> c10(ACTIVATE(z2))
FIRST(z0, z1) -> c11
ACTIVATE(n__terms(z0)) -> c12(TERMS(z0))
ACTIVATE(n__first(z0, z1)) -> c13(FIRST(z0, z1))
ACTIVATE(z0) -> c14
terms(z0) -> cons(recip(sqr(z0)), n__terms(s(z0)))
terms(z0) -> n__terms(z0)
sqr(0') -> 0'
sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
dbl(0') -> 0'
dbl(s(z0)) -> s(s(dbl(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
first(0', z0) -> nil
first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2)))
first(z0, z1) -> n__first(z0, z1)
activate(n__terms(z0)) -> terms(z0)
activate(n__first(z0, z1)) -> first(z0, z1)
activate(z0) -> z0

Types:
TERMS :: 0':s -> c:c1
c :: c2:c3:c4 -> c:c1
SQR :: 0':s -> c2:c3:c4
c1 :: c:c1
0' :: 0':s
c2 :: c2:c3:c4
s :: 0':s -> 0':s
c3 :: c7:c8 -> c2:c3:c4 -> c2:c3:c4
ADD :: 0':s -> 0':s -> c7:c8
sqr :: 0':s -> 0':s
dbl :: 0':s -> 0':s
c4 :: c7:c8 -> c5:c6 -> c2:c3:c4
DBL :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8
c8 :: c7:c8 -> c7:c8
FIRST :: 0':s -> cons:n__terms:n__first:nil -> c9:c10:c11
c9 :: c9:c10:c11
cons :: recip -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c10 :: c12:c13:c14 -> c9:c10:c11
ACTIVATE :: cons:n__terms:n__first:nil -> c12:c13:c14
c11 :: c9:c10:c11
n__terms :: 0':s -> cons:n__terms:n__first:nil
c12 :: c:c1 -> c12:c13:c14
n__first :: 0':s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
c13 :: c9:c10:c11 -> c12:c13:c14
c14 :: c12:c13:c14
terms :: 0':s -> cons:n__terms:n__first:nil
recip :: 0':s -> recip
add :: 0':s -> 0':s -> 0':s
first :: 0':s -> cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
nil :: cons:n__terms:n__first:nil
activate :: cons:n__terms:n__first:nil -> cons:n__terms:n__first:nil
hole_c:c11_15 :: c:c1
hole_0':s2_15 :: 0':s
hole_c2:c3:c43_15 :: c2:c3:c4
hole_c7:c84_15 :: c7:c8
hole_c5:c65_15 :: c5:c6
hole_c9:c10:c116_15 :: c9:c10:c11
hole_cons:n__terms:n__first:nil7_15 :: cons:n__terms:n__first:nil
hole_recip8_15 :: recip
hole_c12:c13:c149_15 :: c12:c13:c14
gen_0':s10_15 :: Nat -> 0':s
gen_c2:c3:c411_15 :: Nat -> c2:c3:c4
gen_c7:c812_15 :: Nat -> c7:c8
gen_c5:c613_15 :: Nat -> c5:c6
gen_cons:n__terms:n__first:nil14_15 :: Nat -> cons:n__terms:n__first:nil


Lemmas:
ADD(gen_0':s10_15(n16_15), gen_0':s10_15(b)) -> gen_c7:c812_15(n16_15), rt in Omega(1 + n16_15)
dbl(gen_0':s10_15(n750_15)) -> gen_0':s10_15(*(2, n750_15)), rt in Omega(0)
DBL(gen_0':s10_15(n1130_15)) -> gen_c5:c613_15(n1130_15), rt in Omega(1 + n1130_15)
add(gen_0':s10_15(n1574_15), gen_0':s10_15(b)) -> gen_0':s10_15(+(n1574_15, b)), rt in Omega(0)
sqr(gen_0':s10_15(n2933_15)) -> gen_0':s10_15(*(n2933_15, n2933_15)), rt in Omega(0)
SQR(gen_0':s10_15(n3490_15)) -> *15_15, rt in Omega(n3490_15 + n3490_15^3)


Generator Equations:
gen_0':s10_15(0) <=> 0'
gen_0':s10_15(+(x, 1)) <=> s(gen_0':s10_15(x))
gen_c2:c3:c411_15(0) <=> c2
gen_c2:c3:c411_15(+(x, 1)) <=> c3(c7, gen_c2:c3:c411_15(x))
gen_c7:c812_15(0) <=> c7
gen_c7:c812_15(+(x, 1)) <=> c8(gen_c7:c812_15(x))
gen_c5:c613_15(0) <=> c5
gen_c5:c613_15(+(x, 1)) <=> c6(gen_c5:c613_15(x))
gen_cons:n__terms:n__first:nil14_15(0) <=> n__terms(0')
gen_cons:n__terms:n__first:nil14_15(+(x, 1)) <=> cons(recip(0'), gen_cons:n__terms:n__first:nil14_15(x))


The following defined symbols remain to be analysed:
activate